| L(s) = 1 | − 0.200i·5-s − 8.51·7-s + 7.55i·11-s − 16.3·13-s − 6.02i·17-s + 0.379·19-s + 28.1i·23-s + 24.9·25-s + 41.4i·29-s + 13.5·31-s + 1.70i·35-s + 4.52·37-s + 76.7i·41-s − 1.74·43-s − 76.5i·47-s + ⋯ |
| L(s) = 1 | − 0.0401i·5-s − 1.21·7-s + 0.686i·11-s − 1.26·13-s − 0.354i·17-s + 0.0199·19-s + 1.22i·23-s + 0.998·25-s + 1.42i·29-s + 0.437·31-s + 0.0488i·35-s + 0.122·37-s + 1.87i·41-s − 0.0405·43-s − 1.62i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6311021622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6311021622\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.200iT - 25T^{2} \) |
| 7 | \( 1 + 8.51T + 49T^{2} \) |
| 11 | \( 1 - 7.55iT - 121T^{2} \) |
| 13 | \( 1 + 16.3T + 169T^{2} \) |
| 17 | \( 1 + 6.02iT - 289T^{2} \) |
| 19 | \( 1 - 0.379T + 361T^{2} \) |
| 23 | \( 1 - 28.1iT - 529T^{2} \) |
| 29 | \( 1 - 41.4iT - 841T^{2} \) |
| 31 | \( 1 - 13.5T + 961T^{2} \) |
| 37 | \( 1 - 4.52T + 1.36e3T^{2} \) |
| 41 | \( 1 - 76.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.74T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 85.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 18.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 37.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 44.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 84.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 120.T + 9.40e3T^{2} \) |
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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
−8.483093147200243264825908387481, −7.35369380455471492481006881372, −7.04270772065680800878895491023, −6.22039997252076308384535503123, −5.18882229998049341322708100042, −4.62607976825231070425399316517, −3.39007766099715948196717180085, −2.82850249870745643726319541557, −1.62897847238590431504934472251, −0.18908509851371200287676566606,
0.77936655187668260281845943660, 2.43538095102053208829811906172, 2.97718847146528206830138327717, 4.04354673156307756917378405291, 4.83720067907355642616373863130, 5.94313230812583462168422581007, 6.38903652736885691102691045777, 7.24704230259810366821947535426, 7.965668917271367040385575027685, 8.949116408031888761600048390551
