| L(s) = 1 | + 3-s − 5-s − 3·7-s − 2·9-s − 6·13-s − 15-s + 7·17-s + 5·19-s − 3·21-s − 6·23-s + 25-s − 5·27-s + 5·29-s − 3·31-s + 3·35-s − 3·37-s − 6·39-s − 2·41-s + 4·43-s + 2·45-s − 2·47-s + 2·49-s + 7·51-s + 53-s + 5·57-s + 10·59-s + 7·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s − 1.66·13-s − 0.258·15-s + 1.69·17-s + 1.14·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s − 0.538·31-s + 0.507·35-s − 0.493·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.291·47-s + 2/7·49-s + 0.980·51-s + 0.137·53-s + 0.662·57-s + 1.30·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8632181345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8632181345\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−14.62639252787908, −14.32659247483423, −14.00835283180015, −13.28141659693071, −12.63482697678003, −12.20373650782664, −11.83853787814268, −11.36475199513510, −10.17346223527265, −10.12636288774830, −9.586708032892846, −9.065427817490269, −8.260325046642469, −7.925324687985584, −7.230670234265750, −6.962620779134632, −5.934190589415362, −5.564801504163377, −4.951555826029692, −4.005875066869772, −3.493193833493213, −2.862849156075044, −2.552501950248879, −1.415128747773650, −0.3262114336642870,
0.3262114336642870, 1.415128747773650, 2.552501950248879, 2.862849156075044, 3.493193833493213, 4.005875066869772, 4.951555826029692, 5.564801504163377, 5.934190589415362, 6.962620779134632, 7.230670234265750, 7.925324687985584, 8.260325046642469, 9.065427817490269, 9.586708032892846, 10.12636288774830, 10.17346223527265, 11.36475199513510, 11.83853787814268, 12.20373650782664, 12.63482697678003, 13.28141659693071, 14.00835283180015, 14.32659247483423, 14.62639252787908
