| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 2·7-s − 8-s + 9-s + 3·10-s + 11-s + 12-s + 2·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s − 2·21-s − 22-s − 24-s + 4·25-s + 27-s − 2·28-s − 9·29-s + 3·30-s + 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.301·11-s + 0.288·12-s + 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.436·21-s − 0.213·22-s − 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.377·28-s − 1.67·29-s + 0.547·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211926 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211926 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−13.15760434740473, −12.62967336832767, −12.30050735812936, −11.80569337896131, −11.31783502488162, −11.00547231869550, −10.24403378679486, −9.947221540844265, −9.400240629508268, −9.058226891488456, −8.300430240259225, −8.189562385881192, −7.631255423257510, −7.124524133235353, −6.832066700472850, −6.086817864250548, −5.695424278132191, −4.801301397636960, −4.262118860676028, −3.728040396166144, −3.320523294382201, −2.823016113805703, −2.152365438875589, −1.355771852547123, −0.6878453405315812, 0,
0.6878453405315812, 1.355771852547123, 2.152365438875589, 2.823016113805703, 3.320523294382201, 3.728040396166144, 4.262118860676028, 4.801301397636960, 5.695424278132191, 6.086817864250548, 6.832066700472850, 7.124524133235353, 7.631255423257510, 8.189562385881192, 8.300430240259225, 9.058226891488456, 9.400240629508268, 9.947221540844265, 10.24403378679486, 11.00547231869550, 11.31783502488162, 11.80569337896131, 12.30050735812936, 12.62967336832767, 13.15760434740473
