| L(s) = 1 | + 2-s + 4-s + 2·5-s + 4·7-s + 8-s + 2·10-s + 4·11-s − 2·13-s + 4·14-s + 16-s − 2·17-s + 2·20-s + 4·22-s − 25-s − 2·26-s + 4·28-s − 6·29-s + 8·31-s + 32-s − 2·34-s + 8·35-s + 2·37-s + 2·40-s + 41-s + 4·43-s + 4·44-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 1.35·35-s + 0.328·37-s + 0.316·40-s + 0.156·41-s + 0.609·43-s + 0.603·44-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.17508019676639, −12.53254249597299, −12.03533349479887, −11.71963579140834, −11.15667964479129, −11.03828292504185, −10.23236144366169, −9.860293180333982, −9.331372880586713, −8.886891765396307, −8.341398343278913, −7.779675133996930, −7.403180018101647, −6.779685053416921, −6.297413726946387, −5.825228121425740, −5.443947046333020, −4.746408100644653, −4.345090838493524, −4.179090658311673, −3.102045246146656, −2.778413333330046, −1.879838015995630, −1.657271964998613, −1.207173145789749, 0,
1.207173145789749, 1.657271964998613, 1.879838015995630, 2.778413333330046, 3.102045246146656, 4.179090658311673, 4.345090838493524, 4.746408100644653, 5.443947046333020, 5.825228121425740, 6.297413726946387, 6.779685053416921, 7.403180018101647, 7.779675133996930, 8.341398343278913, 8.886891765396307, 9.331372880586713, 9.860293180333982, 10.23236144366169, 11.03828292504185, 11.15667964479129, 11.71963579140834, 12.03533349479887, 12.53254249597299, 13.17508019676639
