| L(s) = 1 | − 7-s + 5·13-s + 6·17-s − 2·19-s + 3·23-s − 5·25-s − 3·29-s − 5·31-s − 4·37-s + 9·41-s + 43-s − 6·47-s + 49-s − 6·53-s − 9·59-s + 5·61-s + 13·67-s − 3·71-s + 2·73-s − 8·79-s + 3·83-s + 3·89-s − 5·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.38·13-s + 1.45·17-s − 0.458·19-s + 0.625·23-s − 25-s − 0.557·29-s − 0.898·31-s − 0.657·37-s + 1.40·41-s + 0.152·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.17·59-s + 0.640·61-s + 1.58·67-s − 0.356·71-s + 0.234·73-s − 0.900·79-s + 0.329·83-s + 0.317·89-s − 0.524·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.86591053998040, −13.22377036200220, −12.76510931433536, −12.55540650519046, −11.80229566853248, −11.33518459303666, −10.90856712581247, −10.46895649008724, −9.837329149266591, −9.402857916960388, −8.982352669262204, −8.333944851236827, −7.887849213356584, −7.441538035048489, −6.764247962378518, −6.272573971544685, −5.683940028908440, −5.455274357983122, −4.603672060824051, −3.929957949101250, −3.498219345526358, −3.085237256432090, −2.186784917931909, −1.530307159161417, −0.9244988253115035, 0,
0.9244988253115035, 1.530307159161417, 2.186784917931909, 3.085237256432090, 3.498219345526358, 3.929957949101250, 4.603672060824051, 5.455274357983122, 5.683940028908440, 6.272573971544685, 6.764247962378518, 7.441538035048489, 7.887849213356584, 8.333944851236827, 8.982352669262204, 9.402857916960388, 9.837329149266591, 10.46895649008724, 10.90856712581247, 11.33518459303666, 11.80229566853248, 12.55540650519046, 12.76510931433536, 13.22377036200220, 13.86591053998040
