L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 + 1.5i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 1.73·6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (−1 + 1.73i)9-s + (−0.499 − 0.866i)10-s + (−0.866 − 1.5i)11-s + (0.866 − 1.49i)12-s + (−0.866 + 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)18-s + (−0.866 + 1.5i)19-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 + 1.5i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 1.73·6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (−1 + 1.73i)9-s + (−0.499 − 0.866i)10-s + (−0.866 − 1.5i)11-s + (0.866 − 1.49i)12-s + (−0.866 + 0.499i)14-s − 1.73·15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)18-s + (−0.866 + 1.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9316256264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9316256264\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.43687893829596263639896201281, −9.412566038735869920543574734353, −8.650985794147136544651752417665, −8.110633592960892497283750119950, −7.57806794919239333679378125748, −6.02762215296473008932405439649, −5.42905980248740276175652847935, −4.34605026178565157944791001563, −3.53056340558712919735810987180, −2.38681803197878996095866026377,
0.933992518347371382437859584416, 2.00772590926896083104857449116, 2.74926468137979601340692017294, 4.30919980319982109045790872196, 4.85006902970459262700052202817, 6.77601995655622820572825853996, 7.50877297401903533150934892792, 8.006684828853812366728165581945, 8.599199026261938390612874050163, 9.359830996070514641983474373127