| L(s) = 1 | − 0.917·2-s − 2.85·3-s − 1.15·4-s − 0.732·5-s + 2.61·6-s + 4.57·7-s + 2.89·8-s + 5.13·9-s + 0.671·10-s − 4.12·11-s + 3.30·12-s + 13-s − 4.19·14-s + 2.08·15-s − 0.339·16-s − 3.43·17-s − 4.71·18-s + 0.712·19-s + 0.848·20-s − 13.0·21-s + 3.78·22-s + 1.36·23-s − 8.26·24-s − 4.46·25-s − 0.917·26-s − 6.09·27-s − 5.30·28-s + ⋯ |
| L(s) = 1 | − 0.648·2-s − 1.64·3-s − 0.579·4-s − 0.327·5-s + 1.06·6-s + 1.73·7-s + 1.02·8-s + 1.71·9-s + 0.212·10-s − 1.24·11-s + 0.954·12-s + 0.277·13-s − 1.12·14-s + 0.539·15-s − 0.0847·16-s − 0.832·17-s − 1.11·18-s + 0.163·19-s + 0.189·20-s − 2.84·21-s + 0.807·22-s + 0.283·23-s − 1.68·24-s − 0.892·25-s − 0.179·26-s − 1.17·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.917T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 0.712T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 4.53T + 29T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 - 9.83T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 + 7.11T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.98T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−10.92204376421184613922922616996, −10.24500694905063234267603838073, −8.925388777072507912619254780078, −7.939291090071447440032423011749, −7.31535352816942495041293869605, −5.73448395035848341504897131351, −4.98551706407973014355281618771, −4.32670204224815510482353871834, −1.59069704475622030915272896410, 0,
1.59069704475622030915272896410, 4.32670204224815510482353871834, 4.98551706407973014355281618771, 5.73448395035848341504897131351, 7.31535352816942495041293869605, 7.939291090071447440032423011749, 8.925388777072507912619254780078, 10.24500694905063234267603838073, 10.92204376421184613922922616996
