| L(s) = 1 | + 2·5-s + 4·7-s − 11-s + 2·13-s − 2·17-s − 4·19-s − 25-s − 6·29-s + 4·31-s + 8·35-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 9·49-s − 6·53-s − 2·55-s + 12·59-s + 10·61-s + 4·65-s − 4·67-s + 10·73-s − 4·77-s + 12·79-s + 12·83-s − 4·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.269·55-s + 1.56·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s + 1.17·73-s − 0.455·77-s + 1.35·79-s + 1.31·83-s − 0.433·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.964976277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.964976277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.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.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.053895597391142305879666941000, −7.46394001056611669577011074743, −6.50361961579357696893264119982, −5.85179377291294284579229797341, −5.23348766404177363956197396972, −4.47675688904669232859519816445, −3.78120621079530929924808485609, −2.35453029973131530751341134799, −2.01415947096316150087329841902, −0.927267684575568496438184826433,
0.927267684575568496438184826433, 2.01415947096316150087329841902, 2.35453029973131530751341134799, 3.78120621079530929924808485609, 4.47675688904669232859519816445, 5.23348766404177363956197396972, 5.85179377291294284579229797341, 6.50361961579357696893264119982, 7.46394001056611669577011074743, 8.053895597391142305879666941000
