| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s + 13-s + 16-s − 18-s + 2·19-s + 4·22-s + 4·23-s + 24-s − 26-s − 27-s + 2·29-s − 6·31-s − 32-s + 4·33-s + 36-s + 4·37-s − 2·38-s − 39-s + 6·41-s + 6·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.852·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + 0.657·37-s − 0.324·38-s − 0.160·39-s + 0.937·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.17947486162334, −13.36350386493949, −12.99353098156406, −12.56136480737803, −12.05445348581806, −11.39497832308852, −11.01947830108402, −10.61362245954172, −10.23172077264388, −9.548180319323916, −9.123138824656733, −8.650027130784092, −7.924688379420025, −7.436218561699067, −7.260814713264621, −6.393385895756884, −5.886391126128038, −5.485992492947853, −4.840765889629513, −4.274936539947250, −3.466563051758739, −2.796936953323354, −2.312388039704001, −1.406014295078137, −0.7941099650646428, 0,
0.7941099650646428, 1.406014295078137, 2.312388039704001, 2.796936953323354, 3.466563051758739, 4.274936539947250, 4.840765889629513, 5.485992492947853, 5.886391126128038, 6.393385895756884, 7.260814713264621, 7.436218561699067, 7.924688379420025, 8.650027130784092, 9.123138824656733, 9.548180319323916, 10.23172077264388, 10.61362245954172, 11.01947830108402, 11.39497832308852, 12.05445348581806, 12.56136480737803, 12.99353098156406, 13.36350386493949, 14.17947486162334
