| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s + 13-s − 14-s + 16-s + 17-s + 3·18-s − 4·23-s − 26-s + 28-s − 5·29-s − 8·31-s − 32-s − 34-s − 3·36-s + 5·37-s − 5·41-s − 4·43-s + 4·46-s + 4·47-s + 49-s + 52-s + 9·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 0.834·23-s − 0.196·26-s + 0.188·28-s − 0.928·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s − 1/2·36-s + 0.821·37-s − 0.780·41-s − 0.609·43-s + 0.589·46-s + 0.583·47-s + 1/7·49-s + 0.138·52-s + 1.23·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
| 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
−15.00539374032296, −14.53042939762891, −14.05958216177477, −13.40860885190494, −12.96405256159779, −12.05142995333373, −11.88431825182034, −11.27822094376654, −10.69748950420918, −10.43353907187448, −9.526880578301678, −9.195618852362675, −8.633405166441504, −8.102238763621674, −7.601344359615917, −7.125958456997685, −6.197023530134299, −5.918123374133395, −5.269391031368587, −4.585219560863764, −3.610692741978451, −3.303592493789492, −2.233556766495844, −1.895426048125484, −0.8505161946806018, 0,
0.8505161946806018, 1.895426048125484, 2.233556766495844, 3.303592493789492, 3.610692741978451, 4.585219560863764, 5.269391031368587, 5.918123374133395, 6.197023530134299, 7.125958456997685, 7.601344359615917, 8.102238763621674, 8.633405166441504, 9.195618852362675, 9.526880578301678, 10.43353907187448, 10.69748950420918, 11.27822094376654, 11.88431825182034, 12.05142995333373, 12.96405256159779, 13.40860885190494, 14.05958216177477, 14.53042939762891, 15.00539374032296
