| L(s) = 1 | + 3·5-s + 7-s + 2·13-s + 3·17-s − 4·19-s + 6·23-s + 4·25-s − 10·31-s + 3·35-s − 7·37-s + 9·41-s + 5·43-s + 3·47-s + 49-s + 6·53-s − 9·59-s + 8·61-s + 6·65-s + 8·67-s − 12·71-s − 10·73-s + 5·79-s + 9·83-s + 9·85-s + 18·89-s + 2·91-s − 12·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.554·13-s + 0.727·17-s − 0.917·19-s + 1.25·23-s + 4/5·25-s − 1.79·31-s + 0.507·35-s − 1.15·37-s + 1.40·41-s + 0.762·43-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 1.17·59-s + 1.02·61-s + 0.744·65-s + 0.977·67-s − 1.42·71-s − 1.17·73-s + 0.562·79-s + 0.987·83-s + 0.976·85-s + 1.90·89-s + 0.209·91-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052741534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052741534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−10.54409954642245735729574885708, −9.315596521003952447945720863588, −8.932797574998353937782771963979, −7.74428441921683397738851658769, −6.75277875262882864228727566991, −5.80414730335436521294207112036, −5.18945711306403362481099621101, −3.85586732203537612935644613708, −2.49398887598010803443464560998, −1.39408347452929253097038552736,
1.39408347452929253097038552736, 2.49398887598010803443464560998, 3.85586732203537612935644613708, 5.18945711306403362481099621101, 5.80414730335436521294207112036, 6.75277875262882864228727566991, 7.74428441921683397738851658769, 8.932797574998353937782771963979, 9.315596521003952447945720863588, 10.54409954642245735729574885708
