| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s − 2·13-s + 8·19-s − 25-s − 2·29-s − 4·31-s + 4·35-s − 4·37-s − 2·41-s + 4·43-s + 8·47-s − 3·49-s − 2·53-s + 4·55-s − 4·59-s − 4·65-s + 4·67-s + 8·71-s + 6·73-s + 4·77-s − 2·79-s + 6·83-s − 8·89-s − 4·91-s + 16·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s + 1.83·19-s − 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.657·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.539·55-s − 0.520·59-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.455·77-s − 0.225·79-s + 0.658·83-s − 0.847·89-s − 0.419·91-s + 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.68529656521259, −13.18438952049921, −12.55360694804889, −12.08686451259304, −11.72829886230176, −11.14321648393479, −10.79826269556116, −10.05567732086974, −9.758683765816128, −9.192362372140911, −8.996442666261856, −8.168545755974247, −7.651660006473970, −7.329977292885422, −6.689418094457102, −6.142749747612893, −5.518317074190283, −5.192912226548237, −4.771219650801860, −3.856911279484962, −3.562409131823846, −2.648476586268401, −2.212416279325360, −1.446475495818433, −1.128292288402032, 0,
1.128292288402032, 1.446475495818433, 2.212416279325360, 2.648476586268401, 3.562409131823846, 3.856911279484962, 4.771219650801860, 5.192912226548237, 5.518317074190283, 6.142749747612893, 6.689418094457102, 7.329977292885422, 7.651660006473970, 8.168545755974247, 8.996442666261856, 9.192362372140911, 9.758683765816128, 10.05567732086974, 10.79826269556116, 11.14321648393479, 11.72829886230176, 12.08686451259304, 12.55360694804889, 13.18438952049921, 13.68529656521259
