| L(s) = 1 | + 0.872·2-s − 19.2·3-s − 31.2·4-s − 25·5-s − 16.7·6-s + 14.4·7-s − 55.2·8-s + 126.·9-s − 21.8·10-s + 121·11-s + 600.·12-s + 515.·13-s + 12.6·14-s + 480.·15-s + 951.·16-s + 1.42e3·17-s + 110.·18-s − 361·19-s + 780.·20-s − 278.·21-s + 105.·22-s + 4.70e3·23-s + 1.06e3·24-s + 625·25-s + 450.·26-s + 2.24e3·27-s − 452.·28-s + ⋯ |
| L(s) = 1 | + 0.154·2-s − 1.23·3-s − 0.976·4-s − 0.447·5-s − 0.190·6-s + 0.111·7-s − 0.304·8-s + 0.519·9-s − 0.0690·10-s + 0.301·11-s + 1.20·12-s + 0.846·13-s + 0.0172·14-s + 0.551·15-s + 0.929·16-s + 1.19·17-s + 0.0801·18-s − 0.229·19-s + 0.436·20-s − 0.137·21-s + 0.0465·22-s + 1.85·23-s + 0.375·24-s + 0.200·25-s + 0.130·26-s + 0.592·27-s − 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9545792442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9545792442\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 0.872T + 32T^{2} \) |
| 3 | \( 1 + 19.2T + 243T^{2} \) |
| 7 | \( 1 - 14.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 515.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.42e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.52e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.71e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.06e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 352.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.45e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.87e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.14e5T + 8.58e9T^{2} \) |
| 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
−9.186988504397726847535916304166, −8.436249075963456854366615819723, −7.47394160717517348546757271136, −6.47551007883631354781660399366, −5.52082076112510667557957258568, −5.10672171199728397603966239505, −4.01724176889006116129522708626, −3.26283618183725160298936603010, −1.29070908902298079132284832626, −0.51392958900548363945745228525,
0.51392958900548363945745228525, 1.29070908902298079132284832626, 3.26283618183725160298936603010, 4.01724176889006116129522708626, 5.10672171199728397603966239505, 5.52082076112510667557957258568, 6.47551007883631354781660399366, 7.47394160717517348546757271136, 8.436249075963456854366615819723, 9.186988504397726847535916304166
