| L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 4·13-s + 2·17-s − 2·21-s + 2·23-s − 27-s + 2·29-s − 4·31-s − 33-s + 6·37-s − 4·39-s − 6·41-s − 12·43-s + 6·47-s − 3·49-s − 2·51-s + 2·63-s + 4·67-s − 2·69-s − 10·71-s − 2·73-s + 2·77-s + 2·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.436·21-s + 0.417·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.251·63-s + 0.488·67-s − 0.240·69-s − 1.18·71-s − 0.234·73-s + 0.227·77-s + 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369393409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369393409\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.56745256483145, −13.92051279363834, −13.36438123101853, −13.02584233273068, −12.29181650951690, −11.83910525443549, −11.33336468025828, −11.01740774884356, −10.41128806225513, −9.906480688126067, −9.274915772197600, −8.626667579177611, −8.234881326035959, −7.657019507786850, −6.926661550685826, −6.553866439710969, −5.780328483928967, −5.447667595204829, −4.721076881699392, −4.226502816491628, −3.509898767554416, −2.921565000975106, −1.811276946671198, −1.406260708096325, −0.5888729435221523,
0.5888729435221523, 1.406260708096325, 1.811276946671198, 2.921565000975106, 3.509898767554416, 4.226502816491628, 4.721076881699392, 5.447667595204829, 5.780328483928967, 6.553866439710969, 6.926661550685826, 7.657019507786850, 8.234881326035959, 8.626667579177611, 9.274915772197600, 9.906480688126067, 10.41128806225513, 11.01740774884356, 11.33336468025828, 11.83910525443549, 12.29181650951690, 13.02584233273068, 13.36438123101853, 13.92051279363834, 14.56745256483145
