| L(s) = 1 | + 1.15·2-s − 0.675·4-s + 5-s − 4.95·7-s − 3.07·8-s + 1.15·10-s + 11-s + 4.54·13-s − 5.69·14-s − 2.19·16-s + 0.482·17-s − 19-s − 0.675·20-s + 1.15·22-s + 0.817·23-s + 25-s + 5.23·26-s + 3.34·28-s + 3.20·29-s − 5.82·31-s + 3.63·32-s + 0.555·34-s − 4.95·35-s + 5.38·37-s − 1.15·38-s − 3.07·40-s + 3.75·41-s + ⋯ |
| L(s) = 1 | + 0.813·2-s − 0.337·4-s + 0.447·5-s − 1.87·7-s − 1.08·8-s + 0.363·10-s + 0.301·11-s + 1.26·13-s − 1.52·14-s − 0.548·16-s + 0.117·17-s − 0.229·19-s − 0.151·20-s + 0.245·22-s + 0.170·23-s + 0.200·25-s + 1.02·26-s + 0.631·28-s + 0.595·29-s − 1.04·31-s + 0.642·32-s + 0.0952·34-s − 0.836·35-s + 0.885·37-s − 0.186·38-s − 0.486·40-s + 0.587·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 7 | \( 1 + 4.95T + 7T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 - 0.482T + 17T^{2} \) |
| 23 | \( 1 - 0.817T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 + 2.57T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 - 9.35T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 - 9.99T + 73T^{2} \) |
| 79 | \( 1 + 9.09T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
| 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
−6.97475143940144364486801543258, −6.48918572825426378237862869657, −5.90382171041915028786310693064, −5.52911565095814546448711617981, −4.39346021736365210717116907685, −3.82300462253459044908645425304, −3.20472355065296841332712671136, −2.58574074183929101835150815280, −1.16032314729191558403063670878, 0,
1.16032314729191558403063670878, 2.58574074183929101835150815280, 3.20472355065296841332712671136, 3.82300462253459044908645425304, 4.39346021736365210717116907685, 5.52911565095814546448711617981, 5.90382171041915028786310693064, 6.48918572825426378237862869657, 6.97475143940144364486801543258
