| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·13-s − 15-s − 2·17-s − 4·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s + 4·35-s + 10·37-s − 2·39-s + 6·41-s + 12·43-s + 45-s + 4·47-s + 9·49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s + 10·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.676·35-s + 1.64·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.983525986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.983525986\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.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 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 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.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−13.59422444217473, −13.01633909800106, −12.74970178225301, −12.02432831803038, −11.60827683729852, −11.08266324689101, −10.77638218485347, −10.45051244441817, −9.719568353722075, −9.080762777210929, −8.674064973501133, −8.265381394729221, −7.488424365988058, −7.286405081991743, −6.398226251379197, −6.037207367002904, −5.591084541956551, −4.866272830304736, −4.403763704589181, −4.200287844445956, −3.135505449183248, −2.354814907913348, −1.997520199496179, −1.036124961773064, −0.7825105023014514,
0.7825105023014514, 1.036124961773064, 1.997520199496179, 2.354814907913348, 3.135505449183248, 4.200287844445956, 4.403763704589181, 4.866272830304736, 5.591084541956551, 6.037207367002904, 6.398226251379197, 7.286405081991743, 7.488424365988058, 8.265381394729221, 8.674064973501133, 9.080762777210929, 9.719568353722075, 10.45051244441817, 10.77638218485347, 11.08266324689101, 11.60827683729852, 12.02432831803038, 12.74970178225301, 13.01633909800106, 13.59422444217473
