| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s − 17-s − 5·19-s + 20-s − 22-s − 4·23-s + 25-s − 28-s − 5·29-s + 8·31-s + 32-s − 34-s − 35-s + 8·37-s − 5·38-s + 40-s − 3·41-s − 6·43-s − 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 − 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.14·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.188·28-s − 0.928·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s + 1.31·37-s − 0.811·38-s + 0.158·40-s − 0.468·41-s − 0.914·43-s − 0.150·44-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.85858041834939, −13.31319671584372, −13.17037032364140, −12.52475951046113, −12.09059005753177, −11.60564208830186, −11.02382768774195, −10.47336268930070, −10.17348784914426, −9.571563675932370, −9.017625995204662, −8.485250590799718, −7.795948177322217, −7.509337955725824, −6.584108068318625, −6.315910064580567, −5.968109602915363, −5.217519906617179, −4.701523196740460, −4.183153910719773, −3.593633760684176, −2.935944581664855, −2.289562966434062, −1.911113443019881, −0.9375451714859269, 0,
0.9375451714859269, 1.911113443019881, 2.289562966434062, 2.935944581664855, 3.593633760684176, 4.183153910719773, 4.701523196740460, 5.217519906617179, 5.968109602915363, 6.315910064580567, 6.584108068318625, 7.509337955725824, 7.795948177322217, 8.485250590799718, 9.017625995204662, 9.571563675932370, 10.17348784914426, 10.47336268930070, 11.02382768774195, 11.60564208830186, 12.09059005753177, 12.52475951046113, 13.17037032364140, 13.31319671584372, 13.85858041834939
