| L(s) = 1 | − 3-s − 7-s − 2·9-s − 2·11-s − 13-s − 17-s − 3·19-s + 21-s + 4·23-s + 5·27-s − 3·29-s − 31-s + 2·33-s + 8·37-s + 39-s − 12·41-s + 8·43-s − 5·47-s + 49-s + 51-s + 13·53-s + 3·57-s + 9·59-s − 5·61-s + 2·63-s + 10·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 0.218·21-s + 0.834·23-s + 0.962·27-s − 0.557·29-s − 0.179·31-s + 0.348·33-s + 1.31·37-s + 0.160·39-s − 1.87·41-s + 1.21·43-s − 0.729·47-s + 1/7·49-s + 0.140·51-s + 1.78·53-s + 0.397·57-s + 1.17·59-s − 0.640·61-s + 0.251·63-s + 1.22·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192155793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192155793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
| 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
−13.10293460656984, −12.69263553320815, −12.16605488361221, −11.53695474146662, −11.34639934634632, −10.82565020013663, −10.21695794809911, −10.03787123491007, −9.266037306418165, −8.776076897154402, −8.479014323442040, −7.810782960724981, −7.237240315595918, −6.867184818065818, −6.148890843227142, −5.931938509656637, −5.229909269454503, −4.906996359922717, −4.290999486192879, −3.572022605109945, −3.080631152196136, −2.401640283533073, −2.011818344793703, −0.9029903042284461, −0.3944409583410349,
0.3944409583410349, 0.9029903042284461, 2.011818344793703, 2.401640283533073, 3.080631152196136, 3.572022605109945, 4.290999486192879, 4.906996359922717, 5.229909269454503, 5.931938509656637, 6.148890843227142, 6.867184818065818, 7.237240315595918, 7.810782960724981, 8.479014323442040, 8.776076897154402, 9.266037306418165, 10.03787123491007, 10.21695794809911, 10.82565020013663, 11.34639934634632, 11.53695474146662, 12.16605488361221, 12.69263553320815, 13.10293460656984
