| L(s) = 1 | + 0.656·2-s − 1.56·4-s − 5-s − 3.08·7-s − 2.34·8-s − 0.656·10-s − 0.958·11-s − 2.02·14-s + 1.60·16-s + 0.327·17-s + 2.81·19-s + 1.56·20-s − 0.629·22-s + 5.96·23-s + 25-s + 4.83·28-s − 7.96·29-s + 1.94·31-s + 5.73·32-s + 0.215·34-s + 3.08·35-s + 7.39·37-s + 1.85·38-s + 2.34·40-s + 7.06·41-s + 0.739·43-s + 1.50·44-s + ⋯ |
| L(s) = 1 | + 0.464·2-s − 0.784·4-s − 0.447·5-s − 1.16·7-s − 0.828·8-s − 0.207·10-s − 0.289·11-s − 0.540·14-s + 0.400·16-s + 0.0795·17-s + 0.646·19-s + 0.350·20-s − 0.134·22-s + 1.24·23-s + 0.200·25-s + 0.913·28-s − 1.47·29-s + 0.349·31-s + 1.01·32-s + 0.0369·34-s + 0.520·35-s + 1.21·37-s + 0.300·38-s + 0.370·40-s + 1.10·41-s + 0.112·43-s + 0.226·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 0.958T + 11T^{2} \) |
| 17 | \( 1 - 0.327T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 7.06T + 41T^{2} \) |
| 43 | \( 1 - 0.739T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 - 8.46T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 3.66T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6.13T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
| 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
−7.54696433770573254595022467216, −6.77986946662504126001747726126, −5.99706373394735640572275588892, −5.41961678874486884880179240382, −4.63767624082262795605858953427, −3.90298192568451173596831002291, −3.24820598194486547742320965400, −2.64888364957977754546103211301, −1.01028122708758685898081556825, 0,
1.01028122708758685898081556825, 2.64888364957977754546103211301, 3.24820598194486547742320965400, 3.90298192568451173596831002291, 4.63767624082262795605858953427, 5.41961678874486884880179240382, 5.99706373394735640572275588892, 6.77986946662504126001747726126, 7.54696433770573254595022467216
