| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 7·13-s + 6·17-s + 19-s + 21-s + 27-s − 3·29-s − 2·31-s − 33-s − 5·37-s + 7·39-s − 12·41-s + 8·43-s − 3·47-s + 49-s + 6·51-s + 12·53-s + 57-s + 3·59-s − 10·61-s + 63-s − 13·67-s + 12·71-s − 11·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 1.45·17-s + 0.229·19-s + 0.218·21-s + 0.192·27-s − 0.557·29-s − 0.359·31-s − 0.174·33-s − 0.821·37-s + 1.12·39-s − 1.87·41-s + 1.21·43-s − 0.437·47-s + 1/7·49-s + 0.840·51-s + 1.64·53-s + 0.132·57-s + 0.390·59-s − 1.28·61-s + 0.125·63-s − 1.58·67-s + 1.42·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.189158179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.189158179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.99216922268909, −13.31767779634816, −13.07888851801078, −12.36756414264990, −11.82228386358275, −11.47871528829663, −10.68140672157092, −10.47885792787758, −9.931551794174588, −9.215140507093592, −8.730501699094252, −8.463193427119620, −7.695879037459463, −7.539026775227594, −6.733799470727413, −6.165510632610360, −5.502081305733705, −5.259847995211503, −4.305011475137184, −3.783867682595968, −3.324796033526425, −2.811347309176290, −1.765528234787007, −1.463198236660156, −0.6471576657518967,
0.6471576657518967, 1.463198236660156, 1.765528234787007, 2.811347309176290, 3.324796033526425, 3.783867682595968, 4.305011475137184, 5.259847995211503, 5.502081305733705, 6.165510632610360, 6.733799470727413, 7.539026775227594, 7.695879037459463, 8.463193427119620, 8.730501699094252, 9.215140507093592, 9.931551794174588, 10.47885792787758, 10.68140672157092, 11.47871528829663, 11.82228386358275, 12.36756414264990, 13.07888851801078, 13.31767779634816, 13.99216922268909
