| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 3·8-s + 9-s + 2·11-s + 12-s − 13-s + 2·14-s − 16-s + 18-s + 4·19-s − 2·21-s + 2·22-s − 4·23-s + 3·24-s − 26-s − 27-s − 2·28-s + 2·31-s + 5·32-s − 2·33-s − 36-s + 10·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.426·22-s − 0.834·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.359·31-s + 0.883·32-s − 0.348·33-s − 1/6·36-s + 1.64·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 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.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−12.77870167736930, −12.64716442410441, −11.96628558084245, −11.72155240840875, −11.36406237284402, −10.82267252990079, −10.13971923599908, −9.833776016644863, −9.208283884938723, −9.004395194200732, −8.206050330573761, −7.825115588067577, −7.433857106399787, −6.657588594301511, −6.208259479812988, −5.808702255209925, −5.333355378784900, −4.686090784572669, −4.541390124017405, −3.952822005957474, −3.385387180770175, −2.797779413023306, −2.098603286180044, −1.343087374763661, −0.8154676443338672, 0,
0.8154676443338672, 1.343087374763661, 2.098603286180044, 2.797779413023306, 3.385387180770175, 3.952822005957474, 4.541390124017405, 4.686090784572669, 5.333355378784900, 5.808702255209925, 6.208259479812988, 6.657588594301511, 7.433857106399787, 7.825115588067577, 8.206050330573761, 9.004395194200732, 9.208283884938723, 9.833776016644863, 10.13971923599908, 10.82267252990079, 11.36406237284402, 11.72155240840875, 11.96628558084245, 12.64716442410441, 12.77870167736930
