| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 11-s + 2·15-s + 2·17-s − 4·19-s + 21-s − 4·23-s − 25-s − 27-s + 8·29-s + 3·31-s + 33-s + 2·35-s − 10·37-s − 8·41-s − 7·43-s − 2·45-s − 8·47-s − 6·49-s − 2·51-s + 10·53-s + 2·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.538·31-s + 0.174·33-s + 0.338·35-s − 1.64·37-s − 1.24·41-s − 1.06·43-s − 0.298·45-s − 1.16·47-s − 6/7·49-s − 0.280·51-s + 1.37·53-s + 0.269·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.87857234093582, −12.33660505740367, −12.11154326440291, −11.61614157027241, −11.39534773121923, −10.61105315417064, −10.24967465175745, −10.04805937498974, −9.497840547117386, −8.587755320788769, −8.442488142285462, −8.061261708583385, −7.404439024272915, −6.952283887058579, −6.466960002284488, −6.169377881559974, −5.465235824360883, −4.906189395076172, −4.647451340659601, −3.888031096385586, −3.549827923860909, −3.045064428481299, −2.270761928112562, −1.694636178182431, −0.9679095058078785, 0, 0,
0.9679095058078785, 1.694636178182431, 2.270761928112562, 3.045064428481299, 3.549827923860909, 3.888031096385586, 4.647451340659601, 4.906189395076172, 5.465235824360883, 6.169377881559974, 6.466960002284488, 6.952283887058579, 7.404439024272915, 8.061261708583385, 8.442488142285462, 8.587755320788769, 9.497840547117386, 10.04805937498974, 10.24967465175745, 10.61105315417064, 11.39534773121923, 11.61614157027241, 12.11154326440291, 12.33660505740367, 12.87857234093582
