| L(s) = 1 | − 2.54e7·2-s − 8.64e11·3-s + 8.31e13·4-s + 1.67e17·5-s + 2.19e19·6-s − 3.42e20·7-s + 1.21e22·8-s + 5.08e23·9-s − 4.25e24·10-s − 2.47e24·11-s − 7.18e25·12-s − 1.86e26·13-s + 8.71e27·14-s − 1.44e29·15-s − 3.56e29·16-s + 8.08e29·17-s − 1.29e31·18-s − 1.31e31·19-s + 1.39e31·20-s + 2.96e32·21-s + 6.28e31·22-s + 3.96e33·23-s − 1.05e34·24-s + 1.02e34·25-s + 4.74e33·26-s − 2.32e35·27-s − 2.84e34·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 1.76·3-s + 0.147·4-s + 1.25·5-s + 1.89·6-s − 0.676·7-s + 0.913·8-s + 2.12·9-s − 1.34·10-s − 0.0756·11-s − 0.261·12-s − 0.0952·13-s + 0.724·14-s − 2.22·15-s − 1.12·16-s + 0.577·17-s − 2.27·18-s − 0.616·19-s + 0.185·20-s + 1.19·21-s + 0.0810·22-s + 1.72·23-s − 1.61·24-s + 0.577·25-s + 0.102·26-s − 1.98·27-s − 0.0998·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(50-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+49/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(25)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.54e7T + 5.62e14T^{2} \) |
| 3 | \( 1 + 8.64e11T + 2.39e23T^{2} \) |
| 5 | \( 1 - 1.67e17T + 1.77e34T^{2} \) |
| 7 | \( 1 + 3.42e20T + 2.56e41T^{2} \) |
| 11 | \( 1 + 2.47e24T + 1.06e51T^{2} \) |
| 13 | \( 1 + 1.86e26T + 3.83e54T^{2} \) |
| 17 | \( 1 - 8.08e29T + 1.95e60T^{2} \) |
| 19 | \( 1 + 1.31e31T + 4.55e62T^{2} \) |
| 23 | \( 1 - 3.96e33T + 5.30e66T^{2} \) |
| 29 | \( 1 - 4.73e35T + 4.54e71T^{2} \) |
| 31 | \( 1 + 5.52e36T + 1.19e73T^{2} \) |
| 37 | \( 1 - 6.02e37T + 6.94e76T^{2} \) |
| 41 | \( 1 + 4.00e39T + 1.06e79T^{2} \) |
| 43 | \( 1 + 1.57e39T + 1.09e80T^{2} \) |
| 47 | \( 1 + 3.77e40T + 8.56e81T^{2} \) |
| 53 | \( 1 + 7.38e41T + 3.08e84T^{2} \) |
| 59 | \( 1 + 3.19e43T + 5.91e86T^{2} \) |
| 61 | \( 1 - 2.48e43T + 3.02e87T^{2} \) |
| 67 | \( 1 - 5.76e44T + 3.00e89T^{2} \) |
| 71 | \( 1 + 1.66e45T + 5.14e90T^{2} \) |
| 73 | \( 1 - 4.80e45T + 2.00e91T^{2} \) |
| 79 | \( 1 + 4.21e45T + 9.63e92T^{2} \) |
| 83 | \( 1 + 1.20e47T + 1.08e94T^{2} \) |
| 89 | \( 1 - 8.70e47T + 3.31e95T^{2} \) |
| 97 | \( 1 + 7.30e48T + 2.24e97T^{2} \) |
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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
−18.66258266906742905187920213838, −17.42533108564586364924787074572, −16.59357600966871094712988073268, −12.97200104867324834434759329215, −10.72193136381819341197236094422, −9.584881447441338852328025424151, −6.68829227134675823405056170950, −5.19645735666897088606660426356, −1.36617840071946262905158063342, 0,
1.36617840071946262905158063342, 5.19645735666897088606660426356, 6.68829227134675823405056170950, 9.584881447441338852328025424151, 10.72193136381819341197236094422, 12.97200104867324834434759329215, 16.59357600966871094712988073268, 17.42533108564586364924787074572, 18.66258266906742905187920213838
