| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 7-s − 2·9-s + 2·10-s − 2·12-s − 6·13-s + 2·14-s + 15-s − 4·16-s + 3·17-s + 4·18-s − 2·20-s + 21-s − 4·23-s − 4·25-s + 12·26-s + 5·27-s − 2·28-s + 9·29-s − 2·30-s − 4·31-s + 8·32-s − 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s − 1.66·13-s + 0.534·14-s + 0.258·15-s − 16-s + 0.727·17-s + 0.942·18-s − 0.447·20-s + 0.218·21-s − 0.834·23-s − 4/5·25-s + 2.35·26-s + 0.962·27-s − 0.377·28-s + 1.67·29-s − 0.365·30-s − 0.718·31-s + 1.41·32-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 + T \) | |
| 127 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−7.939464462549859679095492427466, −7.34346291516924243052877992369, −6.57401016203390226750400979564, −5.82587680626435124666310564895, −4.95133374026906823148849244894, −4.18684007278661017971278522779, −2.94754357676235370066028536658, −2.19005501988872260932728427944, −0.817029472084123138038885878124, 0,
0.817029472084123138038885878124, 2.19005501988872260932728427944, 2.94754357676235370066028536658, 4.18684007278661017971278522779, 4.95133374026906823148849244894, 5.82587680626435124666310564895, 6.57401016203390226750400979564, 7.34346291516924243052877992369, 7.939464462549859679095492427466
