| L(s) = 1 | − 3-s − 2·4-s − 5-s + 2·7-s + 9-s + 2·11-s + 2·12-s + 13-s + 15-s + 4·16-s − 2·17-s − 6·19-s + 2·20-s − 2·21-s − 7·23-s + 25-s − 27-s − 4·28-s − 6·29-s − 6·31-s − 2·33-s − 2·35-s − 2·36-s − 4·37-s − 39-s + 7·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s − 0.485·17-s − 1.37·19-s + 0.447·20-s − 0.436·21-s − 1.45·23-s + 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 1.07·31-s − 0.348·33-s − 0.338·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s + 1.06·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 47 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 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
−10.04996275179432088924562520439, −9.044692233787574070771377846126, −8.374524162388426432032156238072, −7.50925969593788658157536204568, −6.31864401933706827967488722397, −5.40113086684499259398717159913, −4.35510905222641967521375323639, −3.83995520912045013003442751869, −1.74571211505373944726297969684, 0,
1.74571211505373944726297969684, 3.83995520912045013003442751869, 4.35510905222641967521375323639, 5.40113086684499259398717159913, 6.31864401933706827967488722397, 7.50925969593788658157536204568, 8.374524162388426432032156238072, 9.044692233787574070771377846126, 10.04996275179432088924562520439
