| L(s) = 1 | − 2.75e12·2-s − 2.21e19·3-s + 5.14e24·4-s + 2.52e28·5-s + 6.08e31·6-s − 2.62e34·7-s − 7.51e36·8-s + 4.55e37·9-s − 6.95e40·10-s − 6.46e41·11-s − 1.13e44·12-s + 1.64e45·13-s + 7.22e46·14-s − 5.59e47·15-s + 8.21e48·16-s + 1.28e49·17-s − 1.25e50·18-s − 8.42e51·19-s + 1.30e53·20-s + 5.81e53·21-s + 1.77e54·22-s + 1.57e55·23-s + 1.66e56·24-s + 2.26e56·25-s − 4.52e57·26-s + 8.79e57·27-s − 1.35e59·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.05·3-s + 2.12·4-s + 1.24·5-s + 1.85·6-s − 1.56·7-s − 1.99·8-s + 0.102·9-s − 2.20·10-s − 0.430·11-s − 2.23·12-s + 1.26·13-s + 2.75·14-s − 1.30·15-s + 1.40·16-s + 0.188·17-s − 0.181·18-s − 1.36·19-s + 2.64·20-s + 1.63·21-s + 0.762·22-s + 1.11·23-s + 2.09·24-s + 0.547·25-s − 2.23·26-s + 0.942·27-s − 3.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(82-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+81/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(41)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.75e12T + 2.41e24T^{2} \) |
| 3 | \( 1 + 2.21e19T + 4.43e38T^{2} \) |
| 5 | \( 1 - 2.52e28T + 4.13e56T^{2} \) |
| 7 | \( 1 + 2.62e34T + 2.83e68T^{2} \) |
| 11 | \( 1 + 6.46e41T + 2.25e84T^{2} \) |
| 13 | \( 1 - 1.64e45T + 1.69e90T^{2} \) |
| 17 | \( 1 - 1.28e49T + 4.63e99T^{2} \) |
| 19 | \( 1 + 8.42e51T + 3.79e103T^{2} \) |
| 23 | \( 1 - 1.57e55T + 1.99e110T^{2} \) |
| 29 | \( 1 - 1.02e59T + 2.84e118T^{2} \) |
| 31 | \( 1 - 3.33e59T + 6.31e120T^{2} \) |
| 37 | \( 1 - 8.95e62T + 1.05e127T^{2} \) |
| 41 | \( 1 - 1.61e65T + 4.32e130T^{2} \) |
| 43 | \( 1 - 1.76e66T + 2.04e132T^{2} \) |
| 47 | \( 1 + 8.77e67T + 2.75e135T^{2} \) |
| 53 | \( 1 - 2.20e69T + 4.63e139T^{2} \) |
| 59 | \( 1 - 3.92e71T + 2.74e143T^{2} \) |
| 61 | \( 1 + 7.29e71T + 4.08e144T^{2} \) |
| 67 | \( 1 - 1.78e73T + 8.16e147T^{2} \) |
| 71 | \( 1 + 3.78e74T + 8.95e149T^{2} \) |
| 73 | \( 1 + 1.18e75T + 8.49e150T^{2} \) |
| 79 | \( 1 + 3.54e76T + 5.10e153T^{2} \) |
| 83 | \( 1 + 5.45e76T + 2.78e155T^{2} \) |
| 89 | \( 1 + 1.21e79T + 7.95e157T^{2} \) |
| 97 | \( 1 - 2.69e80T + 8.48e160T^{2} \) |
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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
−16.03072853896225730829932115806, −12.96199690288858906638487912695, −10.94766250230298455378596091531, −10.02125390120601792525795265268, −8.822770557111204310253685643999, −6.59037260215927496764715882889, −5.96762796447820161666854871301, −2.66577113216756351466966996272, −1.08863083218493401003782556460, 0,
1.08863083218493401003782556460, 2.66577113216756351466966996272, 5.96762796447820161666854871301, 6.59037260215927496764715882889, 8.822770557111204310253685643999, 10.02125390120601792525795265268, 10.94766250230298455378596091531, 12.96199690288858906638487912695, 16.03072853896225730829932115806
