| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 3·11-s − 2·13-s + 2·14-s + 16-s + 3·17-s − 19-s + 3·22-s + 6·23-s − 5·25-s − 2·26-s + 2·28-s − 6·29-s + 32-s + 3·34-s + 4·37-s − 38-s + 9·41-s + 43-s + 3·44-s + 6·46-s − 6·47-s − 3·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.904·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.639·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s + 0.176·32-s + 0.514·34-s + 0.657·37-s − 0.162·38-s + 1.40·41-s + 0.152·43-s + 0.452·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.49206882086851, −13.02007974637040, −12.68850998642586, −12.09381399186399, −11.67361591144771, −11.15405009365698, −11.04101093009252, −10.23203724904294, −9.649088963033905, −9.338201967914384, −8.734079462970759, −8.068981717911653, −7.587346155978905, −7.299896974066875, −6.619148234662422, −6.024053504496355, −5.672993794622378, −5.023699395814731, −4.503818602427821, −4.160464907136729, −3.399027968717769, −2.976433934400371, −2.200542447631342, −1.575820570699927, −1.113095659644886, 0,
1.113095659644886, 1.575820570699927, 2.200542447631342, 2.976433934400371, 3.399027968717769, 4.160464907136729, 4.503818602427821, 5.023699395814731, 5.672993794622378, 6.024053504496355, 6.619148234662422, 7.299896974066875, 7.587346155978905, 8.068981717911653, 8.734079462970759, 9.338201967914384, 9.649088963033905, 10.23203724904294, 11.04101093009252, 11.15405009365698, 11.67361591144771, 12.09381399186399, 12.68850998642586, 13.02007974637040, 13.49206882086851
