| L(s) = 1 | − 3-s − 2·5-s + 9-s + 2·11-s − 13-s + 2·15-s − 4·17-s − 4·19-s − 4·23-s − 25-s − 27-s − 6·29-s − 2·33-s − 2·37-s + 39-s − 8·41-s + 6·43-s − 2·45-s + 4·51-s − 10·53-s − 4·55-s + 4·57-s + 12·59-s − 2·61-s + 2·65-s + 2·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.516·15-s − 0.970·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.348·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s + 0.560·51-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.244·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.21839033324437, −12.69753601484611, −12.43247069749627, −11.86299179512137, −11.45912467990029, −11.15764894723800, −10.72078214002931, −10.07466543146920, −9.731746251102580, −9.043392110281915, −8.728494826605627, −8.044392025405605, −7.784221426126345, −7.078453285213711, −6.727228171443445, −6.286288830122405, −5.690669384124213, −5.176534157091535, −4.505642513031544, −4.130944647754573, −3.761603982471756, −3.134369388603047, −2.172963506124051, −1.911242811860492, −1.022126836311001, 0, 0,
1.022126836311001, 1.911242811860492, 2.172963506124051, 3.134369388603047, 3.761603982471756, 4.130944647754573, 4.505642513031544, 5.176534157091535, 5.690669384124213, 6.286288830122405, 6.727228171443445, 7.078453285213711, 7.784221426126345, 8.044392025405605, 8.728494826605627, 9.043392110281915, 9.731746251102580, 10.07466543146920, 10.72078214002931, 11.15764894723800, 11.45912467990029, 11.86299179512137, 12.43247069749627, 12.69753601484611, 13.21839033324437
