| L(s) = 1 | + 7-s − 4·11-s − 4·17-s + 4·23-s − 5·25-s + 4·29-s − 8·31-s + 2·37-s − 4·41-s − 8·43-s − 8·47-s + 49-s − 4·53-s − 8·59-s − 2·61-s + 8·67-s − 12·71-s − 6·73-s − 4·77-s + 8·79-s − 16·83-s − 12·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.970·17-s + 0.834·23-s − 25-s + 0.742·29-s − 1.43·31-s + 0.328·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.549·53-s − 1.04·59-s − 0.256·61-s + 0.977·67-s − 1.42·71-s − 0.702·73-s − 0.455·77-s + 0.900·79-s − 1.75·83-s − 1.27·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.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.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.02575742052162, −12.72341558401133, −12.10047492329826, −11.55228388150484, −11.17344736153924, −10.89588001284526, −10.19020248574252, −10.04141303690676, −9.304648777458249, −8.947844737501620, −8.334677577419048, −8.060048285484551, −7.531251380244165, −7.016953305353860, −6.619005872332141, −5.968500676464525, −5.488942642837980, −5.014904379031761, −4.587319976556721, −4.112463507238007, −3.255483456903385, −3.031656367539309, −2.239650220714117, −1.813939772586466, −1.192886722826172, 0, 0,
1.192886722826172, 1.813939772586466, 2.239650220714117, 3.031656367539309, 3.255483456903385, 4.112463507238007, 4.587319976556721, 5.014904379031761, 5.488942642837980, 5.968500676464525, 6.619005872332141, 7.016953305353860, 7.531251380244165, 8.060048285484551, 8.334677577419048, 8.947844737501620, 9.304648777458249, 10.04141303690676, 10.19020248574252, 10.89588001284526, 11.17344736153924, 11.55228388150484, 12.10047492329826, 12.72341558401133, 13.02575742052162
