| L(s) = 1 | − 2·5-s + 7-s + 2·11-s − 2·13-s + 6·17-s − 6·19-s + 6·23-s − 25-s − 5·29-s + 10·31-s − 2·35-s + 3·37-s + 6·41-s + 2·43-s + 6·47-s − 6·49-s − 12·53-s − 4·55-s + 9·59-s + 6·61-s + 4·65-s + 10·67-s − 5·71-s + 7·73-s + 2·77-s + 12·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s + 1.25·23-s − 1/5·25-s − 0.928·29-s + 1.79·31-s − 0.338·35-s + 0.493·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s − 6/7·49-s − 1.64·53-s − 0.539·55-s + 1.17·59-s + 0.768·61-s + 0.496·65-s + 1.22·67-s − 0.593·71-s + 0.819·73-s + 0.227·77-s + 1.35·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.848974349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.848974349\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.80159338122136, −12.49467607206460, −11.95980896842981, −11.56085919765180, −11.12859081779431, −10.78742061412926, −10.04523084183729, −9.718276240981116, −9.225233783398246, −8.522960955491561, −8.271602428613090, −7.664688386607199, −7.411108905131494, −6.789673351695576, −6.213871194500573, −5.812948580033243, −4.995165823734974, −4.704569040818475, −4.117144172341114, −3.623983851269424, −3.117525715391268, −2.408130078827996, −1.843251597238837, −0.9297082298817079, −0.5882639264686599,
0.5882639264686599, 0.9297082298817079, 1.843251597238837, 2.408130078827996, 3.117525715391268, 3.623983851269424, 4.117144172341114, 4.704569040818475, 4.995165823734974, 5.812948580033243, 6.213871194500573, 6.789673351695576, 7.411108905131494, 7.664688386607199, 8.271602428613090, 8.522960955491561, 9.225233783398246, 9.718276240981116, 10.04523084183729, 10.78742061412926, 11.12859081779431, 11.56085919765180, 11.95980896842981, 12.49467607206460, 12.80159338122136
