| L(s) = 1 | − 4·5-s − 2·7-s + 5·11-s + 2·13-s − 3·17-s − 19-s − 6·23-s + 11·25-s + 2·29-s − 4·31-s + 8·35-s + 8·37-s + 41-s + 7·43-s + 2·47-s − 3·49-s + 4·53-s − 20·55-s − 5·59-s − 8·65-s + 13·67-s + 8·71-s + 3·73-s − 10·77-s + 8·79-s − 12·83-s + 12·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.50·11-s + 0.554·13-s − 0.727·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s + 0.371·29-s − 0.718·31-s + 1.35·35-s + 1.31·37-s + 0.156·41-s + 1.06·43-s + 0.291·47-s − 3/7·49-s + 0.549·53-s − 2.69·55-s − 0.650·59-s − 0.992·65-s + 1.58·67-s + 0.949·71-s + 0.351·73-s − 1.13·77-s + 0.900·79-s − 1.31·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.947136216996747774222177263079, −7.05855847023569634397882006425, −6.57752914914563184966607552461, −5.87384876752837181471562012339, −4.56782218139289099470225711720, −3.90680402780023493184892422662, −3.69093053598722828272656866723, −2.53443103123441373118114200339, −1.10270048872143684179161303450, 0,
1.10270048872143684179161303450, 2.53443103123441373118114200339, 3.69093053598722828272656866723, 3.90680402780023493184892422662, 4.56782218139289099470225711720, 5.87384876752837181471562012339, 6.57752914914563184966607552461, 7.05855847023569634397882006425, 7.947136216996747774222177263079
