| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 2·17-s − 4·19-s − 25-s + 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s + 2·39-s − 41-s − 4·43-s − 2·45-s − 12·47-s − 2·51-s + 6·53-s − 8·55-s − 4·57-s − 4·59-s − 10·61-s − 4·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.156·41-s − 0.609·43-s − 0.298·45-s − 1.75·47-s − 0.280·51-s + 0.824·53-s − 1.07·55-s − 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−12.52576458271238, −12.19676416386030, −11.86653351207019, −11.31928678778028, −11.05152860647907, −10.32954520426650, −10.05295411297447, −9.436384297278621, −8.949248564744482, −8.505326699647431, −8.304286662723773, −7.729779057545532, −7.263765932908127, −6.630442518551831, −6.295975300640926, −6.040883545723984, −4.943915446549702, −4.555687755530121, −4.254415753241129, −3.633927496744391, −3.273716510775845, −2.688152285973370, −1.970583141413864, −1.431926227197804, −0.7985930684633813, 0,
0.7985930684633813, 1.431926227197804, 1.970583141413864, 2.688152285973370, 3.273716510775845, 3.633927496744391, 4.254415753241129, 4.555687755530121, 4.943915446549702, 6.040883545723984, 6.295975300640926, 6.630442518551831, 7.263765932908127, 7.729779057545532, 8.304286662723773, 8.505326699647431, 8.949248564744482, 9.436384297278621, 10.05295411297447, 10.32954520426650, 11.05152860647907, 11.31928678778028, 11.86653351207019, 12.19676416386030, 12.52576458271238
