L(s) = 1 | + (0.228 + 1.39i)2-s + (−1.89 + 0.637i)4-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−1.32 − 2.49i)8-s + (0.5 − 1.32i)10-s + (2.59 − 1.5i)11-s + (−4.58 − 2.64i)13-s + (1.09 − 0.895i)14-s + (3.18 − 2.41i)16-s + 5.29·17-s − 5.29i·19-s + (1.96 + 0.395i)20-s + (2.68 + 3.28i)22-s + (−2.64 + 4.58i)23-s + ⋯ |
L(s) = 1 | + (0.161 + 0.986i)2-s + (−0.947 + 0.318i)4-s + (−0.387 − 0.223i)5-s + (−0.188 − 0.327i)7-s + (−0.467 − 0.883i)8-s + (0.158 − 0.418i)10-s + (0.783 − 0.452i)11-s + (−1.27 − 0.733i)13-s + (0.292 − 0.239i)14-s + (0.796 − 0.604i)16-s + 1.28·17-s − 1.21i·19-s + (0.438 + 0.0884i)20-s + (0.572 + 0.700i)22-s + (−0.551 + 0.955i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03379 - 0.202020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03379 - 0.202020i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.228 - 1.39i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.866 + 0.5i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.58 + 2.64i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 + (2.64 - 4.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.19 + 3i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.29iT - 37T^{2} \) |
| 41 | \( 1 + (-2.64 + 4.58i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 - 5.29i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.06 - 3.5i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.04434095723601956281899244804, −9.702698045553637246959621205574, −8.465117734542201187685203890712, −7.82583458664438973073561019997, −7.01437867385945576188875009757, −6.04438191707973425081932476809, −5.07352311713439518493927257055, −4.14023886830941533861024067631, −3.05039987885075078839393874774, −0.57752701139125380400912269644,
1.55695901593945557337494238349, 2.85683482669158158497628989621, 3.91009289835576112350472433324, 4.82776971574892314446011261974, 5.94353780139311158662624448762, 7.12058943482673258029478313115, 8.160367517784921241687165522828, 9.142333691834840584028504602160, 9.935386098582612311382821527343, 10.47532596944007304763770181972