| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 4·19-s + 21-s − 8·23-s − 25-s − 27-s + 6·29-s − 4·33-s + 2·35-s − 2·37-s − 2·39-s − 10·41-s + 4·43-s − 2·45-s + 49-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s + 6·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 0.917·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 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 |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.34726302101980, −12.30898972613007, −11.80618939585461, −11.35420012742039, −11.06625380280405, −10.40298170804201, −10.05639533180482, −9.612809445039378, −9.027361671453255, −8.525571525247400, −8.096378418513055, −7.817402123508013, −6.936623355955192, −6.692943972216083, −6.303398583644163, −5.887389809618092, −5.177078978483692, −4.692963037542214, −3.980776742421871, −3.854573245352390, −3.458180687323126, −2.519992086226743, −1.954589092147756, −1.308583809288617, −0.6222756828674368, 0,
0.6222756828674368, 1.308583809288617, 1.954589092147756, 2.519992086226743, 3.458180687323126, 3.854573245352390, 3.980776742421871, 4.692963037542214, 5.177078978483692, 5.887389809618092, 6.303398583644163, 6.692943972216083, 6.936623355955192, 7.817402123508013, 8.096378418513055, 8.525571525247400, 9.027361671453255, 9.612809445039378, 10.05639533180482, 10.40298170804201, 11.06625380280405, 11.35420012742039, 11.80618939585461, 12.30898972613007, 12.34726302101980
