| L(s) = 1 | + (−1.20 − 1.01i)2-s + (0.0853 + 0.483i)4-s + (−1.57 − 0.574i)5-s + (0.482 − 2.73i)7-s + (−1.19 + 2.06i)8-s + (1.32 + 2.29i)10-s + (3.90 − 1.41i)11-s + (5.26 − 4.41i)13-s + (−3.36 + 2.81i)14-s + (4.45 − 1.62i)16-s + (−0.488 − 0.845i)17-s + (−1.34 + 2.32i)19-s + (0.143 − 0.812i)20-s + (−6.15 − 2.24i)22-s + (0.280 + 1.58i)23-s + ⋯ |
| L(s) = 1 | + (−0.854 − 0.717i)2-s + (0.0426 + 0.241i)4-s + (−0.705 − 0.256i)5-s + (0.182 − 1.03i)7-s + (−0.420 + 0.729i)8-s + (0.418 + 0.725i)10-s + (1.17 − 0.428i)11-s + (1.46 − 1.22i)13-s + (−0.898 + 0.753i)14-s + (1.11 − 0.405i)16-s + (−0.118 − 0.205i)17-s + (−0.308 + 0.533i)19-s + (0.0320 − 0.181i)20-s + (−1.31 − 0.477i)22-s + (0.0584 + 0.331i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0422477 - 0.725365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0422477 - 0.725365i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.20 + 1.01i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.57 + 0.574i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.482 + 2.73i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.90 + 1.41i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-5.26 + 4.41i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.488 + 0.845i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.34 - 2.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.280 - 1.58i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.30 + 5.28i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.181 - 1.02i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.654 - 1.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.71 - 3.11i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (9.24 - 3.36i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 12.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + (8.50 + 3.09i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.223 - 1.26i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.55 - 2.98i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.81 + 4.87i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.28 + 3.95i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.56 - 2.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.41 - 3.70i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.27 - 3.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.05 - 2.93i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.14425897206466151146913069112, −9.158267888098045476364634723347, −8.307234144468800009185813332917, −7.84553910966044299791240780324, −6.50924944243303140041122981541, −5.52732809607248920608450490651, −4.04871908832715867964025669570, −3.40211153515848667042011619857, −1.54257029198449565021042083506, −0.55188702552386794945113662783,
1.66335153226098011318739431585, 3.46817225966419530066137686943, 4.27363378517147678842032016355, 5.87191026458504705772127091847, 6.65474325554084699693365006806, 7.30861404176569656139226846285, 8.444786517423694500488798085351, 8.933958650059561642133270557834, 9.431380053924878513985772945624, 10.83783819240141506237646202431
