| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 3·11-s − 14-s + 16-s + 4·17-s + 7·19-s + 20-s + 3·22-s + 2·23-s + 25-s + 28-s + 2·29-s − 5·31-s − 32-s − 4·34-s + 35-s + 4·37-s − 7·38-s − 40-s + 9·41-s + 12·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s + 0.223·20-s + 0.639·22-s + 0.417·23-s + 1/5·25-s + 0.188·28-s + 0.371·29-s − 0.898·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.657·37-s − 1.13·38-s − 0.158·40-s + 1.40·41-s + 1.82·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.613357335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.613357335\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.93436158596107, −13.00636412550684, −12.72869709486249, −12.28003136081317, −11.52489568994246, −11.24966657659415, −10.62773265964163, −10.30182964311356, −9.655265250640903, −9.292967412060374, −8.917035481390023, −8.058482336663271, −7.694912592392843, −7.448422251435735, −6.798301747635142, −5.942204210066698, −5.651608473163983, −5.183195057525685, −4.503036683182503, −3.732388452770448, −2.960624648131218, −2.645666132586891, −1.866091937825553, −1.091170601633821, −0.6559015515116423,
0.6559015515116423, 1.091170601633821, 1.866091937825553, 2.645666132586891, 2.960624648131218, 3.732388452770448, 4.503036683182503, 5.183195057525685, 5.651608473163983, 5.942204210066698, 6.798301747635142, 7.448422251435735, 7.694912592392843, 8.058482336663271, 8.917035481390023, 9.292967412060374, 9.655265250640903, 10.30182964311356, 10.62773265964163, 11.24966657659415, 11.52489568994246, 12.28003136081317, 12.72869709486249, 13.00636412550684, 13.93436158596107
