| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 5-s − 2.23·6-s − 3.60·7-s + 8-s + 2.01·9-s + 10-s + 11-s − 2.23·12-s − 0.768·13-s − 3.60·14-s − 2.23·15-s + 16-s − 5.69·17-s + 2.01·18-s − 5.36·19-s + 20-s + 8.07·21-s + 22-s + 1.68·23-s − 2.23·24-s + 25-s − 0.768·26-s + 2.21·27-s − 3.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.913·6-s − 1.36·7-s + 0.353·8-s + 0.670·9-s + 0.316·10-s + 0.301·11-s − 0.646·12-s − 0.213·13-s − 0.963·14-s − 0.578·15-s + 0.250·16-s − 1.38·17-s + 0.474·18-s − 1.22·19-s + 0.223·20-s + 1.76·21-s + 0.213·22-s + 0.351·23-s − 0.456·24-s + 0.200·25-s − 0.150·26-s + 0.425·27-s − 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9658252598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9658252598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 13 | \( 1 + 0.768T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 5.36T + 19T^{2} \) |
| 23 | \( 1 - 1.68T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 - 0.00435T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 - 1.57T + 67T^{2} \) |
| 71 | \( 1 + 8.75T + 71T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 4.57T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
| 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
−7.32198741851551090329619333780, −6.82031835607678170024252519251, −6.32093807470249646843506945239, −5.82275184680662172650520301484, −5.20088788560491540417150518091, −4.34507104592379436846440312037, −3.73032170082535782398193384075, −2.69155884168240532959316686678, −1.89258373889292225453769096900, −0.44108898597962148086342196873,
0.44108898597962148086342196873, 1.89258373889292225453769096900, 2.69155884168240532959316686678, 3.73032170082535782398193384075, 4.34507104592379436846440312037, 5.20088788560491540417150518091, 5.82275184680662172650520301484, 6.32093807470249646843506945239, 6.82031835607678170024252519251, 7.32198741851551090329619333780
