| L(s) = 1 | + 2-s + 4-s + 4·5-s + 4·7-s + 8-s + 4·10-s + 2·11-s − 2·13-s + 4·14-s + 16-s − 2·17-s + 2·19-s + 4·20-s + 2·22-s + 11·25-s − 2·26-s + 4·28-s − 2·29-s + 32-s − 2·34-s + 16·35-s + 4·37-s + 2·38-s + 4·40-s − 6·41-s − 10·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.51·7-s + 0.353·8-s + 1.26·10-s + 0.603·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.894·20-s + 0.426·22-s + 11/5·25-s − 0.392·26-s + 0.755·28-s − 0.371·29-s + 0.176·32-s − 0.342·34-s + 2.70·35-s + 0.657·37-s + 0.324·38-s + 0.632·40-s − 0.937·41-s − 1.52·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.365080419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.365080419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62735421295774645335638863042, −6.69265944578033613099294080985, −6.35362788354985461499174366441, −5.35356885744180803495724283233, −5.12924034341750180180238442817, −4.47212152404542749707456940527, −3.42573528437023625765419608105, −2.38695715760664154900264131006, −1.87977103445283065955695773340, −1.21768411761699883032703079663,
1.21768411761699883032703079663, 1.87977103445283065955695773340, 2.38695715760664154900264131006, 3.42573528437023625765419608105, 4.47212152404542749707456940527, 5.12924034341750180180238442817, 5.35356885744180803495724283233, 6.35362788354985461499174366441, 6.69265944578033613099294080985, 7.62735421295774645335638863042
