| L(s) = 1 | + (0.291 − 2.98i)3-s + 2.23i·5-s − 4.46·7-s + (−8.82 − 1.74i)9-s − 17.8i·11-s + 11.0·13-s + (6.67 + 0.652i)15-s + 0.794i·17-s + 26.5·19-s + (−1.30 + 13.3i)21-s − 14.9i·23-s − 5.00·25-s + (−7.77 + 25.8i)27-s − 5.58i·29-s − 53.1·31-s + ⋯ |
| L(s) = 1 | + (0.0972 − 0.995i)3-s + 0.447i·5-s − 0.637·7-s + (−0.981 − 0.193i)9-s − 1.62i·11-s + 0.846·13-s + (0.445 + 0.0434i)15-s + 0.0467i·17-s + 1.39·19-s + (−0.0619 + 0.634i)21-s − 0.649i·23-s − 0.200·25-s + (−0.287 + 0.957i)27-s − 0.192i·29-s − 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7906956891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7906956891\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.291 + 2.98i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 4.46T + 49T^{2} \) |
| 11 | \( 1 + 17.8iT - 121T^{2} \) |
| 13 | \( 1 - 11.0T + 169T^{2} \) |
| 17 | \( 1 - 0.794iT - 289T^{2} \) |
| 19 | \( 1 - 26.5T + 361T^{2} \) |
| 23 | \( 1 + 14.9iT - 529T^{2} \) |
| 29 | \( 1 + 5.58iT - 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 + 51.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 97.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.02T + 4.48e3T^{2} \) |
| 71 | \( 1 - 57.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 31.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 2.16T + 6.24e3T^{2} \) |
| 83 | \( 1 + 13.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 173. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.6T + 9.40e3T^{2} \) |
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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
−9.201872454425173998216305548043, −8.500994032685757298216708684972, −7.71382230711252723627486590627, −6.73972947372571278497907676274, −6.13734801805896619526393456648, −5.35981101349232979328977589551, −3.43787398599680204296039846405, −3.11034255837504595253132317020, −1.53023045162711131337625681143, −0.24527777706336231126112865406,
1.71008926201489548928966182916, 3.22226461469635449414046701055, 3.97382354035986665690668136251, 5.03940091934524033492258565094, 5.67025557639794909464512855775, 6.95080447921851817822264625650, 7.75879489839944908587736331302, 9.020780599094533896287136582223, 9.332592065075429465071405839232, 10.14724559556455592148873514616
