| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·13-s − 14-s + 16-s − 7·17-s − 19-s − 5·23-s − 3·26-s + 28-s + 5·29-s + 10·31-s − 32-s + 7·34-s − 2·37-s + 38-s − 2·41-s − 6·43-s + 5·46-s − 6·49-s + 3·52-s + 9·53-s − 56-s − 5·58-s + 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.229·19-s − 1.04·23-s − 0.588·26-s + 0.188·28-s + 0.928·29-s + 1.79·31-s − 0.176·32-s + 1.20·34-s − 0.328·37-s + 0.162·38-s − 0.312·41-s − 0.914·43-s + 0.737·46-s − 6/7·49-s + 0.416·52-s + 1.23·53-s − 0.133·56-s − 0.656·58-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.318574687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318574687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.988216069984535719967507033911, −7.09303200097807127134346250469, −6.42866260129575740230423315884, −6.04102999770281185980626761233, −4.88565845827039955594671283195, −4.32947316292901316236828207828, −3.39015869150746428217890802420, −2.41418690914551868178563433250, −1.72274092416142508360726216866, −0.62459580318349033130449223896,
0.62459580318349033130449223896, 1.72274092416142508360726216866, 2.41418690914551868178563433250, 3.39015869150746428217890802420, 4.32947316292901316236828207828, 4.88565845827039955594671283195, 6.04102999770281185980626761233, 6.42866260129575740230423315884, 7.09303200097807127134346250469, 7.988216069984535719967507033911
