| L(s) = 1 | + 5-s + 11-s − 13-s + 2·17-s + 2·19-s + 2·23-s − 4·25-s + 7·29-s + 3·31-s + 2·37-s − 8·41-s − 8·43-s + 3·53-s + 55-s + 9·59-s − 8·61-s − 65-s − 2·67-s + 16·71-s − 6·73-s − 5·79-s − 11·83-s + 2·85-s + 6·89-s + 2·95-s − 97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.29·29-s + 0.538·31-s + 0.328·37-s − 1.24·41-s − 1.21·43-s + 0.412·53-s + 0.134·55-s + 1.17·59-s − 1.02·61-s − 0.124·65-s − 0.244·67-s + 1.89·71-s − 0.702·73-s − 0.562·79-s − 1.20·83-s + 0.216·85-s + 0.635·89-s + 0.205·95-s − 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.728588856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.728588856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−12.41218318773912, −12.04974300909448, −11.62418891701120, −11.30339006628945, −10.55216314041067, −10.17896663985688, −9.829984271859122, −9.466982083179135, −8.850194164869811, −8.364735466529118, −8.040169616604108, −7.425271052324117, −6.885920665942516, −6.545224505889484, −6.031150627987380, −5.416223674979318, −5.111921869452697, −4.527461126540279, −3.974373778604811, −3.370206676180833, −2.886295528440230, −2.357426614939151, −1.632327623399674, −1.191617576809757, −0.4388902393529711,
0.4388902393529711, 1.191617576809757, 1.632327623399674, 2.357426614939151, 2.886295528440230, 3.370206676180833, 3.974373778604811, 4.527461126540279, 5.111921869452697, 5.416223674979318, 6.031150627987380, 6.545224505889484, 6.885920665942516, 7.425271052324117, 8.040169616604108, 8.364735466529118, 8.850194164869811, 9.466982083179135, 9.829984271859122, 10.17896663985688, 10.55216314041067, 11.30339006628945, 11.62418891701120, 12.04974300909448, 12.41218318773912
