| L(s) = 1 | − 2-s − 3·3-s + 4-s + 5-s + 3·6-s + 2·7-s − 8-s + 6·9-s − 10-s + 5·11-s − 3·12-s − 2·13-s − 2·14-s − 3·15-s + 16-s − 2·17-s − 6·18-s − 19-s + 20-s − 6·21-s − 5·22-s + 23-s + 3·24-s − 4·25-s + 2·26-s − 9·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s − 0.316·10-s + 1.50·11-s − 0.866·12-s − 0.554·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s + 0.223·20-s − 1.30·21-s − 1.06·22-s + 0.208·23-s + 0.612·24-s − 4/5·25-s + 0.392·26-s − 1.73·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7640162400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7640162400\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
| 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
−10.14669126342778803736489235158, −9.577776628192651405702334914578, −8.559044458927928476524472169132, −7.39108856539308841353334545016, −6.58836588993074077310145507278, −5.99438793012257440365969906560, −5.00553654034151835981792538210, −4.10727050063470136833639516291, −1.99931662128424111808219573511, −0.886659922354858315472516073397,
0.886659922354858315472516073397, 1.99931662128424111808219573511, 4.10727050063470136833639516291, 5.00553654034151835981792538210, 5.99438793012257440365969906560, 6.58836588993074077310145507278, 7.39108856539308841353334545016, 8.559044458927928476524472169132, 9.577776628192651405702334914578, 10.14669126342778803736489235158
