| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s − 2·13-s + 16-s + 2·17-s + 4·19-s − 2·20-s − 4·22-s − 25-s + 2·26-s + 6·29-s + 8·31-s − 32-s − 2·34-s − 2·37-s − 4·38-s + 2·40-s + 41-s + 4·43-s + 4·44-s + 12·47-s + 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 0.156·41-s + 0.609·43-s + 0.603·44-s + 1.75·47-s + 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820791050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820791050\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 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.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.15640778674659, −14.26983769632590, −14.09440997229101, −13.45059473072749, −12.42032226833533, −12.05524257454516, −11.93137454529127, −11.22573906829905, −10.69030826776216, −9.997693551481347, −9.580167870257720, −9.062186885324906, −8.389225320025989, −7.915375716034463, −7.467965709002796, −6.743906743787197, −6.451969785146217, −5.522173936323076, −4.942226535342508, −4.023636223889263, −3.737780178392896, −2.836587549966208, −2.207136858389373, −1.048230675815574, −0.7120053102529830,
0.7120053102529830, 1.048230675815574, 2.207136858389373, 2.836587549966208, 3.737780178392896, 4.023636223889263, 4.942226535342508, 5.522173936323076, 6.451969785146217, 6.743906743787197, 7.467965709002796, 7.915375716034463, 8.389225320025989, 9.062186885324906, 9.580167870257720, 9.997693551481347, 10.69030826776216, 11.22573906829905, 11.93137454529127, 12.05524257454516, 12.42032226833533, 13.45059473072749, 14.09440997229101, 14.26983769632590, 15.15640778674659
