| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s − 7-s + 8-s + 6·9-s − 2·11-s + 3·12-s + 6·13-s − 14-s + 16-s + 6·18-s + 6·19-s − 3·21-s − 2·22-s + 3·23-s + 3·24-s + 6·26-s + 9·27-s − 28-s + 6·29-s + 32-s − 6·33-s + 6·36-s + 10·37-s + 6·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 0.603·11-s + 0.866·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.41·18-s + 1.37·19-s − 0.654·21-s − 0.426·22-s + 0.625·23-s + 0.612·24-s + 1.17·26-s + 1.73·27-s − 0.188·28-s + 1.11·29-s + 0.176·32-s − 1.04·33-s + 36-s + 1.64·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.962054689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.962054689\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.80356993266284, −15.52125779310197, −14.97330390974003, −14.24062747443716, −13.89293542779472, −13.41704218425355, −12.97763800005070, −12.58308635724459, −11.44948517863991, −11.24781967314803, −10.15708386659373, −9.855174953858536, −9.175263635333105, −8.392046994147379, −8.127853965052771, −7.491718340423907, −6.619808048288900, −6.277201309188058, −5.136449171804947, −4.663089983898508, −3.551865714017272, −3.363382881548711, −2.823921419453068, −1.844588431611611, −1.101010782314825,
1.101010782314825, 1.844588431611611, 2.823921419453068, 3.363382881548711, 3.551865714017272, 4.663089983898508, 5.136449171804947, 6.277201309188058, 6.619808048288900, 7.491718340423907, 8.127853965052771, 8.392046994147379, 9.175263635333105, 9.855174953858536, 10.15708386659373, 11.24781967314803, 11.44948517863991, 12.58308635724459, 12.97763800005070, 13.41704218425355, 13.89293542779472, 14.24062747443716, 14.97330390974003, 15.52125779310197, 15.80356993266284
