| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s − 11-s − 4·13-s − 14-s + 16-s + 17-s + 4·19-s − 2·20-s − 22-s + 8·23-s − 25-s − 4·26-s − 28-s − 8·29-s − 4·31-s + 32-s + 34-s + 2·35-s + 6·37-s + 4·38-s − 2·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.48·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.72973243479295, −14.97730356528937, −14.67031339339609, −14.37738399731304, −13.22703464990632, −13.10863929540916, −12.61223362340329, −11.88920263900093, −11.39378870171897, −11.15098883801445, −10.27449617469442, −9.649761097090090, −9.239831421394749, −8.364866981827315, −7.601804193483846, −7.363094993935581, −6.851293224063010, −5.910526027307855, −5.337708473101999, −4.843092185869481, −4.075118171162095, −3.479869697779075, −2.894595748287047, −2.176810256746363, −1.032850323504756, 0,
1.032850323504756, 2.176810256746363, 2.894595748287047, 3.479869697779075, 4.075118171162095, 4.843092185869481, 5.337708473101999, 5.910526027307855, 6.851293224063010, 7.363094993935581, 7.601804193483846, 8.364866981827315, 9.239831421394749, 9.649761097090090, 10.27449617469442, 11.15098883801445, 11.39378870171897, 11.88920263900093, 12.61223362340329, 13.10863929540916, 13.22703464990632, 14.37738399731304, 14.67031339339609, 14.97730356528937, 15.72973243479295
