| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s − 2·11-s + 2·12-s + 16-s + 6·17-s − 18-s + 5·19-s + 2·22-s − 8·23-s − 2·24-s − 4·27-s − 4·29-s + 5·31-s − 32-s − 4·33-s − 6·34-s + 36-s + 10·37-s − 5·38-s + 3·41-s + 11·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.14·19-s + 0.426·22-s − 1.66·23-s − 0.408·24-s − 0.769·27-s − 0.742·29-s + 0.898·31-s − 0.176·32-s − 0.696·33-s − 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.811·38-s + 0.468·41-s + 1.67·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.561918979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561918979\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| 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
−13.20838576686836, −12.57932410354793, −12.06073654608135, −11.75763203478897, −11.19264951899147, −10.46511011847959, −10.20846858830685, −9.673979949271797, −9.240666985240825, −8.978986650886819, −8.076502322051718, −7.981711609311023, −7.552833881751722, −7.261088951636274, −6.280378016073456, −5.717131575853077, −5.595983166346866, −4.648204923375728, −3.879892403159278, −3.614554693693238, −2.831647385213318, −2.471872964976709, −2.013298334976439, −1.042949501415143, −0.6382100552357311,
0.6382100552357311, 1.042949501415143, 2.013298334976439, 2.471872964976709, 2.831647385213318, 3.614554693693238, 3.879892403159278, 4.648204923375728, 5.595983166346866, 5.717131575853077, 6.280378016073456, 7.261088951636274, 7.552833881751722, 7.981711609311023, 8.076502322051718, 8.978986650886819, 9.240666985240825, 9.673979949271797, 10.20846858830685, 10.46511011847959, 11.19264951899147, 11.75763203478897, 12.06073654608135, 12.57932410354793, 13.20838576686836
