| L(s) = 1 | + 3-s − 4·5-s + 7-s + 9-s − 2·13-s − 4·15-s − 2·17-s + 6·19-s + 21-s + 2·23-s + 11·25-s + 27-s + 6·29-s + 8·31-s − 4·35-s + 6·37-s − 2·39-s − 6·41-s + 10·43-s − 4·45-s − 2·47-s + 49-s − 2·51-s − 12·53-s + 6·57-s − 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 1.03·15-s − 0.485·17-s + 1.37·19-s + 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 1.52·43-s − 0.596·45-s − 0.291·47-s + 1/7·49-s − 0.280·51-s − 1.64·53-s + 0.794·57-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.241186194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241186194\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−14.76992960822922, −14.34586239203452, −13.83623914254317, −13.25287928035258, −12.49704871902279, −12.13879288440415, −11.73745920132580, −11.09625071338666, −10.80965641868874, −9.887298832120353, −9.495647877575361, −8.759290685726093, −8.235317490083637, −7.828083402019586, −7.478240854919605, −6.797016735451237, −6.295020820541967, −5.025136148418864, −4.866002273656108, −4.167639807330517, −3.580481756131769, −2.939018102729304, −2.442880436867326, −1.199848174657156, −0.6011875303859819,
0.6011875303859819, 1.199848174657156, 2.442880436867326, 2.939018102729304, 3.580481756131769, 4.167639807330517, 4.866002273656108, 5.025136148418864, 6.295020820541967, 6.797016735451237, 7.478240854919605, 7.828083402019586, 8.235317490083637, 8.759290685726093, 9.495647877575361, 9.887298832120353, 10.80965641868874, 11.09625071338666, 11.73745920132580, 12.13879288440415, 12.49704871902279, 13.25287928035258, 13.83623914254317, 14.34586239203452, 14.76992960822922
