| L(s) = 1 | + (−0.265 − 1.71i)3-s + (−0.526 + 1.44i)5-s + (−1.90 − 0.335i)7-s + (−2.85 + 0.907i)9-s + (2.51 − 0.916i)11-s + (−2.60 + 2.18i)13-s + (2.61 + 0.517i)15-s + (−1.51 + 0.874i)17-s + (1.89 + 1.09i)19-s + (−0.0697 + 3.34i)21-s + (0.662 + 3.75i)23-s + (2.01 + 1.68i)25-s + (2.31 + 4.65i)27-s + (−3.29 + 3.92i)29-s + (−6.59 + 1.16i)31-s + ⋯ |
| L(s) = 1 | + (−0.153 − 0.988i)3-s + (−0.235 + 0.647i)5-s + (−0.718 − 0.126i)7-s + (−0.953 + 0.302i)9-s + (0.759 − 0.276i)11-s + (−0.721 + 0.605i)13-s + (0.675 + 0.133i)15-s + (−0.367 + 0.212i)17-s + (0.434 + 0.250i)19-s + (−0.0152 + 0.729i)21-s + (0.138 + 0.783i)23-s + (0.402 + 0.337i)25-s + (0.444 + 0.895i)27-s + (−0.611 + 0.728i)29-s + (−1.18 + 0.208i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582037 + 0.471984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582037 + 0.471984i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.265 + 1.71i)T \) |
| good | 5 | \( 1 + (0.526 - 1.44i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.90 + 0.335i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.51 + 0.916i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.60 - 2.18i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.51 - 0.874i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.89 - 1.09i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.662 - 3.75i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.29 - 3.92i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.59 - 1.16i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.25 - 9.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.08 + 3.67i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.33 - 6.40i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.45 + 8.27i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 2.87iT - 53T^{2} \) |
| 59 | \( 1 + (-6.12 - 2.22i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.808 - 4.58i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.65 - 4.35i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0248 - 0.0430i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.04 - 7.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.10 + 1.31i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (10.5 + 8.88i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (10.1 + 5.84i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.2 + 4.10i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.41482508614615999330464784445, −9.421352756811393270351255995814, −8.672304743465289449221763020841, −7.44672237047487382842341447636, −6.99494758924883801643061128744, −6.27837980745677402564190624179, −5.25852170322261869246096783333, −3.75878194809131591973304699592, −2.85330259210334897249403278036, −1.49172885488731663897551903893,
0.37277974986606029343791727459, 2.55828239612232031626585857290, 3.73254803176359347800677488019, 4.56235889202154626138057423473, 5.43118223230583678194047387113, 6.37270436394025672864980453587, 7.46067049563153926434622688111, 8.561848743310911443153607030575, 9.361632844297341917028100290510, 9.725293318283672693148130568907
