| L(s) = 1 | + (0.494 + 1.52i)2-s + (2.08 − 0.442i)3-s + (−0.453 + 0.329i)4-s + (−0.5 + 0.866i)5-s + (1.70 + 2.95i)6-s + (−3.40 − 1.51i)7-s + (1.86 + 1.35i)8-s + (1.40 − 0.623i)9-s + (−1.56 − 0.332i)10-s + (−0.153 − 1.45i)11-s + (−0.799 + 0.887i)12-s + (0.284 + 0.315i)13-s + (0.622 − 5.92i)14-s + (−0.657 + 2.02i)15-s + (−1.48 + 4.57i)16-s + (0.653 − 6.22i)17-s + ⋯ |
| L(s) = 1 | + (0.349 + 1.07i)2-s + (1.20 − 0.255i)3-s + (−0.226 + 0.164i)4-s + (−0.223 + 0.387i)5-s + (0.695 + 1.20i)6-s + (−1.28 − 0.572i)7-s + (0.658 + 0.478i)8-s + (0.466 − 0.207i)9-s + (−0.495 − 0.105i)10-s + (−0.0462 − 0.440i)11-s + (−0.230 + 0.256i)12-s + (0.0788 + 0.0875i)13-s + (0.166 − 1.58i)14-s + (−0.169 + 0.522i)15-s + (−0.371 + 1.14i)16-s + (0.158 − 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48546 + 0.867807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48546 + 0.867807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-2.82 - 4.79i)T \) |
| good | 2 | \( 1 + (-0.494 - 1.52i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-2.08 + 0.442i)T + (2.74 - 1.22i)T^{2} \) |
| 7 | \( 1 + (3.40 + 1.51i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.153 + 1.45i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.284 - 0.315i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.653 + 6.22i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.73 - 4.15i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.817 + 0.593i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.722 + 2.22i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.13 + 3.69i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.92 - 1.89i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-7.62 + 8.46i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.40 - 4.30i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.49 - 2.44i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-6.18 + 1.31i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + (7.53 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.17 - 1.85i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (0.538 + 5.12i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.75 - 16.7i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.89 - 1.25i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-6.36 + 4.62i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.782 + 0.568i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.67042922590396932729543459997, −12.57597331496987331567237941869, −11.00282223529975606620512555778, −9.865258963596400906154753458013, −8.694355627432321931287967321108, −7.62443371239540688125810229555, −6.92430038373631545378828047554, −5.84961903286818934608894786214, −3.99556438355503689435505331509, −2.71640323693753942336019081901,
2.28017708386296571945735198236, 3.31831283263725858630066277414, 4.29568720597180068533756000559, 6.29053109466871100246795333288, 7.82063381552704475356946925042, 8.967095066868300970116349331125, 9.717186479397205969173412192201, 10.73583508550701760538338956047, 12.02902000401257742450449997072, 12.91991958177828258387989402701
