| L(s) = 1 | + 3·3-s − 5-s − 3·7-s + 6·9-s − 5·11-s − 3·15-s + 3·17-s + 5·19-s − 9·21-s + 3·23-s + 25-s + 9·27-s − 5·29-s − 8·31-s − 15·33-s + 3·35-s − 9·37-s + 3·41-s + 43-s − 6·45-s + 12·47-s + 2·49-s + 9·51-s + 2·53-s + 5·55-s + 15·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s − 1.50·11-s − 0.774·15-s + 0.727·17-s + 1.14·19-s − 1.96·21-s + 0.625·23-s + 1/5·25-s + 1.73·27-s − 0.928·29-s − 1.43·31-s − 2.61·33-s + 0.507·35-s − 1.47·37-s + 0.468·41-s + 0.152·43-s − 0.894·45-s + 1.75·47-s + 2/7·49-s + 1.26·51-s + 0.274·53-s + 0.674·55-s + 1.98·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.13003243917240, −15.85077838181311, −15.39051550170522, −14.82260764436979, −14.20996103225701, −13.69716459348745, −13.13427710762386, −12.74427944718670, −12.28928107270716, −11.30280072505563, −10.51792011780137, −10.08605555399279, −9.384078349657869, −9.070266473158654, −8.343068118004063, −7.635814368249521, −7.399197705881712, −6.796214413789209, −5.534150571648597, −5.214711674295602, −3.857760435858255, −3.651474067520079, −2.839138884775337, −2.494931334837357, −1.327480781862806, 0,
1.327480781862806, 2.494931334837357, 2.839138884775337, 3.651474067520079, 3.857760435858255, 5.214711674295602, 5.534150571648597, 6.796214413789209, 7.399197705881712, 7.635814368249521, 8.343068118004063, 9.070266473158654, 9.384078349657869, 10.08605555399279, 10.51792011780137, 11.30280072505563, 12.28928107270716, 12.74427944718670, 13.13427710762386, 13.69716459348745, 14.20996103225701, 14.82260764436979, 15.39051550170522, 15.85077838181311, 16.13003243917240
