| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s − 2·9-s + 11-s − 12-s − 13-s − 2·14-s + 16-s − 2·17-s − 2·18-s + 2·21-s + 22-s − 23-s − 24-s − 26-s + 5·27-s − 2·28-s − 29-s + 2·31-s + 32-s − 33-s − 2·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.436·21-s + 0.213·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s + 0.962·27-s − 0.377·28-s − 0.185·29-s + 0.359·31-s + 0.176·32-s − 0.174·33-s − 0.342·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−16.31334173034947, −15.64176769977991, −15.17409931507486, −14.53989937624309, −13.93506397796498, −13.56955675803503, −12.80590317553559, −12.28808875910580, −11.99701191263336, −11.10604309895532, −10.92318305086012, −10.16840464314919, −9.372852179524395, −9.018473393899320, −8.051568269039843, −7.503364237524891, −6.601574878906347, −6.325582737110527, −5.697499012945942, −5.070526375616634, −4.327466988909981, −3.696630579790931, −2.834284074862033, −2.324064141436127, −1.059881443976198, 0,
1.059881443976198, 2.324064141436127, 2.834284074862033, 3.696630579790931, 4.327466988909981, 5.070526375616634, 5.697499012945942, 6.325582737110527, 6.601574878906347, 7.503364237524891, 8.051568269039843, 9.018473393899320, 9.372852179524395, 10.16840464314919, 10.92318305086012, 11.10604309895532, 11.99701191263336, 12.28808875910580, 12.80590317553559, 13.56955675803503, 13.93506397796498, 14.53989937624309, 15.17409931507486, 15.64176769977991, 16.31334173034947
