| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·7-s + 3·8-s − 2·10-s − 2·11-s − 2·13-s − 3·14-s − 16-s − 4·17-s + 3·19-s − 2·20-s + 2·22-s − 25-s + 2·26-s − 3·28-s + 8·31-s − 5·32-s + 4·34-s + 6·35-s + 2·37-s − 3·38-s + 6·40-s + 4·41-s − 4·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.13·7-s + 1.06·8-s − 0.632·10-s − 0.603·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s − 0.970·17-s + 0.688·19-s − 0.447·20-s + 0.426·22-s − 1/5·25-s + 0.392·26-s − 0.566·28-s + 1.43·31-s − 0.883·32-s + 0.685·34-s + 1.01·35-s + 0.328·37-s − 0.486·38-s + 0.948·40-s + 0.624·41-s − 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.56780207187755, −13.96182043248540, −13.58260972241384, −13.23665451502031, −12.68774289164048, −11.82346083661158, −11.53650027070751, −10.83864673522999, −10.32207965867375, −9.965835939265879, −9.482596564900171, −8.852704303152938, −8.480564820062866, −7.881774437544475, −7.480368847086842, −6.868855666805723, −6.055552462686599, −5.496796140221531, −4.909545354254916, −4.572690909370517, −3.922351922395679, −2.813972811853297, −2.280801730181894, −1.606213298407287, −0.9762688980812933, 0,
0.9762688980812933, 1.606213298407287, 2.280801730181894, 2.813972811853297, 3.922351922395679, 4.572690909370517, 4.909545354254916, 5.496796140221531, 6.055552462686599, 6.868855666805723, 7.480368847086842, 7.881774437544475, 8.480564820062866, 8.852704303152938, 9.482596564900171, 9.965835939265879, 10.32207965867375, 10.83864673522999, 11.53650027070751, 11.82346083661158, 12.68774289164048, 13.23665451502031, 13.58260972241384, 13.96182043248540, 14.56780207187755
