L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.258 + 0.965i)3-s + (−0.866 − 0.499i)4-s − 6-s + (1.67 + 0.448i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)18-s + 1.73i·21-s + (0.258 + 0.965i)23-s + (0.866 + 0.5i)24-s + (−0.707 − 0.707i)27-s + (−1.22 − 1.22i)28-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.258 + 0.965i)3-s + (−0.866 − 0.499i)4-s − 6-s + (1.67 + 0.448i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)18-s + 1.73i·21-s + (0.258 + 0.965i)23-s + (0.866 + 0.5i)24-s + (−0.707 − 0.707i)27-s + (−1.22 − 1.22i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9947326415\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9947326415\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−10.47755778289601806030086640876, −9.492387051047927492376550481426, −8.910079631423548087992266442669, −8.028713959023576754895725418429, −7.62022865433085171003735629871, −6.13099296937079081101371289858, −5.24293942249888439948333190860, −4.72622105173765869168073841621, −3.68245633247113037934359810588, −1.91623440393042965809613716230,
1.25355587885700000831095309657, 2.09737856918546255331902039154, 3.34919362524968072177451099687, 4.56676315763756468429961677014, 5.40739762224978295849964234291, 6.93077021946505671280480442735, 7.69561770967871707850124330363, 8.447606816627508189767210083192, 8.934600161427883444817444426501, 10.24769090655272555788383931732