| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s − 2·11-s − 2·13-s + 2·14-s + 16-s + 4·19-s − 20-s − 2·22-s − 4·23-s + 25-s − 2·26-s + 2·28-s + 6·29-s + 32-s − 2·35-s − 10·37-s + 4·38-s − 40-s − 2·41-s − 4·43-s − 2·44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.176·32-s − 0.338·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.301·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.18564230281215, −15.52567076295338, −15.40903366432007, −14.48465989815343, −14.16078475460205, −13.69103087665401, −12.96260957143844, −12.30071000146735, −11.96708346833292, −11.36862924982214, −10.85025656336296, −10.10297640851464, −9.722191730052369, −8.639802845822832, −8.143336125060609, −7.659417814159381, −6.970540362724435, −6.389904543966837, −5.406175088036657, −5.043656778014709, −4.484098034262486, −3.604970385015700, −3.006544565824569, −2.142708620673343, −1.330513196696605, 0,
1.330513196696605, 2.142708620673343, 3.006544565824569, 3.604970385015700, 4.484098034262486, 5.043656778014709, 5.406175088036657, 6.389904543966837, 6.970540362724435, 7.659417814159381, 8.143336125060609, 8.639802845822832, 9.722191730052369, 10.10297640851464, 10.85025656336296, 11.36862924982214, 11.96708346833292, 12.30071000146735, 12.96260957143844, 13.69103087665401, 14.16078475460205, 14.48465989815343, 15.40903366432007, 15.52567076295338, 16.18564230281215
