| L(s) = 1 | − 2.26·2-s + 2.09·3-s + 3.12·4-s + 5-s − 4.73·6-s − 7-s − 2.54·8-s + 1.37·9-s − 2.26·10-s + 4.08·11-s + 6.53·12-s + 2.26·14-s + 2.09·15-s − 0.491·16-s + 3.59·17-s − 3.11·18-s + 6.98·19-s + 3.12·20-s − 2.09·21-s − 9.23·22-s + 3.83·23-s − 5.31·24-s + 25-s − 3.39·27-s − 3.12·28-s + 8.96·29-s − 4.73·30-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.20·3-s + 1.56·4-s + 0.447·5-s − 1.93·6-s − 0.377·7-s − 0.898·8-s + 0.458·9-s − 0.715·10-s + 1.23·11-s + 1.88·12-s + 0.604·14-s + 0.540·15-s − 0.122·16-s + 0.871·17-s − 0.733·18-s + 1.60·19-s + 0.698·20-s − 0.456·21-s − 1.96·22-s + 0.799·23-s − 1.08·24-s + 0.200·25-s − 0.654·27-s − 0.590·28-s + 1.66·29-s − 0.864·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854112316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854112316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 - 6.98T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 0.901T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.246T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 - 0.353T + 59T^{2} \) |
| 61 | \( 1 + 8.95T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 - 6.90T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 9.49T + 97T^{2} \) |
| 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
−8.204089425865579193277280650724, −7.68501550261347555891031268974, −6.97288299636130758332649050834, −6.37658258596345502948355166609, −5.38548968327225464265531988215, −4.21864338667994819367323285786, −3.13482978649567652961154629312, −2.75074141271802504265464852282, −1.54330084416205860716398238972, −0.962942116926259930756018927419,
0.962942116926259930756018927419, 1.54330084416205860716398238972, 2.75074141271802504265464852282, 3.13482978649567652961154629312, 4.21864338667994819367323285786, 5.38548968327225464265531988215, 6.37658258596345502948355166609, 6.97288299636130758332649050834, 7.68501550261347555891031268974, 8.204089425865579193277280650724
