| L(s) = 1 | + 2·7-s − 2·13-s − 17-s + 4·19-s + 2·23-s − 5·25-s − 6·31-s − 10·41-s + 4·43-s − 4·47-s − 3·49-s + 2·53-s − 4·59-s − 4·67-s − 2·71-s + 14·73-s + 6·79-s + 12·83-s + 2·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.417·23-s − 25-s − 1.07·31-s − 1.56·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.520·59-s − 0.488·67-s − 0.237·71-s + 1.63·73-s + 0.675·79-s + 1.31·83-s + 0.211·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−12.99763039045799, −12.33016225938049, −12.02648281875856, −11.51654857419279, −11.18416937183549, −10.69867782392244, −10.16213122101403, −9.707589251911823, −9.224926622680326, −8.878218332744144, −8.127687984164392, −7.888606358879243, −7.382326897560924, −6.928273134920524, −6.378682699349237, −5.775744709896233, −5.225234468325810, −4.935474867060247, −4.412030425056754, −3.645458309576007, −3.364583509906286, −2.585307076123598, −1.960904154120774, −1.581374403056555, −0.7697513077710186, 0,
0.7697513077710186, 1.581374403056555, 1.960904154120774, 2.585307076123598, 3.364583509906286, 3.645458309576007, 4.412030425056754, 4.935474867060247, 5.225234468325810, 5.775744709896233, 6.378682699349237, 6.928273134920524, 7.382326897560924, 7.888606358879243, 8.127687984164392, 8.878218332744144, 9.224926622680326, 9.707589251911823, 10.16213122101403, 10.69867782392244, 11.18416937183549, 11.51654857419279, 12.02648281875856, 12.33016225938049, 12.99763039045799
