| L(s) = 1 | + (0.314 − 0.729i)2-s + (0.939 + 0.996i)4-s + (−0.127 − 2.18i)5-s + (−3.45 − 0.819i)7-s + (2.51 − 0.915i)8-s + (−1.63 − 0.595i)10-s + (−3.62 − 2.38i)11-s + (−6.02 − 0.703i)13-s + (−1.68 + 2.26i)14-s + (−0.0357 + 0.613i)16-s + (−1.52 + 1.27i)17-s + (−2.70 − 2.26i)19-s + (2.06 − 2.18i)20-s + (−2.88 + 1.89i)22-s + (5.36 − 1.27i)23-s + ⋯ |
| L(s) = 1 | + (0.222 − 0.515i)2-s + (0.469 + 0.498i)4-s + (−0.0570 − 0.978i)5-s + (−1.30 − 0.309i)7-s + (0.888 − 0.323i)8-s + (−0.517 − 0.188i)10-s + (−1.09 − 0.719i)11-s + (−1.67 − 0.195i)13-s + (−0.450 + 0.605i)14-s + (−0.00892 + 0.153i)16-s + (−0.369 + 0.309i)17-s + (−0.619 − 0.520i)19-s + (0.460 − 0.488i)20-s + (−0.614 + 0.404i)22-s + (1.11 − 0.265i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.140886 - 0.899936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140886 - 0.899936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.314 + 0.729i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.127 + 2.18i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (3.45 + 0.819i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (3.62 + 2.38i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (6.02 + 0.703i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (1.52 - 1.27i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 2.26i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-5.36 + 1.27i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (0.388 + 0.521i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-0.732 + 2.44i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.702 - 3.98i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (3.24 + 7.51i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-2.08 + 1.04i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (0.837 + 2.79i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-1.34 - 2.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.21 + 4.08i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (1.80 - 1.91i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.18 + 2.93i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (1.04 + 0.379i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.97 - 3.26i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.09 + 2.54i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-4.95 + 11.4i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-3.58 + 1.30i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.611 - 10.5i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−10.21450536099525978329964702054, −9.204708743066651362756839273109, −8.322308571200282669001686448083, −7.35863994241186310255682536199, −6.61136503574054400457465392513, −5.28469419732402328779678609045, −4.40657479349900212347453655340, −3.18178006598292869838469594011, −2.40619670920102999833980519653, −0.38899934926501208227159025065,
2.33414324887385969238480216784, 2.96322587611417024862081591904, 4.66533843278008237218816775665, 5.52717010756627745104927725534, 6.64382356092085503419045319708, 7.00604741660430978774116509179, 7.77799323243510635337498091895, 9.316240127088175098163560967137, 10.06910085889576865084261592048, 10.53539336619023305429945034361
