| L(s) = 1 | − 2·3-s − 2·5-s + 9-s − 2·11-s − 2·13-s + 4·15-s − 2·17-s + 4·19-s − 8·23-s − 25-s + 4·27-s + 2·29-s + 4·33-s − 6·37-s + 4·39-s + 6·41-s − 6·43-s − 2·45-s − 8·47-s − 7·49-s + 4·51-s + 8·53-s + 4·55-s − 8·57-s − 4·59-s − 4·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.914·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.560·51-s + 1.09·53-s + 0.539·55-s − 1.05·57-s − 0.520·59-s − 0.512·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 97 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.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
−16.04392327465985, −15.67743522342928, −14.84159481456620, −14.43929436125819, −13.64767446952736, −13.22882842797644, −12.38489321338914, −11.95689564786298, −11.77339289893362, −11.13417866216599, −10.56016669296268, −10.07221576995333, −9.489614301736683, −8.642297759090238, −7.982497826854907, −7.660509606533888, −6.882192113908649, −6.353026459537401, −5.661968799539620, −5.129194880613507, −4.576380192179490, −3.886844972576049, −3.137482196070234, −2.306789062159678, −1.300421328017104, 0, 0,
1.300421328017104, 2.306789062159678, 3.137482196070234, 3.886844972576049, 4.576380192179490, 5.129194880613507, 5.661968799539620, 6.353026459537401, 6.882192113908649, 7.660509606533888, 7.982497826854907, 8.642297759090238, 9.489614301736683, 10.07221576995333, 10.56016669296268, 11.13417866216599, 11.77339289893362, 11.95689564786298, 12.38489321338914, 13.22882842797644, 13.64767446952736, 14.43929436125819, 14.84159481456620, 15.67743522342928, 16.04392327465985
