| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s − 2·9-s + 3·10-s + 12-s + 5·13-s − 14-s − 3·15-s + 16-s + 2·18-s − 2·19-s − 3·20-s + 21-s − 24-s + 4·25-s − 5·26-s − 5·27-s + 28-s + 3·30-s − 4·31-s − 32-s − 3·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.288·12-s + 1.38·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.471·18-s − 0.458·19-s − 0.670·20-s + 0.218·21-s − 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.962·27-s + 0.188·28-s + 0.547·30-s − 0.718·31-s − 0.176·32-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83582 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83582 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.59926548823006, −14.01522086345183, −13.48126102485055, −12.96978895478422, −12.33835780347872, −11.70115028575806, −11.46769392687371, −11.01042572715191, −10.59970530144596, −9.882120944694990, −9.259122823649509, −8.718869152391285, −8.274102963628744, −8.142356407244800, −7.611769790528634, −6.817151582029696, −6.505467990406880, −5.721687450805878, −5.109465802798441, −4.351443653387517, −3.759538914135012, −3.232290504258719, −2.887755214585568, −1.655860044549208, −1.504850791511483, 0, 0,
1.504850791511483, 1.655860044549208, 2.887755214585568, 3.232290504258719, 3.759538914135012, 4.351443653387517, 5.109465802798441, 5.721687450805878, 6.505467990406880, 6.817151582029696, 7.611769790528634, 8.142356407244800, 8.274102963628744, 8.718869152391285, 9.259122823649509, 9.882120944694990, 10.59970530144596, 11.01042572715191, 11.46769392687371, 11.70115028575806, 12.33835780347872, 12.96978895478422, 13.48126102485055, 14.01522086345183, 14.59926548823006
