| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 4·11-s − 12-s + 14-s + 16-s + 2·17-s − 18-s − 8·19-s + 21-s − 4·22-s + 4·23-s + 24-s − 27-s − 28-s − 2·29-s − 4·31-s − 32-s − 4·33-s − 2·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.23623433495119, −12.81868841522304, −12.34585666092961, −12.04246165406041, −11.38610955622416, −10.98461086757306, −10.64611686907480, −10.12866165617297, −9.547497075807146, −9.116159117158990, −8.821840368718370, −8.171087953083376, −7.646843386750904, −6.935238835232446, −6.774232887167116, −6.187572535885137, −5.755945516836772, −5.172629300967833, −4.385088086591587, −3.977484422735198, −3.427308735957518, −2.643384435126490, −2.014053395033441, −1.384292590270867, −0.7499394383853038, 0,
0.7499394383853038, 1.384292590270867, 2.014053395033441, 2.643384435126490, 3.427308735957518, 3.977484422735198, 4.385088086591587, 5.172629300967833, 5.755945516836772, 6.187572535885137, 6.774232887167116, 6.935238835232446, 7.646843386750904, 8.171087953083376, 8.821840368718370, 9.116159117158990, 9.547497075807146, 10.12866165617297, 10.64611686907480, 10.98461086757306, 11.38610955622416, 12.04246165406041, 12.34585666092961, 12.81868841522304, 13.23623433495119
