| L(s) = 1 | + (−0.0528 − 0.367i)2-s + (0.909 + 0.415i)3-s + (1.78 − 0.524i)4-s + (1.16 − 1.34i)5-s + (0.104 − 0.356i)6-s + (−2.06 + 1.65i)7-s + (−0.596 − 1.30i)8-s + (0.654 + 0.755i)9-s + (−0.557 − 0.358i)10-s + (2.52 + 0.363i)11-s + (1.84 + 0.264i)12-s + (1.29 − 2.02i)13-s + (0.716 + 0.673i)14-s + (1.62 − 0.741i)15-s + (2.68 − 1.72i)16-s + (2.41 + 0.710i)17-s + ⋯ |
| L(s) = 1 | + (−0.0373 − 0.260i)2-s + (0.525 + 0.239i)3-s + (0.893 − 0.262i)4-s + (0.522 − 0.603i)5-s + (0.0427 − 0.145i)6-s + (−0.781 + 0.623i)7-s + (−0.210 − 0.461i)8-s + (0.218 + 0.251i)9-s + (−0.176 − 0.113i)10-s + (0.762 + 0.109i)11-s + (0.532 + 0.0764i)12-s + (0.360 − 0.560i)13-s + (0.191 + 0.179i)14-s + (0.419 − 0.191i)15-s + (0.671 − 0.431i)16-s + (0.586 + 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02228 - 0.531192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02228 - 0.531192i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (2.06 - 1.65i)T \) |
| 23 | \( 1 + (-2.24 + 4.23i)T \) |
| good | 2 | \( 1 + (0.0528 + 0.367i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.16 + 1.34i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.52 - 0.363i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 2.02i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.41 - 0.710i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (4.98 - 1.46i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (0.540 + 0.158i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (6.42 - 2.93i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.723 + 0.626i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.680 + 0.589i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (6.82 + 3.11i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 4.33iT - 47T^{2} \) |
| 53 | \( 1 + (-4.74 - 7.37i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.20 + 6.54i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.59 - 10.0i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (9.43 - 1.35i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.808 - 5.62i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.02 + 6.90i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (6.76 - 10.5i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.7 + 12.3i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.99 - 6.55i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.0261 + 0.0302i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.70472459829630564869881739015, −10.05162119096395334806926999289, −9.151765000253744182756409858337, −8.518799607681731421786427556367, −7.14386807638034318263710671460, −6.20484354353739141724129792395, −5.41164131781119613742386261932, −3.81251626085804193442110574030, −2.73553587259878981210943369379, −1.52591531228277309462688220497,
1.78212560517362665390880930478, 3.01756260012489939252642541524, 3.92329802657789462765493853314, 5.84576305674520324054574442444, 6.73713668043879786590492287599, 7.06753577730492976831685018120, 8.257405978398494425390046817625, 9.281585276935294630119453403454, 10.14704029437863012231392616095, 11.03819552249593352216949104373
