| L(s) = 1 | − 2·5-s − 7-s − 2·11-s − 2·13-s + 6·17-s + 6·19-s − 6·23-s − 25-s − 5·29-s − 10·31-s + 2·35-s + 3·37-s + 6·41-s − 2·43-s − 6·47-s − 6·49-s − 12·53-s + 4·55-s − 9·59-s + 6·61-s + 4·65-s − 10·67-s + 5·71-s + 7·73-s + 2·77-s − 12·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.37·19-s − 1.25·23-s − 1/5·25-s − 0.928·29-s − 1.79·31-s + 0.338·35-s + 0.493·37-s + 0.937·41-s − 0.304·43-s − 0.875·47-s − 6/7·49-s − 1.64·53-s + 0.539·55-s − 1.17·59-s + 0.768·61-s + 0.496·65-s − 1.22·67-s + 0.593·71-s + 0.819·73-s + 0.227·77-s − 1.35·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.99149578218166, −12.62626013872385, −12.33773042789810, −11.67795687725327, −11.44119626761240, −10.96656588691506, −10.28951376475742, −9.808292292329065, −9.569888828743106, −9.073019848260694, −8.262028210526697, −7.783767599776921, −7.555052285669143, −7.337729470157930, −6.439239077127756, −5.859749113679912, −5.531190686123683, −4.932991621627988, −4.421992937538026, −3.619358366980503, −3.415442343678830, −2.944814757526453, −2.036645759561799, −1.544389960427147, −0.5865149521973320, 0,
0.5865149521973320, 1.544389960427147, 2.036645759561799, 2.944814757526453, 3.415442343678830, 3.619358366980503, 4.421992937538026, 4.932991621627988, 5.531190686123683, 5.859749113679912, 6.439239077127756, 7.337729470157930, 7.555052285669143, 7.783767599776921, 8.262028210526697, 9.073019848260694, 9.569888828743106, 9.808292292329065, 10.28951376475742, 10.96656588691506, 11.44119626761240, 11.67795687725327, 12.33773042789810, 12.62626013872385, 12.99149578218166
