| L(s) = 1 | + (1.36 + 0.380i)2-s + (0.0120 + 0.00727i)4-s + (2.11 − 0.0820i)5-s + (1.64 + 0.257i)7-s + (−1.93 − 2.05i)8-s + (2.92 + 0.692i)10-s + (0.699 + 5.12i)11-s + (3.44 − 5.02i)13-s + (2.15 + 0.979i)14-s + (−1.87 − 3.56i)16-s + (1.16 − 0.584i)17-s + (0.0591 + 1.01i)19-s + (0.0260 + 0.0143i)20-s + (−0.992 + 7.26i)22-s + (0.0433 − 0.112i)23-s + ⋯ |
| L(s) = 1 | + (0.966 + 0.269i)2-s + (0.00602 + 0.00363i)4-s + (0.945 − 0.0366i)5-s + (0.623 + 0.0974i)7-s + (−0.683 − 0.724i)8-s + (0.924 + 0.219i)10-s + (0.210 + 1.54i)11-s + (0.954 − 1.39i)13-s + (0.576 + 0.261i)14-s + (−0.469 − 0.890i)16-s + (0.282 − 0.141i)17-s + (0.0135 + 0.232i)19-s + (0.00583 + 0.00321i)20-s + (−0.211 + 1.54i)22-s + (0.00904 − 0.0234i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.83278 + 0.147024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.83278 + 0.147024i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.36 - 0.380i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-2.11 + 0.0820i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.257i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.699 - 5.12i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 5.02i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 0.584i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.0591 - 1.01i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (-0.0433 + 0.112i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-2.48 - 1.77i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 2.74i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-4.61 - 6.20i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.596 + 2.31i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.38 - 3.07i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (7.20 + 8.25i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.302 + 1.71i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (8.73 - 3.56i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-1.96 + 1.18i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (9.88 - 7.06i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (1.31 + 4.38i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.13 - 0.979i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (10.9 - 10.7i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.80 - 14.7i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-3.04 + 10.1i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (11.8 + 0.461i)T + (96.7 + 7.51i)T^{2} \) |
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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
−10.09504879826261998666983254670, −9.852802907059565724093306591646, −8.692332105998784375828134228504, −7.70019180928350329042557853439, −6.55734655915159953215818217359, −5.77079186657707146692449167857, −5.06163840557399445798450866158, −4.19690751434600583355204628052, −2.90409209804998593140570658582, −1.44720301649706629870029114812,
1.54873454628582567783102743406, 2.88564513541742550320533414536, 3.94454421320081145970255302576, 4.83225893913001816073900910699, 5.99714914006987975733988404230, 6.25618967102377057000239600094, 7.918359897796816622126148481556, 8.812084510257686871409165928186, 9.345369668270393855185820714529, 10.71065024910838268736974249716
