| L(s) = 1 | + (0.642 + 0.766i)2-s + (−1.68 + 0.398i)3-s + (−0.173 + 0.984i)4-s + (1.35 + 1.78i)5-s + (−1.38 − 1.03i)6-s + (3.12 + 1.80i)7-s + (−0.866 + 0.500i)8-s + (2.68 − 1.34i)9-s + (−0.494 + 2.18i)10-s + (2.73 − 1.57i)11-s + (−0.0993 − 1.72i)12-s + (−2.26 + 0.824i)13-s + (0.627 + 3.55i)14-s + (−2.98 − 2.46i)15-s + (−0.939 − 0.342i)16-s + (4.15 − 3.48i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.973 + 0.229i)3-s + (−0.0868 + 0.492i)4-s + (0.604 + 0.796i)5-s + (−0.566 − 0.422i)6-s + (1.18 + 0.682i)7-s + (−0.306 + 0.176i)8-s + (0.894 − 0.447i)9-s + (−0.156 + 0.689i)10-s + (0.825 − 0.476i)11-s + (−0.0286 − 0.499i)12-s + (−0.628 + 0.228i)13-s + (0.167 + 0.950i)14-s + (−0.771 − 0.636i)15-s + (−0.234 − 0.0855i)16-s + (1.00 − 0.845i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.943478 + 1.39256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943478 + 1.39256i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 3 | \( 1 + (1.68 - 0.398i)T \) |
| 5 | \( 1 + (-1.35 - 1.78i)T \) |
| 19 | \( 1 + (3.27 - 2.87i)T \) |
| good | 7 | \( 1 + (-3.12 - 1.80i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.73 + 1.57i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.26 - 0.824i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.15 + 3.48i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.756 - 4.28i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (6.09 + 5.11i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.95 - 4.01i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.463T + 37T^{2} \) |
| 41 | \( 1 + (1.55 + 0.566i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (5.75 - 1.01i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0528 - 0.0443i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.30 + 0.583i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.4 + 8.72i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 7.70i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.11 + 6.80i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.599 - 3.40i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.696 + 1.91i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (5.08 - 13.9i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.86 + 10.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.318 + 0.116i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.07 + 5.09i)T + (16.8 - 95.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42521687747941242256815034791, −10.12950571164308489514950647842, −9.420276813113759559867527047376, −8.155121133612549074607018203711, −7.18516893243449370101054954047, −6.26094820936083894918898815447, −5.55601463225245296749388690189, −4.78989723669875910639758504054, −3.49518645703010732350976984887, −1.83766021612402956424053412942,
1.05512839503239942335422102790, 1.97950952234140776692761286794, 4.18475866932802814222571808916, 4.75338741534051673842248391925, 5.62105468302858163246471023109, 6.62420258470238202589708520735, 7.71089081798033645760234189689, 8.799127612823295210450062816960, 10.04744938491410237317558022584, 10.46114003840550062314314366934
