| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 16-s + 4·17-s + 18-s + 4·19-s + 22-s + 4·23-s + 24-s + 4·26-s + 27-s − 2·29-s + 2·31-s + 32-s + 33-s + 4·34-s + 36-s + 6·37-s + 4·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.905791509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.905791509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.82923255334378, −13.57391488913525, −13.18626780250684, −12.50721201426370, −12.10383165136182, −11.60939415442734, −10.99396794658606, −10.63376940951328, −9.997362813088776, −9.264032719683295, −9.145818598298458, −8.303117780392416, −7.780727276621888, −7.441094203529030, −6.718057988086340, −6.214230273144909, −5.640436386173551, −5.152144869239393, −4.418567933998982, −3.873744903939564, −3.342206198441564, −2.896851288077310, −2.162738418670122, −1.269940939761253, −0.8912493993671892,
0.8912493993671892, 1.269940939761253, 2.162738418670122, 2.896851288077310, 3.342206198441564, 3.873744903939564, 4.418567933998982, 5.152144869239393, 5.640436386173551, 6.214230273144909, 6.718057988086340, 7.441094203529030, 7.780727276621888, 8.303117780392416, 9.145818598298458, 9.264032719683295, 9.997362813088776, 10.63376940951328, 10.99396794658606, 11.60939415442734, 12.10383165136182, 12.50721201426370, 13.18626780250684, 13.57391488913525, 13.82923255334378
