| L(s) = 1 | + (0.492 + 1.35i)2-s + (−0.333 + 1.69i)3-s + (−0.0577 + 0.0484i)4-s + (0.440 + 2.49i)5-s + (−2.46 + 0.386i)6-s + (−1.60 − 2.10i)7-s + (2.40 + 1.38i)8-s + (−2.77 − 1.13i)9-s + (−3.16 + 1.82i)10-s + (2.06 + 0.364i)11-s + (−0.0631 − 0.114i)12-s + (0.0331 − 0.0909i)13-s + (2.05 − 3.21i)14-s + (−4.39 − 0.0834i)15-s + (−0.719 + 4.08i)16-s + (−3.94 − 6.82i)17-s + ⋯ |
| L(s) = 1 | + (0.348 + 0.957i)2-s + (−0.192 + 0.981i)3-s + (−0.0288 + 0.0242i)4-s + (0.196 + 1.11i)5-s + (−1.00 + 0.157i)6-s + (−0.608 − 0.793i)7-s + (0.848 + 0.490i)8-s + (−0.926 − 0.377i)9-s + (−1.00 + 0.577i)10-s + (0.623 + 0.110i)11-s + (−0.0182 − 0.0330i)12-s + (0.00918 − 0.0252i)13-s + (0.547 − 0.858i)14-s + (−1.13 − 0.0215i)15-s + (−0.179 + 1.02i)16-s + (−0.956 − 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.584227 + 1.27785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.584227 + 1.27785i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.333 - 1.69i)T \) |
| 7 | \( 1 + (1.60 + 2.10i)T \) |
| good | 2 | \( 1 + (-0.492 - 1.35i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.440 - 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 0.364i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.0331 + 0.0909i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.94 + 6.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.53 - 0.887i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.82 - 4.56i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.573 - 1.57i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.916 - 1.09i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.88 + 3.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.60 - 0.946i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 5.87i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.23 - 2.71i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + (-1.28 - 7.27i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.80 + 9.30i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.6 + 3.86i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (11.9 - 6.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (12.0 + 6.94i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 - 2.11i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.67 - 3.15i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.28 + 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.0 + 2.65i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.45428970673571098171268895140, −11.59093890495967999819763802723, −10.88060808179168759287548990988, −10.05155332377496731897548233442, −9.069174957220106259662739628676, −7.24164759648745404975100067909, −6.78448336330722845634038943257, −5.62942806217587960525408338032, −4.40515264043686539503688396227, −3.04833762327746762751464674276,
1.39677344465967811263909321627, 2.71837098208884301031687567896, 4.38946886351229764960704945480, 5.85367826902358029144156670456, 6.86130321077056000418883745979, 8.382432776173260968902230823705, 9.101920093938881676653298415648, 10.55870101146310671073177997688, 11.63105103120326537060935225904, 12.32284866247722638194597453967
