| L(s) = 1 | − 2·3-s − 5-s + 5·7-s + 9-s − 11-s − 13-s + 2·15-s + 6·17-s − 4·19-s − 10·21-s − 3·23-s − 4·25-s + 4·27-s − 29-s − 6·31-s + 2·33-s − 5·35-s − 2·37-s + 2·39-s + 3·41-s + 43-s − 45-s + 8·47-s + 18·49-s − 12·51-s − 2·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 2.18·21-s − 0.625·23-s − 4/5·25-s + 0.769·27-s − 0.185·29-s − 1.07·31-s + 0.348·33-s − 0.845·35-s − 0.328·37-s + 0.320·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 18/7·49-s − 1.68·51-s − 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242768581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242768581\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.64352639790481947594057724651, −7.24156362857415409315542499382, −6.12030848458202332465561799855, −5.52119719417681011443147104598, −5.14980421685056277826976405959, −4.35556416974134246937863103232, −3.74665656181255240413935221249, −2.41153986547938051973550694739, −1.59212135468018264296057310488, −0.59060058210832542103735297581,
0.59060058210832542103735297581, 1.59212135468018264296057310488, 2.41153986547938051973550694739, 3.74665656181255240413935221249, 4.35556416974134246937863103232, 5.14980421685056277826976405959, 5.52119719417681011443147104598, 6.12030848458202332465561799855, 7.24156362857415409315542499382, 7.64352639790481947594057724651
