| L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 10-s + 4·11-s + 2·13-s − 14-s − 16-s − 2·17-s + 20-s + 4·22-s + 25-s + 2·26-s + 28-s − 6·29-s − 8·31-s + 5·32-s − 2·34-s + 35-s − 10·37-s + 3·40-s + 2·41-s − 4·43-s − 4·44-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.169·35-s − 1.64·37-s + 0.474·40-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.33990690844528, −13.15195019860652, −12.50905132502744, −12.17400315723191, −11.73803975047171, −11.16213583768625, −10.82259849226443, −10.10912986317711, −9.540645226534336, −9.084932043997369, −8.772836564300704, −8.377108767723588, −7.554234893883151, −7.025080256629428, −6.663412861897484, −5.991518111487427, −5.567249751086226, −5.114688352199313, −4.253257028933419, −4.063711392103728, −3.493414171011658, −3.191784335579817, −2.211909275815286, −1.589158080316110, −0.7192511120740970, 0,
0.7192511120740970, 1.589158080316110, 2.211909275815286, 3.191784335579817, 3.493414171011658, 4.063711392103728, 4.253257028933419, 5.114688352199313, 5.567249751086226, 5.991518111487427, 6.663412861897484, 7.025080256629428, 7.554234893883151, 8.377108767723588, 8.772836564300704, 9.084932043997369, 9.540645226534336, 10.10912986317711, 10.82259849226443, 11.16213583768625, 11.73803975047171, 12.17400315723191, 12.50905132502744, 13.15195019860652, 13.33990690844528
