| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s + 2·11-s − 14-s + 16-s + 17-s + 3·18-s − 6·19-s − 20-s − 2·22-s + 25-s + 28-s − 32-s − 34-s − 35-s − 3·36-s − 8·37-s + 6·38-s + 40-s + 6·41-s − 12·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s + 0.188·28-s − 0.176·32-s − 0.171·34-s − 0.169·35-s − 1/2·36-s − 1.31·37-s + 0.973·38-s + 0.158·40-s + 0.937·41-s − 1.82·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−9.144207297103801424929075933754, −8.526681954963473153095352870535, −7.973896066069189426034490500224, −6.92070298303289303701996257401, −6.18381592600271152011704913955, −5.13647791409863770800062770583, −3.98624329158796758204853588605, −2.90844442802267130691488838714, −1.65593440852082818572844070628, 0,
1.65593440852082818572844070628, 2.90844442802267130691488838714, 3.98624329158796758204853588605, 5.13647791409863770800062770583, 6.18381592600271152011704913955, 6.92070298303289303701996257401, 7.973896066069189426034490500224, 8.526681954963473153095352870535, 9.144207297103801424929075933754
