| L(s) = 1 | − 4·5-s + 2·7-s + 11-s − 2·13-s + 2·17-s + 6·19-s − 4·23-s + 11·25-s − 6·29-s − 4·31-s − 8·35-s − 6·37-s − 10·41-s − 6·43-s + 8·47-s − 3·49-s − 4·55-s − 4·59-s − 6·61-s + 8·65-s − 8·67-s − 2·73-s + 2·77-s + 10·79-s − 12·83-s − 8·85-s − 4·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 1.37·19-s − 0.834·23-s + 11/5·25-s − 1.11·29-s − 0.718·31-s − 1.35·35-s − 0.986·37-s − 1.56·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.539·55-s − 0.520·59-s − 0.768·61-s + 0.992·65-s − 0.977·67-s − 0.234·73-s + 0.227·77-s + 1.12·79-s − 1.31·83-s − 0.867·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.847930541996233473124247216175, −8.057124954523707004822208987593, −7.53251376909507826007962870225, −6.96045911731745643617967557305, −5.52630761902370115381086518039, −4.77930282088861382836755069859, −3.85624161784864461163790617342, −3.18953257582755724401307416707, −1.55587956177828296160422082958, 0,
1.55587956177828296160422082958, 3.18953257582755724401307416707, 3.85624161784864461163790617342, 4.77930282088861382836755069859, 5.52630761902370115381086518039, 6.96045911731745643617967557305, 7.53251376909507826007962870225, 8.057124954523707004822208987593, 8.847930541996233473124247216175
