| L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s + 4·11-s − 2·13-s − 15-s + 2·17-s + 3·19-s + 21-s − 6·23-s − 4·25-s + 5·27-s − 5·29-s − 4·31-s − 4·33-s − 35-s + 6·37-s + 2·39-s + 3·41-s + 8·43-s − 2·45-s + 2·47-s − 6·49-s − 2·51-s − 11·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.688·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.962·27-s − 0.928·29-s − 0.718·31-s − 0.696·33-s − 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.468·41-s + 1.21·43-s − 0.298·45-s + 0.291·47-s − 6/7·49-s − 0.280·51-s − 1.51·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 59 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−8.018755445011978815149977634888, −7.40961699026055564212526317651, −6.40387843928083411347729073969, −5.94264930407483059532376201400, −5.36336736026187235479038459757, −4.28681391729088540393374354665, −3.49734312353215216887343182177, −2.46296394936642547160226286791, −1.36824680440498460384564598194, 0,
1.36824680440498460384564598194, 2.46296394936642547160226286791, 3.49734312353215216887343182177, 4.28681391729088540393374354665, 5.36336736026187235479038459757, 5.94264930407483059532376201400, 6.40387843928083411347729073969, 7.40961699026055564212526317651, 8.018755445011978815149977634888
