| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s − 7-s + 8-s − 2·9-s + 3·10-s + 12-s + 13-s − 14-s + 3·15-s + 16-s − 3·17-s − 2·18-s + 2·19-s + 3·20-s − 21-s + 6·23-s + 24-s + 4·25-s + 26-s − 5·27-s − 28-s − 6·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.458·19-s + 0.670·20-s − 0.218·21-s + 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.962·27-s − 0.188·28-s − 1.11·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.236078616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.236078616\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.58773217464922, −14.07702242278609, −13.59170606567655, −13.24787142054943, −12.96756924944614, −12.22739166918333, −11.48333152996086, −11.17423737569557, −10.48491231808636, −9.948526528308783, −9.347184982315644, −8.968937875298708, −8.476857066596813, −7.628964961394736, −7.079623711006307, −6.470324967012246, −5.861283828523974, −5.579833047101127, −4.900249635308443, −4.140051180873458, −3.440581890550940, −2.774201236769465, −2.398820467943402, −1.692519858864826, −0.7513332372587130,
0.7513332372587130, 1.692519858864826, 2.398820467943402, 2.774201236769465, 3.440581890550940, 4.140051180873458, 4.900249635308443, 5.579833047101127, 5.861283828523974, 6.470324967012246, 7.079623711006307, 7.628964961394736, 8.476857066596813, 8.968937875298708, 9.347184982315644, 9.948526528308783, 10.48491231808636, 11.17423737569557, 11.48333152996086, 12.22739166918333, 12.96756924944614, 13.24787142054943, 13.59170606567655, 14.07702242278609, 14.58773217464922
