| L(s) = 1 | + 2·5-s − 3·9-s − 2·13-s − 17-s − 4·19-s − 25-s + 6·29-s + 6·37-s + 6·41-s − 12·43-s − 6·45-s + 8·47-s + 2·53-s − 4·59-s + 2·61-s − 4·65-s + 12·67-s − 2·73-s + 8·79-s + 9·81-s − 12·83-s − 2·85-s − 10·89-s − 8·95-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 0.554·13-s − 0.242·17-s − 0.917·19-s − 1/5·25-s + 1.11·29-s + 0.986·37-s + 0.937·41-s − 1.82·43-s − 0.894·45-s + 1.16·47-s + 0.274·53-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 1.46·67-s − 0.234·73-s + 0.900·79-s + 81-s − 1.31·83-s − 0.216·85-s − 1.05·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.47960623133162, −14.24783233937217, −13.78930430085228, −13.09539730668684, −12.86634975428889, −12.02895837382478, −11.74762416272974, −11.04711755384481, −10.58581307042651, −10.04533067789943, −9.523727451061522, −9.065473215252562, −8.385385680436822, −8.104891451370087, −7.282960430817521, −6.620783551788983, −6.186575496056792, −5.678291742064313, −5.100888857768071, −4.495277622288897, −3.814896002881014, −2.925937531208070, −2.451414119382233, −1.931159500041836, −0.9303038128017697, 0,
0.9303038128017697, 1.931159500041836, 2.451414119382233, 2.925937531208070, 3.814896002881014, 4.495277622288897, 5.100888857768071, 5.678291742064313, 6.186575496056792, 6.620783551788983, 7.282960430817521, 8.104891451370087, 8.385385680436822, 9.065473215252562, 9.523727451061522, 10.04533067789943, 10.58581307042651, 11.04711755384481, 11.74762416272974, 12.02895837382478, 12.86634975428889, 13.09539730668684, 13.78930430085228, 14.24783233937217, 14.47960623133162
