| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 4·7-s − 3·8-s + 9-s + 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s − 15-s − 16-s + 18-s + 6·19-s − 20-s − 4·21-s − 2·22-s + 3·24-s + 25-s + 2·26-s − 27-s − 4·28-s + 10·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.872·21-s − 0.426·22-s + 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.85·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.862334036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862334036\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−10.58910239343321954289350590528, −9.906595972704294682133625947834, −8.666178337426129024514963047000, −8.076056699728009010373918938735, −6.78753511563713634027620301812, −5.64926104374539828357492883103, −5.07780048273782775953918115396, −4.37531349869513784629265744240, −2.93359921888063408929918150795, −1.23147154777002850576678018749,
1.23147154777002850576678018749, 2.93359921888063408929918150795, 4.37531349869513784629265744240, 5.07780048273782775953918115396, 5.64926104374539828357492883103, 6.78753511563713634027620301812, 8.076056699728009010373918938735, 8.666178337426129024514963047000, 9.906595972704294682133625947834, 10.58910239343321954289350590528
