| L(s) = 1 | − 2·4-s − 7-s + 2·13-s + 4·16-s + 8·19-s + 2·28-s − 7·31-s − 37-s − 13·43-s − 6·49-s − 4·52-s + 14·61-s − 8·64-s + 11·67-s − 10·73-s − 16·76-s − 4·79-s − 2·91-s − 19·97-s + 101-s + 103-s + 107-s + 109-s − 4·112-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 0.554·13-s + 16-s + 1.83·19-s + 0.377·28-s − 1.25·31-s − 0.164·37-s − 1.98·43-s − 6/7·49-s − 0.554·52-s + 1.79·61-s − 64-s + 1.34·67-s − 1.17·73-s − 1.83·76-s − 0.450·79-s − 0.209·91-s − 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.377·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393632001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393632001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
| 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
−13.86647832941044, −13.48282369878440, −13.03387751071529, −12.67363989636794, −11.97423590805660, −11.53368034020537, −11.05056345235184, −10.31727620569321, −9.773472127379377, −9.606546311106265, −8.949651681567522, −8.418200660425572, −8.045480675188490, −7.285684212207632, −6.917006858905959, −6.142865949282590, −5.504108434003828, −5.217399636623294, −4.589392819105511, −3.781745388095299, −3.454027587955755, −2.934793636630393, −1.856046152548841, −1.197590059050107, −0.4288940790535789,
0.4288940790535789, 1.197590059050107, 1.856046152548841, 2.934793636630393, 3.454027587955755, 3.781745388095299, 4.589392819105511, 5.217399636623294, 5.504108434003828, 6.142865949282590, 6.917006858905959, 7.285684212207632, 8.045480675188490, 8.418200660425572, 8.949651681567522, 9.606546311106265, 9.773472127379377, 10.31727620569321, 11.05056345235184, 11.53368034020537, 11.97423590805660, 12.67363989636794, 13.03387751071529, 13.48282369878440, 13.86647832941044
