| L(s) = 1 | − 3·5-s + 4·11-s − 13-s + 17-s − 4·19-s − 6·23-s + 4·25-s + 3·29-s + 5·31-s + 8·37-s + 3·41-s − 8·43-s + 47-s − 12·55-s + 11·59-s + 14·61-s + 3·65-s + 2·67-s − 10·71-s − 4·73-s − 4·79-s − 11·83-s − 3·85-s − 4·89-s + 12·95-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.20·11-s − 0.277·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s + 0.468·41-s − 1.21·43-s + 0.145·47-s − 1.61·55-s + 1.43·59-s + 1.79·61-s + 0.372·65-s + 0.244·67-s − 1.18·71-s − 0.468·73-s − 0.450·79-s − 1.20·83-s − 0.325·85-s − 0.423·89-s + 1.23·95-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.81392810464485, −13.34602116535062, −12.62879048257856, −12.34449810908372, −11.76786990233208, −11.47761337196051, −11.19867363815957, −10.21240343921702, −10.07842813804712, −9.435777969088286, −8.733294856984598, −8.284663477077121, −8.078682257546353, −7.325408547791092, −6.905403336547974, −6.363721578583583, −5.897699973910121, −5.122165520549632, −4.362295575614130, −4.125211539664890, −3.732338867366812, −2.924712616359917, −2.343867361765340, −1.482296668648512, −0.7703961459600005, 0,
0.7703961459600005, 1.482296668648512, 2.343867361765340, 2.924712616359917, 3.732338867366812, 4.125211539664890, 4.362295575614130, 5.122165520549632, 5.897699973910121, 6.363721578583583, 6.905403336547974, 7.325408547791092, 8.078682257546353, 8.284663477077121, 8.733294856984598, 9.435777969088286, 10.07842813804712, 10.21240343921702, 11.19867363815957, 11.47761337196051, 11.76786990233208, 12.34449810908372, 12.62879048257856, 13.34602116535062, 13.81392810464485
