| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 2·11-s + 12-s − 4·13-s − 16-s − 6·17-s + 18-s − 6·19-s − 2·22-s + 8·23-s + 3·24-s − 4·26-s − 27-s − 29-s + 6·31-s + 5·32-s + 2·33-s − 6·34-s − 36-s − 6·37-s − 6·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.426·22-s + 1.66·23-s + 0.612·24-s − 0.784·26-s − 0.192·27-s − 0.185·29-s + 1.07·31-s + 0.883·32-s + 0.348·33-s − 1.02·34-s − 1/6·36-s − 0.986·37-s − 0.973·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.70092409699140, −13.32194690498614, −13.00035500676697, −12.66958974767345, −11.99596443512206, −11.68984095045700, −11.01894350830010, −10.55821251790740, −10.15741016759592, −9.420566229808062, −9.100044931576983, −8.394014589481579, −8.160750000085181, −7.072428621178614, −6.890886187222153, −6.363643310937901, −5.582726386184237, −5.246485513763349, −4.614887944877146, −4.422648884927740, −3.756488840837513, −2.790737521627808, −2.583626162373048, −1.686639075647048, −0.5944785510513148, 0,
0.5944785510513148, 1.686639075647048, 2.583626162373048, 2.790737521627808, 3.756488840837513, 4.422648884927740, 4.614887944877146, 5.246485513763349, 5.582726386184237, 6.363643310937901, 6.890886187222153, 7.072428621178614, 8.160750000085181, 8.394014589481579, 9.100044931576983, 9.420566229808062, 10.15741016759592, 10.55821251790740, 11.01894350830010, 11.68984095045700, 11.99596443512206, 12.66958974767345, 13.00035500676697, 13.32194690498614, 13.70092409699140
