| L(s) = 1 | − 2·5-s + 2·7-s − 4·11-s − 6·13-s + 2·17-s − 4·19-s + 8·23-s − 25-s − 8·29-s − 8·31-s − 4·35-s + 2·37-s − 2·41-s − 4·43-s − 3·49-s + 2·53-s + 8·55-s − 4·59-s + 6·61-s + 12·65-s + 8·67-s + 6·73-s − 8·77-s + 4·79-s − 12·83-s − 4·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 3/7·49-s + 0.274·53-s + 1.07·55-s − 0.520·59-s + 0.768·61-s + 1.48·65-s + 0.977·67-s + 0.702·73-s − 0.911·77-s + 0.450·79-s − 1.31·83-s − 0.433·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.02479953026432, −12.74638370942448, −12.25993027001375, −11.75841625808853, −11.20855628846736, −10.97500333643745, −10.51749418796165, −9.886978425306611, −9.433428009646575, −8.973374742098520, −8.236942699610586, −7.921839639484288, −7.634302048319009, −6.954464156103042, −6.878919876740177, −5.711860695241581, −5.303515853137866, −5.055240738166496, −4.429048895312298, −3.929897893486231, −3.265314864888964, −2.733114016186371, −2.106810439732456, −1.624352319428695, −0.5440222370024102, 0,
0.5440222370024102, 1.624352319428695, 2.106810439732456, 2.733114016186371, 3.265314864888964, 3.929897893486231, 4.429048895312298, 5.055240738166496, 5.303515853137866, 5.711860695241581, 6.878919876740177, 6.954464156103042, 7.634302048319009, 7.921839639484288, 8.236942699610586, 8.973374742098520, 9.433428009646575, 9.886978425306611, 10.51749418796165, 10.97500333643745, 11.20855628846736, 11.75841625808853, 12.25993027001375, 12.74638370942448, 13.02479953026432
