L(s) = 1 | + (1.26 − 1.18i)3-s + (2.87 − 1.65i)5-s + (1.73 + 2i)7-s + (0.186 − 2.99i)9-s + (−1.65 + 2.87i)11-s − 4·13-s + (1.65 − 5.5i)15-s + (2.87 + 1.65i)17-s + (−6.06 + 3.5i)19-s + (4.55 + 0.469i)21-s + (−1.65 − 2.87i)23-s + (3 − 5.19i)25-s + (−3.31 − 4.00i)27-s − 6.63i·29-s + (2.59 + 1.5i)31-s + ⋯ |
L(s) = 1 | + (0.728 − 0.684i)3-s + (1.28 − 0.741i)5-s + (0.654 + 0.755i)7-s + (0.0620 − 0.998i)9-s + (−0.500 + 0.866i)11-s − 1.10·13-s + (0.428 − 1.42i)15-s + (0.696 + 0.402i)17-s + (−1.39 + 0.802i)19-s + (0.994 + 0.102i)21-s + (−0.345 − 0.598i)23-s + (0.600 − 1.03i)25-s + (−0.638 − 0.769i)27-s − 1.23i·29-s + (0.466 + 0.269i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85606 - 0.666694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85606 - 0.666694i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.26 + 1.18i)T \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 5 | \( 1 + (-2.87 + 1.65i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.65 - 2.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-2.87 - 1.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.06 - 3.5i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.65 + 2.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 + (-2.59 - 1.5i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.63iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + (4.97 + 8.61i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.87 + 1.65i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.65 + 2.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + (2.87 - 1.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.80963040011588447625334938123, −10.06272078983271674066758087114, −9.683430031926522313819079117788, −8.468072615510332062825765215879, −7.976052546883421897822271752457, −6.56102035460628574932270518009, −5.59209064552753986881497208402, −4.52341667919293087115619729582, −2.43556928282328463914073216731, −1.79182818441912778562778588356,
2.12536284104950622157825801596, 3.16053331028231878747144342697, 4.64769253056344577794311766035, 5.61098542921114388033877614432, 6.97142537941494286267431751878, 7.917431876009141345461226009148, 9.013426088299489883027739399497, 9.953511988890905556162597710191, 10.53594240218987384512506815960, 11.21570173732679800291035215845