| L(s) = 1 | + 2·3-s + 7-s + 9-s + 2·17-s + 2·19-s + 2·21-s + 8·23-s − 4·27-s + 2·29-s − 4·31-s + 6·37-s − 2·41-s + 8·43-s − 4·47-s + 49-s + 4·51-s + 10·53-s + 4·57-s − 6·59-s + 4·61-s + 63-s − 12·67-s + 16·69-s + 14·73-s + 8·79-s − 11·81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 1.66·23-s − 0.769·27-s + 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s + 1.92·69-s + 1.63·73-s + 0.900·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.017092709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.017092709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−9.016353831659563256316815976220, −7.955680497262477941126609567507, −7.59496919133107363658161608903, −6.68922441578017008264572726362, −5.62940121509393392778473313445, −4.86690703180998964190536046386, −3.83051098046556122685042241853, −3.06551026181718468415456617058, −2.29037174921539130023849707486, −1.08114356407297375974866877413,
1.08114356407297375974866877413, 2.29037174921539130023849707486, 3.06551026181718468415456617058, 3.83051098046556122685042241853, 4.86690703180998964190536046386, 5.62940121509393392778473313445, 6.68922441578017008264572726362, 7.59496919133107363658161608903, 7.955680497262477941126609567507, 9.016353831659563256316815976220
