| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s − 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s − 22-s − 24-s + 25-s + 27-s − 2·29-s + 30-s − 32-s + 33-s − 2·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s + 0.174·33-s − 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.61832550467839, −14.33574498843674, −13.67942139030831, −13.02520992976214, −12.71608848081319, −11.89933281250546, −11.46822233644963, −11.29672241759251, −10.21150357262708, −10.02989131331294, −9.552526037056165, −8.843208427921527, −8.420898133458836, −8.002445198930514, −7.363579130365818, −6.972395972919971, −6.391441355693329, −5.595464166882251, −5.038337051200255, −4.298512836978001, −3.472394524221525, −3.250451232438825, −2.385262793925395, −1.625693331032433, −1.001682898599446, 0,
1.001682898599446, 1.625693331032433, 2.385262793925395, 3.250451232438825, 3.472394524221525, 4.298512836978001, 5.038337051200255, 5.595464166882251, 6.391441355693329, 6.972395972919971, 7.363579130365818, 8.002445198930514, 8.420898133458836, 8.843208427921527, 9.552526037056165, 10.02989131331294, 10.21150357262708, 11.29672241759251, 11.46822233644963, 11.89933281250546, 12.71608848081319, 13.02520992976214, 13.67942139030831, 14.33574498843674, 14.61832550467839
