| L(s) = 1 | + (−6.47 + 4.70i)2-s + (66.2 + 48.1i)3-s + (19.7 − 60.8i)4-s + 520.·5-s − 655.·6-s + (−313. + 966. i)7-s + (158. + 486. i)8-s + (1.39e3 + 4.29e3i)9-s + (−3.36e3 + 2.44e3i)10-s + (1.81e3 − 5.59e3i)11-s + (4.24e3 − 3.08e3i)12-s + (6.72e3 + 4.88e3i)13-s + (−2.51e3 − 7.72e3i)14-s + (3.44e4 + 2.50e4i)15-s + (−3.31e3 − 2.40e3i)16-s + (−9.41e3 − 2.89e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (1.41 + 1.02i)3-s + (0.154 − 0.475i)4-s + 1.86·5-s − 1.23·6-s + (−0.345 + 1.06i)7-s + (0.109 + 0.336i)8-s + (0.638 + 1.96i)9-s + (−1.06 + 0.773i)10-s + (0.411 − 1.26i)11-s + (0.708 − 0.514i)12-s + (0.849 + 0.617i)13-s + (−0.244 − 0.752i)14-s + (2.63 + 1.91i)15-s + (−0.202 − 0.146i)16-s + (−0.464 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0365 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0365 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.17977 + 2.26097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17977 + 2.26097i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 - 4.70i)T \) |
| 31 | \( 1 + (1.58e5 + 4.78e4i)T \) |
| good | 3 | \( 1 + (-66.2 - 48.1i)T + (675. + 2.07e3i)T^{2} \) |
| 5 | \( 1 - 520.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (313. - 966. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 11 | \( 1 + (-1.81e3 + 5.59e3i)T + (-1.57e7 - 1.14e7i)T^{2} \) |
| 13 | \( 1 + (-6.72e3 - 4.88e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (9.41e3 + 2.89e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-2.60e3 + 1.89e3i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + (9.98e3 + 3.07e4i)T + (-2.75e9 + 2.00e9i)T^{2} \) |
| 29 | \( 1 + (8.01e4 - 5.82e4i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 37 | \( 1 + 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.32e5 - 9.60e4i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + (4.85e5 - 3.52e5i)T + (8.39e10 - 2.58e11i)T^{2} \) |
| 47 | \( 1 + (5.59e5 + 4.06e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (1.10e5 + 3.41e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.78e5 + 2.02e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + 1.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.74e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.37e6 + 4.22e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (9.76e5 - 3.00e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-1.27e6 - 3.92e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-3.50e6 + 2.54e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 + (2.59e5 - 8.00e5i)T + (-3.57e13 - 2.59e13i)T^{2} \) |
| 97 | \( 1 + (1.97e6 - 6.07e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−13.97409987930360195509984047072, −13.39750144834358179777098946268, −11.04271074462872405042439938052, −9.729504117609403148097724578996, −9.125185581431858566108572331976, −8.635969831415702695799178221332, −6.43292265598971452181539380867, −5.25492846720228444491760483905, −3.06396396372544217575541355735, −1.92053288885717408032035536405,
1.41221160950058464352949069790, 1.97383967820491983886888889473, 3.56721430512396867859287540688, 6.34214135004023359009711946768, 7.34430130497163148549559748261, 8.698310832593741770368430053639, 9.652920827463879227230680088773, 10.46545611890605141195712844997, 12.73361922049168194996831958385, 13.19908479882873848427985181860
