| L(s) = 1 | − 5-s − 2·7-s − 6·11-s + 2·13-s − 2·19-s + 6·23-s + 25-s − 6·29-s − 8·31-s + 2·35-s + 2·37-s + 6·41-s − 43-s + 6·47-s − 3·49-s + 6·53-s + 6·55-s + 6·59-s + 8·61-s − 2·65-s + 4·67-s − 4·73-s + 12·77-s − 8·79-s + 6·89-s − 4·91-s + 2·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.80·11-s + 0.554·13-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 0.809·55-s + 0.781·59-s + 1.02·61-s − 0.248·65-s + 0.488·67-s − 0.468·73-s + 1.36·77-s − 0.900·79-s + 0.635·89-s − 0.419·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.47734942465619, −14.86728009274820, −14.48815796228183, −13.46041971337806, −13.26217469339236, −12.67258547606533, −12.53928823683128, −11.38969436441714, −11.12548247828596, −10.62629464482050, −10.04003187539832, −9.446010323675636, −8.782139713863848, −8.375735167512915, −7.525493778863526, −7.310395283989753, −6.602929919840055, −5.697877701920944, −5.488138911062562, −4.672905530878900, −3.909705432884918, −3.331554877928333, −2.680536924746885, −2.020629905321123, −0.8025077502696198, 0,
0.8025077502696198, 2.020629905321123, 2.680536924746885, 3.331554877928333, 3.909705432884918, 4.672905530878900, 5.488138911062562, 5.697877701920944, 6.602929919840055, 7.310395283989753, 7.525493778863526, 8.375735167512915, 8.782139713863848, 9.446010323675636, 10.04003187539832, 10.62629464482050, 11.12548247828596, 11.38969436441714, 12.53928823683128, 12.67258547606533, 13.26217469339236, 13.46041971337806, 14.48815796228183, 14.86728009274820, 15.47734942465619
