| L(s) = 1 | + 2-s + 4-s − 3·5-s − 2·7-s + 8-s − 3·10-s − 5·11-s − 7·13-s − 2·14-s + 16-s + 17-s + 7·19-s − 3·20-s − 5·22-s − 4·23-s + 4·25-s − 7·26-s − 2·28-s + 8·29-s − 31-s + 32-s + 34-s + 6·35-s − 6·37-s + 7·38-s − 3·40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.755·7-s + 0.353·8-s − 0.948·10-s − 1.50·11-s − 1.94·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 1.60·19-s − 0.670·20-s − 1.06·22-s − 0.834·23-s + 4/5·25-s − 1.37·26-s − 0.377·28-s + 1.48·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 1.01·35-s − 0.986·37-s + 1.13·38-s − 0.474·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| 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
−10.27687222773840834971210474196, −9.784601496694799073053795901658, −8.173685798257603138181014683880, −7.56447641405875879501291182791, −6.85816503906409217391202980861, −5.37639162318012550383848847990, −4.72776639720081804092969473957, −3.43583600078258724736543379400, −2.66483243126869907657421826942, 0,
2.66483243126869907657421826942, 3.43583600078258724736543379400, 4.72776639720081804092969473957, 5.37639162318012550383848847990, 6.85816503906409217391202980861, 7.56447641405875879501291182791, 8.173685798257603138181014683880, 9.784601496694799073053795901658, 10.27687222773840834971210474196
