| L(s) = 1 | − 5-s − 7-s + 5·11-s − 13-s + 17-s + 6·19-s + 6·23-s − 4·25-s − 6·29-s − 4·31-s + 35-s + 11·37-s + 9·43-s + 4·47-s + 49-s + 7·53-s − 5·55-s + 12·59-s + 6·61-s + 65-s − 13·67-s + 4·71-s − 13·73-s − 5·77-s − 15·79-s + 13·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.50·11-s − 0.277·13-s + 0.242·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s − 1.11·29-s − 0.718·31-s + 0.169·35-s + 1.80·37-s + 1.37·43-s + 0.583·47-s + 1/7·49-s + 0.961·53-s − 0.674·55-s + 1.56·59-s + 0.768·61-s + 0.124·65-s − 1.58·67-s + 0.474·71-s − 1.52·73-s − 0.569·77-s − 1.68·79-s + 1.42·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256499336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256499336\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.05979315629819, −15.20395478742646, −14.69906034448138, −14.45392214626381, −13.55094172682949, −13.21290316006802, −12.42641449621318, −11.98253433499736, −11.32062203784890, −11.16431183925421, −10.09798967323697, −9.550062224504322, −9.181657898665781, −8.600326372617793, −7.599117101976216, −7.320054575173864, −6.747901410446061, −5.763555229628079, −5.554041615702348, −4.384944685624263, −3.966342413478203, −3.285494458505112, −2.520648726179307, −1.413302833807636, −0.6972433810103567,
0.6972433810103567, 1.413302833807636, 2.520648726179307, 3.285494458505112, 3.966342413478203, 4.384944685624263, 5.554041615702348, 5.763555229628079, 6.747901410446061, 7.320054575173864, 7.599117101976216, 8.600326372617793, 9.181657898665781, 9.550062224504322, 10.09798967323697, 11.16431183925421, 11.32062203784890, 11.98253433499736, 12.42641449621318, 13.21290316006802, 13.55094172682949, 14.45392214626381, 14.69906034448138, 15.20395478742646, 16.05979315629819
