| L(s) = 1 | + (−6.12 + 5.14i)2-s + (39.9 + 14.5i)3-s + (11.1 − 63.0i)4-s + (−77.2 − 437. i)5-s + (−319. + 116. i)6-s + (−813. + 1.40e3i)7-s + (256. + 443. i)8-s + (−290. − 243. i)9-s + (2.72e3 + 2.28e3i)10-s + (2.07e3 + 3.59e3i)11-s + (1.36e3 − 2.35e3i)12-s + (−9.33e3 + 3.39e3i)13-s + (−2.26e3 − 1.28e4i)14-s + (3.28e3 − 1.86e4i)15-s + (−3.84e3 − 1.40e3i)16-s + (−1.87e4 + 1.57e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.854 + 0.310i)3-s + (0.0868 − 0.492i)4-s + (−0.276 − 1.56i)5-s + (−0.604 + 0.219i)6-s + (−0.896 + 1.55i)7-s + (0.176 + 0.306i)8-s + (−0.132 − 0.111i)9-s + (0.861 + 0.723i)10-s + (0.469 + 0.813i)11-s + (0.227 − 0.393i)12-s + (−1.17 + 0.428i)13-s + (−0.220 − 1.24i)14-s + (0.251 − 1.42i)15-s + (−0.234 − 0.0855i)16-s + (−0.923 + 0.775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0901i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0190290 + 0.421304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0190290 + 0.421304i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.12 - 5.14i)T \) |
| 19 | \( 1 + (7.38e3 - 2.89e4i)T \) |
| good | 3 | \( 1 + (-39.9 - 14.5i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (77.2 + 437. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (813. - 1.40e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.07e3 - 3.59e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (9.33e3 - 3.39e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.87e4 - 1.57e4i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.31e4 + 7.47e4i)T + (-3.19e9 - 1.16e9i)T^{2} \) |
| 29 | \( 1 + (7.06e4 + 5.92e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-3.46e3 + 5.99e3i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.36e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.16e4 + 4.23e3i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.19e5 - 6.76e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-4.22e5 - 3.54e5i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + (1.18e5 - 6.70e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (1.02e6 - 8.58e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-4.92e5 + 2.79e6i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (2.65e6 + 2.22e6i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-7.60e5 - 4.31e6i)T + (-8.54e12 + 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-1.17e6 - 4.27e5i)T + (8.46e12 + 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-3.34e5 - 1.21e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.59e6 + 4.48e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-4.33e6 + 1.57e6i)T + (3.38e13 - 2.84e13i)T^{2} \) |
| 97 | \( 1 + (9.55e6 - 8.01e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−15.35529580516328524726131068383, −14.66951419052603096218291233090, −12.74513844944267044667120937600, −12.10558426347845145540044520569, −9.569727265314620103195658821303, −9.098769379032809936660185937540, −8.194549499190423765760350020346, −6.14082008934573784140851424907, −4.48483093806406505362041686115, −2.20469227296889151093330545866,
0.18432723398659221429320047963, 2.66870060741994531259104732089, 3.56281142932978085552666535011, 6.95967449454650048292533682932, 7.46805189467717359156801444943, 9.273581941005497088558926150169, 10.51614597607559162554080223867, 11.34608146041850227018041132272, 13.34584180000393763383310669832, 13.97209842858397426532933280488
