Singularities of Møller-Plesset series: example "a30"

[n₁, n₂, n₃] approximant is defined as a solution of the quadratic equation
A(z)f² + B(z)f + C(z) = 0
with polynomial coefficients A(z), B(z) and C(z) of degree n₃, n₂ and n₁ respectively.

Square-root singularities are determined as zeroes of the discriminant
D(z) = B²(z) - 4A(z)C(z).
The weight c of the singularity z_c is defined so that
f ~ c(1 - z/z_c)^1/2 at z -> z_c.
The weight is calculated by formula
c = 1/2[-z(D/A²)']^1/2
where r. h. s. of the above equation is evaluated at z = z_c.

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1388

0.169
Singularities of quadratic [1, 0, 0] approximant

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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3914

0.257

2

125.4315

2.44 i

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Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5112

0.365
Singularities of quadratic [1, 1, 1] approximant

2

-20.3923

0.841

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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.9687

0.0485
Singularities of quadratic [2, 1, 1] approximant

2

1.1285

0.0564 i

3

2.1886

7.6

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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4418

0.279
Singularities of quadratic [2, 2, 1] approximant

2

-6.6717 + 0.7311 i

1.18 + 0.432 i

3

-6.6717 - 0.7311 i

1.18 - 0.432 i

4

46.0893

1.91 i

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Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5599

0.758
Singularities of quadratic [2, 2, 2] approximant

2

2.6831

0.595 i

3

4.3729

9.01

4

-6.8695

0.242

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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5572 + 0.3191 i

0.247 - 0.238 i
Singularities of quadratic [3, 2, 2] approximant

2

1.5572 - 0.3191 i

0.247 + 0.238 i

3

2.0903

0.419

4

-5.0965

0.151

5

-12.4177

0.172 i

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Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5925 + 0.3212 i

0.311 - 0.224 i
Singularities of quadratic [3, 3, 2] approximant

2

1.5925 - 0.3212 i

0.311 + 0.224 i

3

2.2479

0.607

4

-4.8397

0.127

5

-11.1844

0.147 i

6

1515.8553

2.58 i

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Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5566 + 0.2954 i

0.208 - 0.339 i
Singularities of quadratic [3, 3, 3] approximant

2

1.5566 - 0.2954 i

0.208 + 0.339 i

3

2.0368

0.366

4

-4.8779 + 0.3642 i

0.182 + 0.317 i

5

-4.8779 - 0.3642 i

0.182 - 0.317 i

6

-9.942

0.242

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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-1.441

0.0175
Singularities of quadratic [4, 3, 3] approximant

2

-1.4434

0.0176 i

3

1.5793 + 0.3222 i

0.29 - 0.22 i

4

1.5793 - 0.3222 i

0.29 + 0.22 i

5

2.1795

0.524

6

-5.2201

0.179

7

-11.9573

0.179 i

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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5934 + 0.3625 i

0.276 - 0.0573 i
Singularities of quadratic [4, 4, 3] approximant

2

1.5934 - 0.3625 i

0.276 + 0.0573 i

3

2.3112

0.972

4

-2.8768

0.0164

5

-3.3527

0.0167 i

6

-5.2728 + 5.294 i

0.0866 - 0.0528 i

7

-5.2728 - 5.294 i

0.0866 + 0.0528 i

8

49.8787

4.23 i

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Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-1.173 + 0.5955 i

0.0000344 - 0.000436 i
Singularities of quadratic [4, 4, 4] approximant

2

-1.173 - 0.5955 i

0.0000344 + 0.000436 i

3

-1.1816 + 0.5944 i

0.000435 + 0.0000382 i

4

-1.1816 - 0.5944 i

0.000435 - 0.0000382 i

5

1.5975 + 0.3997 i

0.213 + 0.0621 i

6

1.5975 - 0.3997 i

0.213 - 0.0621 i

7

2.1802

0.861

8

-3.8669

0.0165

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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5855 + 0.2983 i

0.166 - 0.362 i
Singularities of quadratic [5, 4, 4] approximant

2

1.5855 - 0.2983 i

0.166 + 0.362 i

3

-2.811 + 0.1542 i

0.00693 + 0.00749 i

4

-2.811 - 0.1542 i

0.00693 - 0.00749 i

5

2.8974 + 0.4075 i

0.668 + 1.42 i

6

2.8974 - 0.4075 i

0.668 - 1.42 i

7

-4.5793

0.025

8

6.688 + 4.3766 i

0.681 - 0.0983 i

9

6.688 - 4.3766 i

0.681 + 0.0983 i

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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1154 + 0.0011 i

