| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 4·13-s − 15-s − 4·17-s − 2·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 8·31-s + 2·35-s + 2·37-s + 4·39-s + 10·41-s + 10·43-s + 45-s − 12·47-s − 3·49-s + 4·51-s − 6·53-s + 12·59-s + 10·61-s + 2·63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.970·17-s − 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 1.56·41-s + 1.52·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s + 1.56·59-s + 1.28·61-s + 0.251·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.99620863601131, −13.08053935298003, −12.87759107584801, −12.56875584394703, −11.72608602773791, −11.36741329457721, −11.07040156764675, −10.52870696551122, −9.887291126972421, −9.460845891048017, −9.142152250128563, −8.290799086037663, −7.933689148813684, −7.237717317598015, −6.982282037762111, −6.238736737150227, −5.664325041981048, −5.354529505121164, −4.670454401793322, −4.274465346999036, −3.659990592421966, −2.730856329085030, −2.031500512231700, −1.827397206938806, −0.7703192652054214, 0,
0.7703192652054214, 1.827397206938806, 2.031500512231700, 2.730856329085030, 3.659990592421966, 4.274465346999036, 4.670454401793322, 5.354529505121164, 5.664325041981048, 6.238736737150227, 6.982282037762111, 7.237717317598015, 7.933689148813684, 8.290799086037663, 9.142152250128563, 9.460845891048017, 9.887291126972421, 10.52870696551122, 11.07040156764675, 11.36741329457721, 11.72608602773791, 12.56875584394703, 12.87759107584801, 13.08053935298003, 13.99620863601131
