| L(s) = 1 | − 4·4-s + (−4.5 − 2.17i)5-s + 13.0i·7-s − 9·9-s − 3·11-s + 16·16-s − 30.5i·17-s − 19·19-s + (18 + 8.71i)20-s + 34.8i·23-s + (15.5 + 19.6i)25-s − 52.3i·28-s + (28.5 − 58.8i)35-s + 36·36-s + 13.0i·43-s + 12·44-s + ⋯ |
| L(s) = 1 | − 4-s + (−0.900 − 0.435i)5-s + 1.86i·7-s − 9-s − 0.272·11-s + 16-s − 1.79i·17-s − 19-s + (0.900 + 0.435i)20-s + 1.51i·23-s + (0.619 + 0.784i)25-s − 1.86i·28-s + (0.814 − 1.68i)35-s + 36-s + 0.304i·43-s + 0.272·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0622267 + 0.271240i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0622267 + 0.271240i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.5 + 2.17i)T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 4T^{2} \) |
| 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 - 13.0iT - 49T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 30.5iT - 289T^{2} \) |
| 23 | \( 1 - 34.8iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 56.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 103T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 143. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 139. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−14.25157432388585649567356766633, −13.01776673187754855297668165522, −12.05242044382348750409410876140, −11.37248360954775766794190211647, −9.366359671506851662218323206288, −8.826988218231914666856229004933, −7.85984485776406111952369264797, −5.72380528146169868913489511550, −4.84872842311756052007724712634, −3.00563147960642771695247642133,
0.21694977185179685344681100031, 3.63104032948032955842137086463, 4.48375704412604770621758707447, 6.43275358082324250978925055312, 7.87179491560391212294171419485, 8.578296919043166736706408665356, 10.43740418812664159808126451517, 10.75816691137317208333703983399, 12.39799822921906044180539089534, 13.39187719624353869399446367513
