| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s − 14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s − 21-s − 4·22-s − 23-s + 24-s + 25-s − 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 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.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.20628706077910, −13.59875543144581, −13.43552661322746, −12.69737345809749, −12.46609898317890, −11.93949892615106, −11.22975978193443, −10.66482126757026, −10.26032485214607, −9.653496237292497, −9.417105909263039, −8.539916999052313, −8.135457287679649, −7.418126496220017, −7.174531708768344, −6.526225561723915, −5.753774810131013, −5.386617651543202, −4.918679122961710, −4.232740837303174, −3.434064583467599, −3.127707731063421, −2.398529160813633, −1.992685240318772, −1.070211440120763, 0,
1.070211440120763, 1.992685240318772, 2.398529160813633, 3.127707731063421, 3.434064583467599, 4.232740837303174, 4.918679122961710, 5.386617651543202, 5.753774810131013, 6.526225561723915, 7.174531708768344, 7.418126496220017, 8.135457287679649, 8.539916999052313, 9.417105909263039, 9.653496237292497, 10.26032485214607, 10.66482126757026, 11.22975978193443, 11.93949892615106, 12.46609898317890, 12.69737345809749, 13.43552661322746, 13.59875543144581, 14.20628706077910
