| L(s) = 1 | − 5-s + 6·13-s + 2·17-s − 2·19-s − 6·23-s + 25-s − 6·29-s + 4·31-s + 8·37-s + 8·41-s − 43-s − 6·47-s − 7·49-s − 6·53-s + 4·59-s + 14·61-s − 6·65-s + 4·67-s + 8·71-s − 4·73-s − 12·79-s − 2·83-s − 2·85-s − 14·89-s + 2·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.66·13-s + 0.485·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.31·37-s + 1.24·41-s − 0.152·43-s − 0.875·47-s − 49-s − 0.824·53-s + 0.520·59-s + 1.79·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.468·73-s − 1.35·79-s − 0.219·83-s − 0.216·85-s − 1.48·89-s + 0.205·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.92188448503515, −13.10073873117531, −12.89998299178907, −12.45834196033508, −11.64656111428865, −11.31500378008132, −11.10529860659818, −10.41567770164938, −9.729968507490039, −9.607483196272645, −8.689786975992405, −8.359988266850911, −7.956900606537439, −7.484819186669703, −6.681458505199553, −6.318646931547282, −5.783215255867967, −5.340652980632795, −4.417467674733085, −4.080101674078547, −3.592011528329693, −2.971289211559745, −2.233800209611962, −1.504496335981810, −0.8943912991130857, 0,
0.8943912991130857, 1.504496335981810, 2.233800209611962, 2.971289211559745, 3.592011528329693, 4.080101674078547, 4.417467674733085, 5.340652980632795, 5.783215255867967, 6.318646931547282, 6.681458505199553, 7.484819186669703, 7.956900606537439, 8.359988266850911, 8.689786975992405, 9.607483196272645, 9.729968507490039, 10.41567770164938, 11.10529860659818, 11.31500378008132, 11.64656111428865, 12.45834196033508, 12.89998299178907, 13.10073873117531, 13.92188448503515
