| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 5·13-s − 14-s + 16-s − 17-s + 6·19-s + 20-s − 8·23-s − 4·25-s + 5·26-s − 28-s − 4·29-s − 6·31-s + 32-s − 34-s − 35-s − 5·37-s + 6·38-s + 40-s + 4·41-s − 43-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s + 0.223·20-s − 1.66·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 0.742·29-s − 1.07·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s − 0.821·37-s + 0.973·38-s + 0.158·40-s + 0.624·41-s − 0.152·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.22492393427609, −12.59265092750582, −12.22322523854321, −11.69559776739010, −11.20679685371315, −10.95398596745421, −10.24942529411363, −9.835062651660678, −9.462709455671916, −8.890874799440840, −8.347282241246401, −7.824464501756800, −7.343540949165245, −6.838656402048518, −6.195375093154924, −5.916929413540272, −5.472558072652839, −5.052569419253360, −4.155446933222635, −3.723599980481977, −3.541894899965361, −2.741770707414256, −2.054701049876586, −1.668356294945720, −0.9264478535996806, 0,
0.9264478535996806, 1.668356294945720, 2.054701049876586, 2.741770707414256, 3.541894899965361, 3.723599980481977, 4.155446933222635, 5.052569419253360, 5.472558072652839, 5.916929413540272, 6.195375093154924, 6.838656402048518, 7.343540949165245, 7.824464501756800, 8.347282241246401, 8.890874799440840, 9.462709455671916, 9.835062651660678, 10.24942529411363, 10.95398596745421, 11.20679685371315, 11.69559776739010, 12.22322523854321, 12.59265092750582, 13.22492393427609