0.0222 - 0.0225 i
Singularities of quadratic [5, 5, 4] approximant

2

1.1154 - 0.0011 i

0.0222 + 0.0225 i

3

1.6072 + 0.38 i

0.272 + 0.0118 i

4

1.6072 - 0.38 i

0.272 - 0.0118 i

5

2.354

1.31

6

-2.994

0.0184

7

-3.6344

0.0187 i

8

-4.7614 + 5.7464 i

0.0781 - 0.0566 i

9

-4.7614 - 5.7464 i

0.0781 + 0.0566 i

10

35.0255

8.99 i

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Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4998 + 0.0802 i

0.0963 - 0.0681 i
Singularities of quadratic [5, 5, 5] approximant

2

1.4998 - 0.0802 i

0.0963 + 0.0681 i

3

1.7641 + 0.4726 i

0.144 - 0.305 i

4

1.7641 - 0.4726 i

0.144 + 0.305 i

5

2.1589

2.72

6

-3.2032

0.0188

7

-3.6926 + 1.6387 i

0.00601 - 0.0172 i

8

-3.6926 - 1.6387 i

0.00601 + 0.0172 i

9

-3.2824 + 2.6542 i

0.028 + 0.00901 i

10

-3.2824 - 2.6542 i

0.028 - 0.00901 i

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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-1.221

0.0000785
Singularities of quadratic [6, 5, 5] approximant

2

-1.2218

0.0000785 i

3

1.4767 + 0.1329 i

0.0468 - 0.0143 i

4

1.4767 - 0.1329 i

0.0468 + 0.0143 i

5

1.7459

0.394

6

1.8179 + 0.6416 i

0.12 + 0.0249 i

7

1.8179 - 0.6416 i

0.12 - 0.0249 i

8

-2.733

0.00335

9

-5.3269

0.0108 i

10

-1.7864 + 5.5903 i

0.0271 + 0.0247 i

11

-1.7864 - 5.5903 i

0.0271 - 0.0247 i

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Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4297 + 0.022 i

0.101 - 0.137 i
Singularities of quadratic [6, 6, 5] approximant

2

1.4297 - 0.022 i

0.101 + 0.137 i

3

1.6579 + 0.4269 i

0.176 + 0.157 i

4

1.6579 - 0.4269 i

0.176 - 0.157 i

5

2.4747

8.67

6

-2.9675

0.00524

7

-2.7124 + 1.6161 i

0.000715 + 0.0049 i

8

-2.7124 - 1.6161 i

0.000715 - 0.0049 i

9

-2.7035 + 2.1193 i

0.00663 - 0.00088 i

10

-2.7035 - 2.1193 i

0.00663 + 0.00088 i

11

18.0867

4.45 i

12

-57.7998

1.46 i

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Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.2544 + 0.e-5 i

6.27e-8 - 6.27e-8 i
Singularities of quadratic [6, 6, 6] approximant

2

0.2544 - 0.e-5 i

6.27e-8 + 6.27e-8 i

3

1.4488

0.139

4

1.5142

0.765 i

5

1.6549 + 0.4684 i

0.0684 + 0.111 i

6

1.6549 - 0.4684 i

0.0684 - 0.111 i

7

2.5653

5.24e4

8

-2.8514 + 0.814 i

0.004 + 0.00279 i

9

-2.8514 - 0.814 i

0.004 - 0.00279 i

10

-3.2502 + 1.5582 i

0.00425 - 0.0069 i

11

-3.2502 - 1.5582 i

0.00425 + 0.0069 i

12

-6.5728

0.235

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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-0.4278 + 0.e-5 i

3.68e-7 + 3.68e-7 i
Singularities of quadratic [7, 6, 6] approximant

2

-0.4278 - 0.e-5 i

3.68e-7 - 3.68e-7 i

3

1.3852

0.0349

4

1.4994

0.0931 i

5

1.5775 + 0.4526 i

0.0601 + 0.0246 i

6

1.5775 - 0.4526 i

0.0601 - 0.0246 i

7

-3.0933

0.0158

8

-2.2854 + 2.8671 i

0.00722 - 0.00361 i

9

-2.2854 - 2.8671 i

0.00722 + 0.00361 i

10

3.9442 + 1.1436 i

0.183 - 0.199 i

11

3.9442 - 1.1436 i

0.183 + 0.199 i

12

-3.7911 + 2.7039 i

0.000483 + 0.00862 i

13

-3.7911 - 2.7039 i

0.000483 - 0.00862 i

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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-0.2539 + 0.4889 i

