| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 2·17-s + 4·21-s − 25-s − 27-s + 2·29-s − 4·33-s − 8·35-s − 10·37-s − 2·39-s − 6·41-s − 8·43-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s − 6·53-s + 8·55-s − 4·59-s + 14·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s + 0.485·17-s + 0.872·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.35·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 1.07·55-s − 0.520·59-s + 1.79·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 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.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 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 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.73843795258438, −13.55923806684211, −13.09114625476947, −12.40497322223474, −12.05609664881432, −11.77524305253062, −10.86624138758283, −10.54628066945354, −9.976473784258070, −9.534957489528887, −9.307705603035321, −8.604646150908013, −8.108564897164839, −7.103679911289232, −6.771230176588057, −6.450794994505206, −5.878479997766452, −5.517315268078909, −4.846261179160598, −4.027869531518425, −3.510436935239108, −3.141133847756612, −2.161007708606574, −1.575682124678353, −0.8766774043828788, 0,
0.8766774043828788, 1.575682124678353, 2.161007708606574, 3.141133847756612, 3.510436935239108, 4.027869531518425, 4.846261179160598, 5.517315268078909, 5.878479997766452, 6.450794994505206, 6.771230176588057, 7.103679911289232, 8.108564897164839, 8.604646150908013, 9.307705603035321, 9.534957489528887, 9.976473784258070, 10.54628066945354, 10.86624138758283, 11.77524305253062, 12.05609664881432, 12.40497322223474, 13.09114625476947, 13.55923806684211, 13.73843795258438
