L(s) = 1 | + 2-s − 2.40·3-s + 4-s + 3.31·5-s − 2.40·6-s − 1.55·7-s + 8-s + 2.79·9-s + 3.31·10-s + 3.17·11-s − 2.40·12-s + 0.560·13-s − 1.55·14-s − 7.98·15-s + 16-s + 5.55·17-s + 2.79·18-s − 5.83·19-s + 3.31·20-s + 3.73·21-s + 3.17·22-s − 3.23·23-s − 2.40·24-s + 6.00·25-s + 0.560·26-s + 0.486·27-s − 1.55·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 1.48·5-s − 0.983·6-s − 0.586·7-s + 0.353·8-s + 0.932·9-s + 1.04·10-s + 0.957·11-s − 0.695·12-s + 0.155·13-s − 0.414·14-s − 2.06·15-s + 0.250·16-s + 1.34·17-s + 0.659·18-s − 1.33·19-s + 0.741·20-s + 0.815·21-s + 0.677·22-s − 0.673·23-s − 0.491·24-s + 1.20·25-s + 0.109·26-s + 0.0937·27-s − 0.293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.085512049\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.085512049\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 569 | \( 1 + T \) |
good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 0.560T + 13T^{2} \) |
| 17 | \( 1 - 5.55T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 5.51T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.862652242277106704369119333572, −9.462227830847586132291447167286, −8.024705730355314003997277419143, −6.59918256777526410510037313047, −6.27160778042210285565852813016, −5.77559645893023325507669588852, −4.88613083859291292764962894333, −3.83022521596060443581669250039, −2.41699613964447022462132701662, −1.13748743925885564345874510952,
1.13748743925885564345874510952, 2.41699613964447022462132701662, 3.83022521596060443581669250039, 4.88613083859291292764962894333, 5.77559645893023325507669588852, 6.27160778042210285565852813016, 6.59918256777526410510037313047, 8.024705730355314003997277419143, 9.462227830847586132291447167286, 9.862652242277106704369119333572