| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s − 19-s − 21-s + 8·23-s + 25-s − 27-s + 6·29-s + 4·33-s − 35-s − 10·37-s + 2·39-s − 2·41-s + 8·43-s − 45-s + 49-s + 2·51-s + 2·53-s + 4·55-s + 57-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 1.21·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.539·55-s + 0.132·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.43779338015317, −14.94871283549706, −14.27001859960360, −13.73036154674520, −13.05066344434713, −12.70144713561435, −12.12278735642156, −11.62953298339934, −10.96627919627667, −10.62880724215735, −10.19411764630032, −9.423269074325683, −8.683375647221192, −8.369783626166977, −7.508835869537258, −7.173888917466723, −6.629731937596188, −5.748681183441785, −5.223857051166670, −4.711078980832804, −4.248801903740052, −3.190865417799002, −2.698432872965347, −1.827767608736262, −0.8478002969863034, 0,
0.8478002969863034, 1.827767608736262, 2.698432872965347, 3.190865417799002, 4.248801903740052, 4.711078980832804, 5.223857051166670, 5.748681183441785, 6.629731937596188, 7.173888917466723, 7.508835869537258, 8.369783626166977, 8.683375647221192, 9.423269074325683, 10.19411764630032, 10.62880724215735, 10.96627919627667, 11.62953298339934, 12.12278735642156, 12.70144713561435, 13.05066344434713, 13.73036154674520, 14.27001859960360, 14.94871283549706, 15.43779338015317
