| L(s) = 1 | + 7-s − 4·11-s + 3·13-s + 7·17-s − 6·19-s − 9·23-s + 3·29-s + 7·31-s + 10·37-s + 41-s + 13·43-s + 2·47-s + 49-s + 53-s + 11·59-s − 13·61-s + 8·71-s + 8·73-s − 4·77-s − 4·79-s + 7·83-s + 14·89-s + 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 0.832·13-s + 1.69·17-s − 1.37·19-s − 1.87·23-s + 0.557·29-s + 1.25·31-s + 1.64·37-s + 0.156·41-s + 1.98·43-s + 0.291·47-s + 1/7·49-s + 0.137·53-s + 1.43·59-s − 1.66·61-s + 0.949·71-s + 0.936·73-s − 0.455·77-s − 0.450·79-s + 0.768·83-s + 1.48·89-s + 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.800357515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800357515\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.76751724272259, −13.31127337540656, −12.66176506612781, −12.36426250870125, −11.86584098378191, −11.23702203089293, −10.75739947560159, −10.24546933235617, −10.02481115602940, −9.323556785265545, −8.633569861839650, −8.112593585574785, −7.857197109458434, −7.477354638783344, −6.481042729627550, −5.990728637823967, −5.801481256646129, −4.979971714364845, −4.420116951481485, −3.927636207683456, −3.273290862413292, −2.439737222408215, −2.187521019445872, −1.123072027134165, −0.5882560773375481,
0.5882560773375481, 1.123072027134165, 2.187521019445872, 2.439737222408215, 3.273290862413292, 3.927636207683456, 4.420116951481485, 4.979971714364845, 5.801481256646129, 5.990728637823967, 6.481042729627550, 7.477354638783344, 7.857197109458434, 8.112593585574785, 8.633569861839650, 9.323556785265545, 10.02481115602940, 10.24546933235617, 10.75739947560159, 11.23702203089293, 11.86584098378191, 12.36426250870125, 12.66176506612781, 13.31127337540656, 13.76751724272259
