| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s + 2·10-s − 2·12-s + 13-s + 2·14-s − 15-s − 4·16-s − 4·17-s + 2·18-s + 6·19-s + 2·20-s − 21-s + 6·23-s + 25-s + 2·26-s − 27-s + 2·28-s − 29-s − 2·30-s − 3·31-s − 8·32-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 + 1/3·9-s + 0.632·10-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.471·18-s + 1.37·19-s + 0.447·20-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.365·30-s − 0.538·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.658490576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.658490576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−13.24355951122634, −12.89089705447836, −12.38459623484663, −11.93439667722892, −11.38926812054928, −11.02666934201930, −10.80132507514154, −9.890654422906308, −9.518210906695126, −8.942995926842339, −8.539443937488964, −7.687638288864748, −7.175699453272776, −6.737297377037960, −6.227976741264998, −5.755574893933568, −5.210277623275836, −4.837200081738843, −4.557668439297334, −3.570467250783506, −3.413335827260376, −2.641718633530152, −1.969431766275170, −1.376669218380102, −0.5013962154106849,
0.5013962154106849, 1.376669218380102, 1.969431766275170, 2.641718633530152, 3.413335827260376, 3.570467250783506, 4.557668439297334, 4.837200081738843, 5.210277623275836, 5.755574893933568, 6.227976741264998, 6.737297377037960, 7.175699453272776, 7.687638288864748, 8.539443937488964, 8.942995926842339, 9.518210906695126, 9.890654422906308, 10.80132507514154, 11.02666934201930, 11.38926812054928, 11.93439667722892, 12.38459623484663, 12.89089705447836, 13.24355951122634
