| L(s) = 1 | + (−0.520 + 1.31i)2-s + (−3.05 − 1.26i)3-s + (−1.45 − 1.36i)4-s + (0.290 − 0.147i)5-s + (3.25 − 3.36i)6-s + (1.49 + 2.43i)7-s + (2.56 − 1.20i)8-s + (5.63 + 5.63i)9-s + (0.0432 + 0.458i)10-s + (3.50 + 0.275i)11-s + (2.72 + 6.03i)12-s + (−0.883 − 3.67i)13-s + (−3.97 + 0.692i)14-s + (−1.07 + 0.0845i)15-s + (0.246 + 3.99i)16-s + (1.39 − 1.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.368 + 0.929i)2-s + (−1.76 − 0.731i)3-s + (−0.728 − 0.684i)4-s + (0.129 − 0.0660i)5-s + (1.33 − 1.37i)6-s + (0.563 + 0.919i)7-s + (0.905 − 0.425i)8-s + (1.87 + 1.87i)9-s + (0.0136 + 0.144i)10-s + (1.05 + 0.0831i)11-s + (0.785 + 1.74i)12-s + (−0.244 − 1.02i)13-s + (−1.06 + 0.185i)14-s + (−0.277 + 0.0218i)15-s + (0.0617 + 0.998i)16-s + (0.339 − 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.560049 + 0.202586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.560049 + 0.202586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.520 - 1.31i)T \) |
| 41 | \( 1 + (-4.73 + 4.31i)T \) |
| good | 3 | \( 1 + (3.05 + 1.26i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.290 + 0.147i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.49 - 2.43i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-3.50 - 0.275i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.883 + 3.67i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-1.39 + 1.19i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.69 - 0.406i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-3.85 + 2.80i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.55 - 5.59i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-1.45 - 4.48i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.02 - 3.15i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.501 + 0.0794i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (0.324 - 0.529i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-4.94 + 5.78i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (3.00 + 4.13i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.64 - 0.735i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.0526 - 0.668i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.641 + 8.15i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (9.60 - 9.60i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.664 + 1.60i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 16.4iT - 83T^{2} \) |
| 89 | \( 1 + (-1.48 + 0.908i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.594 + 7.55i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.71618066223460324145077368740, −12.03990512363041529914279900240, −11.02028210752325175610937696236, −10.00893562225311125681821560305, −8.644845145884982289520214210268, −7.41417414010626101848768461405, −6.52889096751345710670694534795, −5.52062050495263560574360522805, −4.92038789410970293262860891729, −1.22656673541460844242776939400,
1.10283124093274656823374183464, 4.03764219555029609733273249341, 4.62283819865769332380448244794, 6.16796543229051659649478272357, 7.41416239650132369928936143519, 9.228318791439125469929030709222, 10.02718319023561214704148527834, 10.83325623738787035503993751342, 11.69283799660674566310450163245, 12.01698345645501881181863391069
