| L(s) = 1 | + 5-s + 7-s − 6·11-s − 13-s + 4·19-s + 25-s − 4·31-s + 35-s + 4·37-s − 6·41-s + 10·43-s + 6·47-s + 49-s + 6·53-s − 6·55-s + 10·61-s − 65-s + 4·67-s − 6·71-s + 2·73-s − 6·77-s − 4·79-s − 6·83-s − 6·89-s − 91-s + 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.917·19-s + 1/5·25-s − 0.718·31-s + 0.169·35-s + 0.657·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 0.809·55-s + 1.28·61-s − 0.124·65-s + 0.488·67-s − 0.712·71-s + 0.234·73-s − 0.683·77-s − 0.450·79-s − 0.658·83-s − 0.635·89-s − 0.104·91-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.06699511892475, −12.65693708608620, −12.19290743503688, −11.59590117043078, −11.15638686077140, −10.71252508737008, −10.24264234126898, −9.915413202058426, −9.372736369672873, −8.866043191521988, −8.311224908346361, −7.895532101822328, −7.352717603386587, −7.140380294078689, −6.376953255105052, −5.645230873064829, −5.432846031601944, −5.100682121938572, −4.367690657461832, −3.883883094307886, −3.084179322433231, −2.587224320160400, −2.269530856701832, −1.465782812964553, −0.7849696068408040, 0,
0.7849696068408040, 1.465782812964553, 2.269530856701832, 2.587224320160400, 3.084179322433231, 3.883883094307886, 4.367690657461832, 5.100682121938572, 5.432846031601944, 5.645230873064829, 6.376953255105052, 7.140380294078689, 7.352717603386587, 7.895532101822328, 8.311224908346361, 8.866043191521988, 9.372736369672873, 9.915413202058426, 10.24264234126898, 10.71252508737008, 11.15638686077140, 11.59590117043078, 12.19290743503688, 12.65693708608620, 13.06699511892475
