| L(s) = 1 | − 5-s − 7-s + 4·11-s + 13-s + 2·17-s − 19-s + 7·23-s − 4·25-s + 5·29-s − 9·31-s + 35-s − 2·37-s − 2·41-s + 43-s − 9·47-s + 49-s − 3·53-s − 4·55-s + 14·61-s − 65-s + 10·67-s + 14·71-s + 3·73-s − 4·77-s + 5·79-s − 5·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.229·19-s + 1.45·23-s − 4/5·25-s + 0.928·29-s − 1.61·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s − 1.31·47-s + 1/7·49-s − 0.412·53-s − 0.539·55-s + 1.79·61-s − 0.124·65-s + 1.22·67-s + 1.66·71-s + 0.351·73-s − 0.455·77-s + 0.562·79-s − 0.548·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726250254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726250254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.659289985713125227592623388003, −7.955642571147852594550953093808, −6.98922741538970774574128366775, −6.59017971264705563997918730771, −5.62211935026791674353981798082, −4.77787393100694552394235825339, −3.75236091962915980617404931771, −3.33196922399387879734203086692, −1.96825721966258824948558752660, −0.804468343646160551982995326924,
0.804468343646160551982995326924, 1.96825721966258824948558752660, 3.33196922399387879734203086692, 3.75236091962915980617404931771, 4.77787393100694552394235825339, 5.62211935026791674353981798082, 6.59017971264705563997918730771, 6.98922741538970774574128366775, 7.955642571147852594550953093808, 8.659289985713125227592623388003
