| L(s) = 1 | − 2·2-s + 2·4-s + 11-s − 3·13-s − 4·16-s + 19-s − 2·22-s + 2·23-s + 6·26-s + 8·29-s − 5·31-s + 8·32-s + 3·37-s − 2·38-s − 8·41-s + 11·43-s + 2·44-s − 4·46-s − 4·47-s − 6·52-s + 6·53-s − 16·58-s + 6·59-s − 6·61-s + 10·62-s − 8·64-s − 13·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.301·11-s − 0.832·13-s − 16-s + 0.229·19-s − 0.426·22-s + 0.417·23-s + 1.17·26-s + 1.48·29-s − 0.898·31-s + 1.41·32-s + 0.493·37-s − 0.324·38-s − 1.24·41-s + 1.67·43-s + 0.301·44-s − 0.589·46-s − 0.583·47-s − 0.832·52-s + 0.824·53-s − 2.10·58-s + 0.781·59-s − 0.768·61-s + 1.27·62-s − 64-s − 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.76388827867476, −13.34318141426531, −12.77313513251621, −12.15024576775584, −11.75500341672442, −11.31775027033270, −10.64857230242041, −10.26939375308967, −9.953598302231016, −9.305192907886635, −8.896034805129284, −8.606724732087883, −7.879801987995405, −7.432116208937228, −7.119375536247939, −6.508470872534720, −5.920629817181529, −5.241060091321013, −4.563275527292437, −4.249857163919452, −3.245508984593413, −2.739842627551642, −2.046096799496780, −1.404590227565743, −0.7636746506542952, 0,
0.7636746506542952, 1.404590227565743, 2.046096799496780, 2.739842627551642, 3.245508984593413, 4.249857163919452, 4.563275527292437, 5.241060091321013, 5.920629817181529, 6.508470872534720, 7.119375536247939, 7.432116208937228, 7.879801987995405, 8.606724732087883, 8.896034805129284, 9.305192907886635, 9.953598302231016, 10.26939375308967, 10.64857230242041, 11.31775027033270, 11.75500341672442, 12.15024576775584, 12.77313513251621, 13.34318141426531, 13.76388827867476
