| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 2·13-s + 2·14-s + 16-s − 6·17-s − 20-s + 8·23-s + 25-s + 2·26-s + 2·28-s + 6·29-s + 4·31-s + 32-s − 6·34-s − 2·35-s − 10·37-s − 40-s − 12·41-s − 4·43-s + 8·46-s − 8·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1.64·37-s − 0.158·40-s − 1.87·41-s − 0.609·43-s + 1.17·46-s − 1.16·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 263 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.49979606405022, −15.29990475351233, −14.65506635746652, −14.01230986755599, −13.60443312078274, −13.07424696242198, −12.54117626855405, −11.75128087386406, −11.55764499719659, −10.83527427105495, −10.58389931719114, −9.734105754650888, −8.798075101032388, −8.550225751514733, −7.946332212530579, −7.095272776418525, −6.681862832825706, −6.185271262238697, −5.085220439636469, −4.834325156269179, −4.340864187517149, −3.293830406234435, −3.029635244432001, −1.876494438048476, −1.328047920349183, 0,
1.328047920349183, 1.876494438048476, 3.029635244432001, 3.293830406234435, 4.340864187517149, 4.834325156269179, 5.085220439636469, 6.185271262238697, 6.681862832825706, 7.095272776418525, 7.946332212530579, 8.550225751514733, 8.798075101032388, 9.734105754650888, 10.58389931719114, 10.83527427105495, 11.55764499719659, 11.75128087386406, 12.54117626855405, 13.07424696242198, 13.60443312078274, 14.01230986755599, 14.65506635746652, 15.29990475351233, 15.49979606405022
