| L(s) = 1 | − 2·5-s − 4·11-s + 3·13-s + 2·17-s − 5·19-s + 23-s − 25-s − 2·29-s − 7·31-s − 11·37-s − 6·41-s + 9·43-s − 12·47-s + 8·53-s + 8·55-s − 6·59-s + 10·61-s − 6·65-s + 3·67-s − 8·71-s − 73-s − 17·79-s + 14·83-s − 4·85-s − 6·89-s + 10·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 0.832·13-s + 0.485·17-s − 1.14·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.25·31-s − 1.80·37-s − 0.937·41-s + 1.37·43-s − 1.75·47-s + 1.09·53-s + 1.07·55-s − 0.781·59-s + 1.28·61-s − 0.744·65-s + 0.366·67-s − 0.949·71-s − 0.117·73-s − 1.91·79-s + 1.53·83-s − 0.433·85-s − 0.635·89-s + 1.02·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.50755302524129, −13.27710868223494, −12.81674030240826, −12.30195467946847, −11.90606461228003, −11.18415149008788, −11.04418319770395, −10.38071251804651, −10.16169492094810, −9.379005063559908, −8.767590767660591, −8.417279764405471, −8.024045450666017, −7.412316352850737, −7.096973226546830, −6.440938075349409, −5.786440757845978, −5.374893648025048, −4.834704261404731, −4.161593930397432, −3.625435120125512, −3.320542739109490, −2.488275218269712, −1.895979287952339, −1.189590652532774, 0, 0,
1.189590652532774, 1.895979287952339, 2.488275218269712, 3.320542739109490, 3.625435120125512, 4.161593930397432, 4.834704261404731, 5.374893648025048, 5.786440757845978, 6.440938075349409, 7.096973226546830, 7.412316352850737, 8.024045450666017, 8.417279764405471, 8.767590767660591, 9.379005063559908, 10.16169492094810, 10.38071251804651, 11.04418319770395, 11.18415149008788, 11.90606461228003, 12.30195467946847, 12.81674030240826, 13.27710868223494, 13.50755302524129
