| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s + 15-s − 2·17-s + 4·19-s + 21-s − 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 35-s + 2·37-s − 2·41-s + 4·43-s − 45-s − 12·47-s + 49-s + 2·51-s − 2·53-s − 4·55-s − 4·57-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.539·55-s − 0.529·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.154719561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154719561\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 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.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 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.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.58283936291164, −12.10063364681298, −11.89085027774767, −11.36434082178822, −11.17570403725984, −10.28472126622114, −10.02249503068806, −9.672009576453726, −8.987276474729750, −8.638153086756306, −8.031219104702337, −7.625443123919130, −6.922313609612141, −6.628100500120699, −6.189624731146690, −5.774264206032213, −4.950784436394867, −4.623806017407563, −4.124714767074835, −3.527075276558003, −3.095986623746275, −2.360906735836148, −1.647707438388449, −0.9595849014555699, −0.5013423096045785,
0.5013423096045785, 0.9595849014555699, 1.647707438388449, 2.360906735836148, 3.095986623746275, 3.527075276558003, 4.124714767074835, 4.623806017407563, 4.950784436394867, 5.774264206032213, 6.189624731146690, 6.628100500120699, 6.922313609612141, 7.625443123919130, 8.031219104702337, 8.638153086756306, 8.987276474729750, 9.672009576453726, 10.02249503068806, 10.28472126622114, 11.17570403725984, 11.36434082178822, 11.89085027774767, 12.10063364681298, 12.58283936291164
