| L(s) = 1 | − 5-s − 4·7-s − 2·11-s − 6·13-s − 2·17-s + 5·19-s + 4·23-s − 4·25-s − 10·29-s − 5·31-s + 4·35-s − 10·37-s + 2·41-s − 4·43-s + 12·47-s + 9·49-s − 11·53-s + 2·55-s − 5·59-s + 6·65-s + 5·67-s + 12·71-s − 3·73-s + 8·77-s − 17·79-s − 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.603·11-s − 1.66·13-s − 0.485·17-s + 1.14·19-s + 0.834·23-s − 4/5·25-s − 1.85·29-s − 0.898·31-s + 0.676·35-s − 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s − 1.51·53-s + 0.269·55-s − 0.650·59-s + 0.744·65-s + 0.610·67-s + 1.42·71-s − 0.351·73-s + 0.911·77-s − 1.91·79-s − 1.31·83-s + 0.216·85-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 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.97765749258683, −12.69402191656231, −12.42223456639379, −11.79189167291588, −11.36646086630755, −10.83890050714824, −10.30991556992682, −9.804131583208375, −9.421848205450007, −9.200287864922051, −8.520980565388041, −7.761128706695606, −7.403479664866190, −7.078418072815878, −6.700369029257342, −5.812814179967842, −5.480858865383889, −5.079345797381592, −4.317722982787283, −3.787978958462562, −3.215149341420899, −2.875275536468443, −2.206613948967947, −1.567386791297360, −0.4088115323298838, 0,
0.4088115323298838, 1.567386791297360, 2.206613948967947, 2.875275536468443, 3.215149341420899, 3.787978958462562, 4.317722982787283, 5.079345797381592, 5.480858865383889, 5.812814179967842, 6.700369029257342, 7.078418072815878, 7.403479664866190, 7.761128706695606, 8.520980565388041, 9.200287864922051, 9.421848205450007, 9.804131583208375, 10.30991556992682, 10.83890050714824, 11.36646086630755, 11.79189167291588, 12.42223456639379, 12.69402191656231, 12.97765749258683
