| L(s) = 1 | − 2·7-s + 6·11-s + 2·13-s − 6·17-s + 4·19-s − 8·23-s − 8·31-s − 2·37-s + 6·41-s − 4·43-s − 4·47-s − 3·49-s + 6·53-s + 6·59-s − 6·61-s + 4·71-s − 12·73-s − 12·77-s − 8·79-s + 12·83-s − 14·89-s − 4·91-s − 8·97-s − 8·101-s + 6·103-s + 12·107-s + 2·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.80·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 1.43·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 0.824·53-s + 0.781·59-s − 0.768·61-s + 0.474·71-s − 1.40·73-s − 1.36·77-s − 0.900·79-s + 1.31·83-s − 1.48·89-s − 0.419·91-s − 0.812·97-s − 0.796·101-s + 0.591·103-s + 1.16·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.40105149003537460202689767130, −6.81881958726475072238658115521, −6.21870853936590950115599381464, −5.70247311601237723416335801798, −4.53034043099275947376332448716, −3.87735021616818969685702865169, −3.36909142000314021540051752147, −2.16723442732855924766083584253, −1.33599208770880145337476104754, 0,
1.33599208770880145337476104754, 2.16723442732855924766083584253, 3.36909142000314021540051752147, 3.87735021616818969685702865169, 4.53034043099275947376332448716, 5.70247311601237723416335801798, 6.21870853936590950115599381464, 6.81881958726475072238658115521, 7.40105149003537460202689767130
