| L(s) = 1 | − 2·11-s − 13-s + 8·17-s + 7·19-s − 6·23-s − 4·29-s + 8·31-s + 7·37-s + 2·41-s − 4·43-s − 12·47-s + 4·53-s − 12·59-s − 3·61-s − 9·67-s − 12·71-s − 9·73-s + 17·79-s − 4·83-s + 8·89-s + 11·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.277·13-s + 1.94·17-s + 1.60·19-s − 1.25·23-s − 0.742·29-s + 1.43·31-s + 1.15·37-s + 0.312·41-s − 0.609·43-s − 1.75·47-s + 0.549·53-s − 1.56·59-s − 0.384·61-s − 1.09·67-s − 1.42·71-s − 1.05·73-s + 1.91·79-s − 0.439·83-s + 0.847·89-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077983786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077983786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.31250098113443, −12.65103628114503, −12.08132572488260, −11.85962501589635, −11.50584985821747, −10.69930330287298, −10.24613374418951, −9.881820901406757, −9.506243696730177, −9.019461866767176, −8.023134467481017, −7.923777501627473, −7.641222189472773, −6.958577586329004, −6.180172078283831, −5.901195626802424, −5.262769825898643, −4.930279964517083, −4.210453205189924, −3.606227063483856, −2.908691626110111, −2.797870775365210, −1.668651265308271, −1.275492165127748, −0.4282235762889945,
0.4282235762889945, 1.275492165127748, 1.668651265308271, 2.797870775365210, 2.908691626110111, 3.606227063483856, 4.210453205189924, 4.930279964517083, 5.262769825898643, 5.901195626802424, 6.180172078283831, 6.958577586329004, 7.641222189472773, 7.923777501627473, 8.023134467481017, 9.019461866767176, 9.506243696730177, 9.881820901406757, 10.24613374418951, 10.69930330287298, 11.50584985821747, 11.85962501589635, 12.08132572488260, 12.65103628114503, 13.31250098113443
