| L(s) = 1 | + 2·5-s + 2·7-s + 6·11-s − 2·13-s + 6·23-s − 25-s + 10·29-s − 2·31-s + 4·35-s + 6·37-s − 6·41-s + 8·43-s − 3·49-s − 10·53-s + 12·55-s − 8·59-s + 14·61-s − 4·65-s − 4·67-s + 2·71-s + 14·73-s + 12·77-s + 10·79-s + 8·83-s + 10·89-s − 4·91-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.80·11-s − 0.554·13-s + 1.25·23-s − 1/5·25-s + 1.85·29-s − 0.359·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 3/7·49-s − 1.37·53-s + 1.61·55-s − 1.04·59-s + 1.79·61-s − 0.496·65-s − 0.488·67-s + 0.237·71-s + 1.63·73-s + 1.36·77-s + 1.12·79-s + 0.878·83-s + 1.05·89-s − 0.419·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.502152406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.502152406\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.30148449679278, −12.69246363635830, −12.29596869142147, −11.83970722899405, −11.23235282024666, −11.07213923543914, −10.31335065632876, −9.772149313451254, −9.511511754697727, −8.873155542699482, −8.667874135811210, −7.862395112429708, −7.469682917853618, −6.758894809517329, −6.318823955563283, −6.119652560354700, −5.143720256630445, −4.892473370186430, −4.386874118162831, −3.653025878427570, −3.127436617699589, −2.306826906717174, −1.899330352287956, −1.159559535493410, −0.7674994168623107,
0.7674994168623107, 1.159559535493410, 1.899330352287956, 2.306826906717174, 3.127436617699589, 3.653025878427570, 4.386874118162831, 4.892473370186430, 5.143720256630445, 6.119652560354700, 6.318823955563283, 6.758894809517329, 7.469682917853618, 7.862395112429708, 8.667874135811210, 8.873155542699482, 9.511511754697727, 9.772149313451254, 10.31335065632876, 11.07213923543914, 11.23235282024666, 11.83970722899405, 12.29596869142147, 12.69246363635830, 13.30148449679278
