L(s) = 1 | + (−0.899 + 1.09i)2-s + (0.555 + 0.831i)3-s + (−0.382 − 1.96i)4-s + (0.713 + 3.58i)5-s + (−1.40 − 0.141i)6-s + (−0.499 − 1.20i)7-s + (2.48 + 1.34i)8-s + (−0.382 + 0.923i)9-s + (−4.55 − 2.44i)10-s + (0.165 + 0.110i)11-s + (1.41 − 1.40i)12-s + (−0.763 + 3.83i)13-s + (1.76 + 0.539i)14-s + (−2.58 + 2.58i)15-s + (−3.70 + 1.50i)16-s + (−3.40 − 3.40i)17-s + ⋯ |
L(s) = 1 | + (−0.635 + 0.771i)2-s + (0.320 + 0.480i)3-s + (−0.191 − 0.981i)4-s + (0.319 + 1.60i)5-s + (−0.574 − 0.0576i)6-s + (−0.188 − 0.456i)7-s + (0.879 + 0.476i)8-s + (−0.127 + 0.307i)9-s + (−1.44 − 0.773i)10-s + (0.0497 + 0.0332i)11-s + (0.409 − 0.406i)12-s + (−0.211 + 1.06i)13-s + (0.472 + 0.144i)14-s + (−0.667 + 0.667i)15-s + (−0.926 + 0.375i)16-s + (−0.824 − 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.400865 + 0.835544i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.400865 + 0.835544i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.899 - 1.09i)T \) |
| 3 | \( 1 + (-0.555 - 0.831i)T \) |
good | 5 | \( 1 + (-0.713 - 3.58i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.499 + 1.20i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.165 - 0.110i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (0.763 - 3.83i)T + (-12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (3.40 + 3.40i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.60 - 0.319i)T + (17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-5.45 - 2.25i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (4.85 - 3.24i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 7.27iT - 31T^{2} \) |
| 37 | \( 1 + (-10.9 + 2.18i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.05 - 1.67i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-4.87 + 7.30i)T + (-16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-7.45 - 7.45i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.21 - 4.15i)T + (20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (1.52 + 7.68i)T + (-54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (0.775 + 1.16i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (2.61 + 3.91i)T + (-25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (3.70 + 8.95i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.717 - 1.73i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (7.84 - 7.84i)T - 79iT^{2} \) |
| 83 | \( 1 + (8.03 + 1.59i)T + (76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (9.80 - 4.06i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 6.53iT - 97T^{2} \) |
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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
−13.44277325064472241494314841429, −11.25706219156803627864221505223, −10.86185039401310161998674962333, −9.644422403971298346313980519030, −9.205343227374405316456305698600, −7.45757036000761819743043897171, −7.00332258945355786122062542556, −5.83494172895362497856173496813, −4.21881920925903999976686063069, −2.50156277704827264716036050042,
1.06336801348087981768907239312, 2.62651117447409197108138122103, 4.36481501149676709477500036381, 5.73772252612494056676031170861, 7.46129115364995220396535020177, 8.592045352529709412847775716417, 8.964415172179109565114609890728, 10.03545372828784477569139063366, 11.29740685220005056843844759637, 12.46784662144360177888438280272