| L(s) = 1 | − 2·5-s + 2·7-s − 2·13-s + 2·17-s − 25-s − 2·29-s + 4·31-s − 4·35-s + 6·37-s − 6·41-s + 4·43-s − 4·47-s − 3·49-s − 2·53-s − 6·59-s + 6·61-s + 4·65-s − 8·67-s + 6·71-s − 6·73-s − 12·79-s + 12·83-s − 4·85-s − 14·89-s − 4·91-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.554·13-s + 0.485·17-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 0.781·59-s + 0.768·61-s + 0.496·65-s − 0.977·67-s + 0.712·71-s − 0.702·73-s − 1.35·79-s + 1.31·83-s − 0.433·85-s − 1.48·89-s − 0.419·91-s − 1.42·97-s + 0.0995·101-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
| \(L(1)\) |
\(\approx\) |
\(1.156065447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156065447\) |
| \(L(\frac{3}{2})\) |
|
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.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−12.80650643769886, −12.43680494740957, −11.86288585960659, −11.63527122113126, −11.16199831803005, −10.74058596008642, −10.10082822512687, −9.694764632428568, −9.252846848952330, −8.416078414250868, −8.297793718770154, −7.700650905314721, −7.409296335519462, −6.856112220607039, −6.204027565782462, −5.710620440445741, −5.054885428017778, −4.683658784254307, −4.149798504202013, −3.674159356004947, −2.990219899811183, −2.513799631070484, −1.698984786670997, −1.195734860120032, −0.3099813117585431,
0.3099813117585431, 1.195734860120032, 1.698984786670997, 2.513799631070484, 2.990219899811183, 3.674159356004947, 4.149798504202013, 4.683658784254307, 5.054885428017778, 5.710620440445741, 6.204027565782462, 6.856112220607039, 7.409296335519462, 7.700650905314721, 8.297793718770154, 8.416078414250868, 9.252846848952330, 9.694764632428568, 10.10082822512687, 10.74058596008642, 11.16199831803005, 11.63527122113126, 11.86288585960659, 12.43680494740957, 12.80650643769886
