| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 4·11-s + 14-s + 16-s + 17-s − 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s + 28-s + 4·29-s + 6·31-s + 32-s + 34-s − 35-s + 6·37-s − 4·38-s − 40-s − 6·41-s + 4·43-s + 4·44-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 + 1.20·11-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.188·28-s + 0.742·29-s + 1.07·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.986·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−12.95997943758414, −12.51504525978128, −12.17962439808624, −11.64625225449758, −11.42972505530959, −10.92934022408791, −10.30031365807253, −9.848643686818795, −9.500367485484162, −8.635550796258602, −8.336830485644380, −7.977435545593971, −7.352975906270135, −6.763205339902228, −6.353303717542115, −6.025089427794888, −5.404140622990317, −4.589403460763426, −4.427293511013996, −3.953407099082417, −3.370112823262862, −2.789145977154686, −2.108757125379317, −1.558525282586419, −0.9270968487055710, 0,
0.9270968487055710, 1.558525282586419, 2.108757125379317, 2.789145977154686, 3.370112823262862, 3.953407099082417, 4.427293511013996, 4.589403460763426, 5.404140622990317, 6.025089427794888, 6.353303717542115, 6.763205339902228, 7.352975906270135, 7.977435545593971, 8.336830485644380, 8.635550796258602, 9.500367485484162, 9.848643686818795, 10.30031365807253, 10.92934022408791, 11.42972505530959, 11.64625225449758, 12.17962439808624, 12.51504525978128, 12.95997943758414
