| L(s) = 1 | + (1.23 − 3.80i)2-s + (−0.732 − 0.531i)3-s + (−12.9 − 9.40i)4-s + (54.7 + 11.4i)5-s + (−2.92 + 2.12i)6-s + 133.·7-s + (−51.7 + 37.6i)8-s + (−74.8 − 230. i)9-s + (111. − 193. i)10-s + (89.0 − 274. i)11-s + (4.47 + 13.7i)12-s + (−99.4 − 306. i)13-s + (164. − 506. i)14-s + (−33.9 − 37.4i)15-s + (79.1 + 243. i)16-s + (176. − 128. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.0469 − 0.0341i)3-s + (−0.404 − 0.293i)4-s + (0.978 + 0.205i)5-s + (−0.0332 + 0.0241i)6-s + 1.02·7-s + (−0.286 + 0.207i)8-s + (−0.307 − 0.947i)9-s + (0.351 − 0.613i)10-s + (0.221 − 0.683i)11-s + (0.00896 + 0.0276i)12-s + (−0.163 − 0.502i)13-s + (0.224 − 0.691i)14-s + (−0.0389 − 0.0430i)15-s + (0.0772 + 0.237i)16-s + (0.148 − 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.62207 - 1.39206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62207 - 1.39206i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 5 | \( 1 + (-54.7 - 11.4i)T \) |
| good | 3 | \( 1 + (0.732 + 0.531i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 133.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-89.0 + 274. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (99.4 + 306. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-176. + 128. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-270. + 196. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-381. + 1.17e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-2.83e3 - 2.06e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.31e3 - 1.68e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.85e3 - 8.79e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-5.99e3 - 1.84e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.43e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.14e3 + 1.55e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.59e4 + 1.15e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.15e4 - 3.55e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (2.34e3 - 7.21e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-5.08e4 + 3.69e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (5.76e4 + 4.18e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-7.99e3 + 2.45e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-8.01e3 - 5.82e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (7.13e4 - 5.18e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (4.19e4 - 1.29e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.06e5 - 7.74e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.26789413940398440849570739288, −13.14667138616970069681118863219, −11.86441093669182668144517793642, −10.86622685657920989612696995282, −9.643422873630013586434975932430, −8.422923152483270988482344843331, −6.36241818130189363180156285545, −5.03570719298765819844229552534, −3.02966535939864992165640379484, −1.21010794852882549823721704543,
1.95194612345219113547217206088, 4.63693374947571710561907628562, 5.67023665827406217981849076789, 7.30581257903836950111320558496, 8.582253395643839564659948478859, 9.898893405693999736701447855104, 11.32767489388068104957644551680, 12.77945696179773846001052169257, 13.99121315053205534596652994862, 14.51936079651612454052040453403
