| L(s) = 1 | − 2-s + 4-s − 3·5-s + 2·7-s − 8-s + 3·10-s − 4·13-s − 2·14-s + 16-s + 6·17-s − 7·19-s − 3·20-s − 9·23-s + 4·25-s + 4·26-s + 2·28-s − 9·29-s + 2·31-s − 32-s − 6·34-s − 6·35-s − 4·37-s + 7·38-s + 3·40-s + 6·41-s − 4·43-s + 9·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.755·7-s − 0.353·8-s + 0.948·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 1.60·19-s − 0.670·20-s − 1.87·23-s + 4/5·25-s + 0.784·26-s + 0.377·28-s − 1.67·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 1.01·35-s − 0.657·37-s + 1.13·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s + 1.32·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
| 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
−10.51435934841690623906670741078, −9.697252328270977284624171046662, −8.465132934113824858756487265843, −7.85400592111218694139788018379, −7.32800712279579443871219132970, −5.93961037827028489073777534125, −4.62390384313555204390288516348, −3.60634322109716502135704894763, −1.99374822115735108795241652513, 0,
1.99374822115735108795241652513, 3.60634322109716502135704894763, 4.62390384313555204390288516348, 5.93961037827028489073777534125, 7.32800712279579443871219132970, 7.85400592111218694139788018379, 8.465132934113824858756487265843, 9.697252328270977284624171046662, 10.51435934841690623906670741078
