| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 13-s − 2·15-s − 2·17-s + 21-s + 8·23-s − 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 2·35-s − 6·37-s + 39-s + 6·41-s + 4·43-s + 2·45-s + 12·47-s + 49-s + 2·51-s + 6·53-s + 8·55-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s − 0.485·17-s + 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.338·35-s − 0.986·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022206816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022206816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 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.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−7.41512018265474608753943673366, −7.03598212609499461290739745852, −6.34698031070445576045554008995, −5.71499991063855109737138294815, −5.18700821288935041391736940657, −4.24108370448484738444213066966, −3.58476219702193668905634150659, −2.50616564768885017644691599531, −1.69370046694456703781223481963, −0.73296002896299097918050911403,
0.73296002896299097918050911403, 1.69370046694456703781223481963, 2.50616564768885017644691599531, 3.58476219702193668905634150659, 4.24108370448484738444213066966, 5.18700821288935041391736940657, 5.71499991063855109737138294815, 6.34698031070445576045554008995, 7.03598212609499461290739745852, 7.41512018265474608753943673366
