| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 4·7-s + 8-s + 9-s + 2·10-s + 11-s + 12-s − 13-s + 4·14-s + 2·15-s + 16-s − 8·17-s + 18-s − 6·19-s + 2·20-s + 4·21-s + 22-s − 6·23-s + 24-s − 25-s − 26-s + 27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 1.37·19-s + 0.447·20-s + 0.872·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606923730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606923730\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−10.31160160089939555137362853114, −9.223837190946646176988675459038, −8.484705470801542137562478640847, −7.63986618398519576100110925605, −6.57651112815013131067556299490, −5.76124847444114544900893975199, −4.60896546773282466465581973181, −4.09464619000295058481802737653, −2.26326736979752848209934129716, −1.92814877972623226631135598752,
1.92814877972623226631135598752, 2.26326736979752848209934129716, 4.09464619000295058481802737653, 4.60896546773282466465581973181, 5.76124847444114544900893975199, 6.57651112815013131067556299490, 7.63986618398519576100110925605, 8.484705470801542137562478640847, 9.223837190946646176988675459038, 10.31160160089939555137362853114
