| L(s) = 1 | + (−1.32 − 1.11i)2-s + (0.173 + 0.984i)4-s + (−1.62 − 0.592i)5-s + (0.347 − 1.96i)7-s + (−0.866 + 1.5i)8-s + (1.49 + 2.59i)10-s + (−3.25 + 1.18i)11-s + (−0.766 + 0.642i)13-s + (−2.65 + 2.22i)14-s + (4.69 − 1.71i)16-s + (2.59 + 4.5i)17-s + (−1 + 1.73i)19-s + (0.300 − 1.70i)20-s + (5.63 + 2.05i)22-s + (−0.601 − 3.41i)23-s + ⋯ |
| L(s) = 1 | + (−0.938 − 0.787i)2-s + (0.0868 + 0.492i)4-s + (−0.727 − 0.264i)5-s + (0.131 − 0.744i)7-s + (−0.306 + 0.530i)8-s + (0.474 + 0.821i)10-s + (−0.981 + 0.357i)11-s + (−0.212 + 0.178i)13-s + (−0.709 + 0.595i)14-s + (1.17 − 0.427i)16-s + (0.630 + 1.09i)17-s + (−0.229 + 0.397i)19-s + (0.0672 − 0.381i)20-s + (1.20 + 0.437i)22-s + (−0.125 − 0.711i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.402504 + 0.0953953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402504 + 0.0953953i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.32 + 1.11i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.62 + 0.592i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.347 + 1.96i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.25 - 1.18i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.601 + 3.41i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.32 + 1.11i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.38 - 7.87i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.30 + 4.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.87 - 0.684i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.20 - 6.82i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-13.0 - 4.73i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.21 - 6.89i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.66 - 6.42i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.19 + 9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 1.28i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-10.6 - 8.90i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.59 - 4.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.87 - 0.684i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.41160870556686193075326022784, −9.898724052846833132139568021067, −8.698364501282562822141936219527, −8.067965565226054716729837797395, −7.43099349340776597879193163614, −6.04379869952827258845493667054, −4.83761830684622737736359246003, −3.81727831172371027334532033561, −2.50794967097364466926721323671, −1.14808522248618584994817858270,
0.33661693654214458371783424770, 2.59480105021525445966175096067, 3.71355973512997442452007726470, 5.21565396152939948687550191200, 6.01409831255299123634591723961, 7.28301308714599193714133452798, 7.69465843364561131954413711575, 8.435833160099019833627883981993, 9.350638208343330914954523855052, 9.991521626495780893517734415262
