| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 6·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 8·29-s + 8·31-s − 4·35-s + 2·37-s + 2·41-s + 4·43-s − 3·49-s − 2·53-s − 8·55-s − 4·59-s + 6·61-s − 12·65-s − 8·67-s + 6·73-s + 8·77-s − 4·79-s − 12·83-s − 4·85-s + 2·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.676·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s − 0.274·53-s − 1.07·55-s − 0.520·59-s + 0.768·61-s − 1.48·65-s − 0.977·67-s + 0.702·73-s + 0.911·77-s − 0.450·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.196038587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.196038587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−12.85626043912333, −12.68895856863219, −12.05859380060758, −11.60287054379761, −11.02367365828077, −10.42343872406221, −10.01911145346969, −9.814220066216503, −9.329105377486091, −8.804655000199972, −8.239208077579074, −7.559884107817087, −7.315914840289748, −6.696706614068140, −6.248547928585865, −5.726336930009956, −5.095531171173751, −4.823214385648330, −4.369086337083835, −3.275173788184790, −2.837415353834947, −2.608020047829361, −1.976825262972015, −1.034222997628412, −0.4495998438705779,
0.4495998438705779, 1.034222997628412, 1.976825262972015, 2.608020047829361, 2.837415353834947, 3.275173788184790, 4.369086337083835, 4.823214385648330, 5.095531171173751, 5.726336930009956, 6.248547928585865, 6.696706614068140, 7.315914840289748, 7.559884107817087, 8.239208077579074, 8.804655000199972, 9.329105377486091, 9.814220066216503, 10.01911145346969, 10.42343872406221, 11.02367365828077, 11.60287054379761, 12.05859380060758, 12.68895856863219, 12.85626043912333
