| L(s) = 1 | + (−0.644 + 2.40i)2-s + (−2.98 + 0.325i)3-s + (−1.90 − 1.10i)4-s + (−4.49 + 2.18i)5-s + (1.13 − 7.38i)6-s + (0.0176 − 0.0658i)7-s + (−3.16 + 3.16i)8-s + (8.78 − 1.94i)9-s + (−2.34 − 12.2i)10-s + (4.62 + 8.00i)11-s + (6.05 + 2.66i)12-s + (3.38 + 12.6i)13-s + (0.146 + 0.0848i)14-s + (12.7 − 7.97i)15-s + (−9.97 − 17.2i)16-s + (13.6 + 13.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.322 + 1.20i)2-s + (−0.994 + 0.108i)3-s + (−0.477 − 0.275i)4-s + (−0.899 + 0.436i)5-s + (0.189 − 1.23i)6-s + (0.00251 − 0.00940i)7-s + (−0.395 + 0.395i)8-s + (0.976 − 0.216i)9-s + (−0.234 − 1.22i)10-s + (0.420 + 0.727i)11-s + (0.504 + 0.222i)12-s + (0.260 + 0.972i)13-s + (0.0104 + 0.00606i)14-s + (0.847 − 0.531i)15-s + (−0.623 − 1.08i)16-s + (0.800 + 0.800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0491108 + 0.564866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0491108 + 0.564866i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.98 - 0.325i)T \) |
| 5 | \( 1 + (4.49 - 2.18i)T \) |
| good | 2 | \( 1 + (0.644 - 2.40i)T + (-3.46 - 2i)T^{2} \) |
| 7 | \( 1 + (-0.0176 + 0.0658i)T + (-42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-4.62 - 8.00i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.38 - 12.6i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (-13.6 - 13.6i)T + 289iT^{2} \) |
| 19 | \( 1 - 5.80iT - 361T^{2} \) |
| 23 | \( 1 + (11.7 + 43.8i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-6.37 + 3.67i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (8.35 - 14.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-36.0 - 36.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (31.1 - 53.9i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-9.70 - 2.59i)T + (1.60e3 + 924.5i)T^{2} \) |
| 47 | \( 1 + (-11.9 + 44.6i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-28.0 + 28.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-16.3 - 9.44i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.69 - 4.66i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (47.9 - 12.8i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 56.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (54.7 - 54.7i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + (40.9 - 23.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-60.0 - 16.0i)T + (5.96e3 + 3.44e3i)T^{2} \) |
| 89 | \( 1 + 59.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-40.2 + 150. i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−16.41610485136487920329203869956, −15.22098025146395128718103721358, −14.47850108062775776690986966165, −12.35606805804342945422704255746, −11.55625432193573358919275907672, −10.14206147323265477743783319116, −8.385148029936179519696838238789, −7.06988424431806399276593217708, −6.23522115812561672606229575223, −4.39720143779429275875860143917,
0.795229857257298267408465620886, 3.66504858273279736086663974858, 5.64184156435829661821638535780, 7.54829395329739799993887171031, 9.291261165765907413481180295063, 10.64520839171645848795906758663, 11.55609163526893630908789483307, 12.18262667387813036371484044908, 13.33950194778751446339121887448, 15.43805320884605233817249170204