2.44e-7 + 4.e-8 i
Singularities of quadratic [7, 7, 6] approximant

2

-0.2539 - 0.4889 i

2.44e-7 - 4.e-8 i

3

-0.2539 + 0.4889 i

4.e-8 - 2.44e-7 i

4

-0.2539 - 0.4889 i

4.e-8 + 2.44e-7 i

5

1.0336 + 0.0022 i

0.000139 - 0.000137 i

6

1.0336 - 0.0022 i

0.000139 + 0.000137 i

7

1.4468

0.0193

8

1.8305 + 0.697 i

0.0404 + 0.0382 i

9

1.8305 - 0.697 i

0.0404 - 0.0382 i

10

-2.3396 + 0.6743 i

0.000511 + 0.000222 i

11

-2.3396 - 0.6743 i

0.000511 - 0.000222 i

12

-3.6688

0.00131

13

-6.5053

0.0178 i

14

14.9597

4.58 i

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Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1388	0.169

Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3914	0.257
2	125.4315	2.44 i

Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5112	0.365
2	-20.3923	0.841

Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.9687	0.0485
2	1.1285	0.0564 i
3	2.1886	7.6

Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4418	0.279
2	-6.6717 + 0.7311 i	1.18 + 0.432 i
3	-6.6717 - 0.7311 i	1.18 - 0.432 i
4	46.0893	1.91 i

Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5599	0.758
2	2.6831	0.595 i
3	4.3729	9.01
4	-6.8695	0.242

Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5572 + 0.3191 i	0.247 - 0.238 i
2	1.5572 - 0.3191 i	0.247 + 0.238 i
3	2.0903	0.419
4	-5.0965	0.151
5	-12.4177	0.172 i

Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5925 + 0.3212 i	0.311 - 0.224 i
2	1.5925 - 0.3212 i	0.311 + 0.224 i
3	2.2479	0.607
4	-4.8397	0.127
5	-11.1844	0.147 i
6	1515.8553	2.58 i

Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5566 + 0.2954 i	0.208 - 0.339 i
2	1.5566 - 0.2954 i	0.208 + 0.339 i
3	2.0368	0.366
4	-4.8779 + 0.3642 i	0.182 + 0.317 i
5	-4.8779 - 0.3642 i	0.182 - 0.317 i
6	-9.942	0.242

Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.441	0.0175
2	-1.4434	0.0176 i
3	1.5793 + 0.3222 i	0.29 - 0.22 i
4	1.5793 - 0.3222 i	0.29 + 0.22 i
5	2.1795	0.524
6	-5.2201	0.179
7	-11.9573	0.179 i

Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5934 + 0.3625 i	0.276 - 0.0573 i
2	1.5934 - 0.3625 i	0.276 + 0.0573 i
3	2.3112	0.972
4	-2.8768	0.0164
5	-3.3527	0.0167 i
6	-5.2728 + 5.294 i	0.0866 - 0.0528 i
7	-5.2728 - 5.294 i	0.0866 + 0.0528 i
8	49.8787	4.23 i

Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.173 + 0.5955 i	0.0000344 - 0.000436 i
2	-1.173 - 0.5955 i	0.0000344 + 0.000436 i
3	-1.1816 + 0.5944 i	0.000435 + 0.0000382 i
4	-1.1816 - 0.5944 i	0.000435 - 0.0000382 i
5	1.5975 + 0.3997 i	0.213 + 0.0621 i
6	1.5975 - 0.3997 i	0.213 - 0.0621 i
7	2.1802	0.861
8	-3.8669	0.0165

Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5855 + 0.2983 i	0.166 - 0.362 i
2	1.5855 - 0.2983 i	0.166 + 0.362 i
3	-2.811 + 0.1542 i	0.00693 + 0.00749 i
4	-2.811 - 0.1542 i	0.00693 - 0.00749 i
5	2.8974 + 0.4075 i	0.668 + 1.42 i
6	2.8974 - 0.4075 i	0.668 - 1.42 i
7	-4.5793	0.025
8	6.688 + 4.3766 i	0.681 - 0.0983 i
9	6.688 - 4.3766 i	0.681 + 0.0983 i

Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1154 + 0.0011 i	0.0222 - 0.0225 i
2	1.1154 - 0.0011 i	0.0222 + 0.0225 i
3	1.6072 + 0.38 i	0.272 + 0.0118 i
4	1.6072 - 0.38 i	0.272 - 0.0118 i
5	2.354	1.31
6	-2.994	0.0184
7	-3.6344	0.0187 i
8	-4.7614 + 5.7464 i	0.0781 - 0.0566 i
9	-4.7614 - 5.7464 i	0.0781 + 0.0566 i
10	35.0255	8.99 i

Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4998 + 0.0802 i	0.0963 - 0.0681 i
2	1.4998 - 0.0802 i	0.0963 + 0.0681 i
3	1.7641 + 0.4726 i	0.144 - 0.305 i
4	1.7641 - 0.4726 i	0.144 + 0.305 i
5	2.1589	2.72
6	-3.2032	0.0188
7	-3.6926 + 1.6387 i	0.00601 - 0.0172 i
8	-3.6926 - 1.6387 i	0.00601 + 0.0172 i
9	-3.2824 + 2.6542 i	0.028 + 0.00901 i
10	-3.2824 - 2.6542 i	0.028 - 0.00901 i

Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.221	0.0000785
2	-1.2218	0.0000785 i
3	1.4767 + 0.1329 i	0.0468 - 0.0143 i
4	1.4767 - 0.1329 i	0.0468 + 0.0143 i
5	1.7459	0.394
6	1.8179 + 0.6416 i	0.12 + 0.0249 i
7	1.8179 - 0.6416 i	0.12 - 0.0249 i
8	-2.733	0.00335
9	-5.3269	0.0108 i
10	-1.7864 + 5.5903 i	0.0271 + 0.0247 i
11	-1.7864 - 5.5903 i	0.0271 - 0.0247 i

Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4297 + 0.022 i	0.101 - 0.137 i
2	1.4297 - 0.022 i	0.101 + 0.137 i
3	1.6579 + 0.4269 i	0.176 + 0.157 i
4	1.6579 - 0.4269 i	0.176 - 0.157 i
5	2.4747	8.67
6	-2.9675	0.00524
7	-2.7124 + 1.6161 i	0.000715 + 0.0049 i
8	-2.7124 - 1.6161 i	0.000715 - 0.0049 i
9	-2.7035 + 2.1193 i	0.00663 - 0.00088 i
10	-2.7035 - 2.1193 i	0.00663 + 0.00088 i
11	18.0867	4.45 i
12	-57.7998	1.46 i

Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.2544 + 0.e-5 i	6.27e-8 - 6.27e-8 i
2	0.2544 - 0.e-5 i	6.27e-8 + 6.27e-8 i
3	1.4488	0.139
4	1.5142	0.765 i
5	1.6549 + 0.4684 i	0.0684 + 0.111 i
6	1.6549 - 0.4684 i	0.0684 - 0.111 i
7	2.5653	5.24e4
8	-2.8514 + 0.814 i	0.004 + 0.00279 i
9	-2.8514 - 0.814 i	0.004 - 0.00279 i
10	-3.2502 + 1.5582 i	0.00425 - 0.0069 i
11	-3.2502 - 1.5582 i	0.00425 + 0.0069 i
12	-6.5728	0.235

Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-0.4278 + 0.e-5 i	3.68e-7 + 3.68e-7 i
2	-0.4278 - 0.e-5 i	3.68e-7 - 3.68e-7 i
3	1.3852	0.0349
4	1.4994	0.0931 i
5	1.5775 + 0.4526 i	0.0601 + 0.0246 i
6	1.5775 - 0.4526 i	0.0601 - 0.0246 i
7	-3.0933	0.0158
8	-2.2854 + 2.8671 i	0.00722 - 0.00361 i
9	-2.2854 - 2.8671 i	0.00722 + 0.00361 i
10	3.9442 + 1.1436 i	0.183 - 0.199 i
11	3.9442 - 1.1436 i	0.183 + 0.199 i
12	-3.7911 + 2.7039 i	0.000483 + 0.00862 i
13	-3.7911 - 2.7039 i	0.000483 - 0.00862 i

Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-0.2539 + 0.4889 i	2.44e-7 + 4.e-8 i
2	-0.2539 - 0.4889 i	2.44e-7 - 4.e-8 i
3	-0.2539 + 0.4889 i	4.e-8 - 2.44e-7 i
4	-0.2539 - 0.4889 i	4.e-8 + 2.44e-7 i
5	1.0336 + 0.0022 i	0.000139 - 0.000137 i
6	1.0336 - 0.0022 i	0.000139 + 0.000137 i
7	1.4468	0.0193
8	1.8305 + 0.697 i	0.0404 + 0.0382 i
9	1.8305 - 0.697 i	0.0404 - 0.0382 i
10	-2.3396 + 0.6743 i	0.000511 + 0.000222 i
11	-2.3396 - 0.6743 i	0.000511 - 0.000222 i
12	-3.6688	0.00131
13	-6.5053	0.0178 i
14	14.9597	4.58 i