| L(s) = 1 | − 3-s + 4·7-s + 9-s − 13-s + 2·17-s + 4·19-s − 4·21-s + 8·23-s − 27-s − 2·29-s + 8·31-s + 2·37-s + 39-s − 6·41-s − 12·43-s + 9·49-s − 2·51-s + 10·53-s − 4·57-s + 10·61-s + 4·63-s + 4·67-s − 8·69-s + 16·71-s + 6·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.160·39-s − 0.937·41-s − 1.82·43-s + 9/7·49-s − 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.28·61-s + 0.503·63-s + 0.488·67-s − 0.963·69-s + 1.89·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.168927005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.168927005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−14.19148383725745, −13.87076615706167, −13.24052397198859, −12.76638706630016, −11.99260117500858, −11.71465506399207, −11.35479845404575, −10.81572122232758, −10.25141107212596, −9.758996476308454, −9.186656164159299, −8.385188668445802, −8.146817945786576, −7.528595763645736, −6.869712778459607, −6.581479188673564, −5.534638868898203, −5.115386606558101, −4.987203844694063, −4.156521118035649, −3.466782841693392, −2.715075513991765, −1.946848890786606, −1.198870876233714, −0.7218811306776762,
0.7218811306776762, 1.198870876233714, 1.946848890786606, 2.715075513991765, 3.466782841693392, 4.156521118035649, 4.987203844694063, 5.115386606558101, 5.534638868898203, 6.581479188673564, 6.869712778459607, 7.528595763645736, 8.146817945786576, 8.385188668445802, 9.186656164159299, 9.758996476308454, 10.25141107212596, 10.81572122232758, 11.35479845404575, 11.71465506399207, 11.99260117500858, 12.76638706630016, 13.24052397198859, 13.87076615706167, 14.19148383725745
