| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 4·11-s − 12-s + 2·13-s + 14-s + 16-s + 2·17-s + 18-s + 19-s − 21-s + 4·22-s + 8·23-s − 24-s + 2·26-s − 27-s + 28-s + 6·29-s + 32-s − 4·33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.482786598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.482786598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.51867728260580, −15.14500207548764, −14.46560453181681, −14.11536588935266, −13.51302076027206, −12.80824616968759, −12.45981246915299, −11.75183374043180, −11.34449503974275, −10.96177987581784, −10.25574544295321, −9.565563409215779, −8.965053901628547, −8.326097769849767, −7.497173734189755, −7.024109124342449, −6.283097808656807, −5.962928314735584, −5.127093245635378, −4.593165353118220, −3.999448388615093, −3.254999487295893, −2.510689166179134, −1.324309929416895, −0.9650481553391330,
0.9650481553391330, 1.324309929416895, 2.510689166179134, 3.254999487295893, 3.999448388615093, 4.593165353118220, 5.127093245635378, 5.962928314735584, 6.283097808656807, 7.024109124342449, 7.497173734189755, 8.326097769849767, 8.965053901628547, 9.565563409215779, 10.25574544295321, 10.96177987581784, 11.34449503974275, 11.75183374043180, 12.45981246915299, 12.80824616968759, 13.51302076027206, 14.11536588935266, 14.46560453181681, 15.14500207548764, 15.51867728260580
