| 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
−12.46784662144360177888438280272, −11.29740685220005056843844759637, −10.03545372828784477569139063366, −8.964415172179109565114609890728, −8.592045352529709412847775716417, −7.46129115364995220396535020177, −5.73772252612494056676031170861, −4.36481501149676709477500036381, −2.62651117447409197108138122103, −1.06336801348087981768907239312,
2.50156277704827264716036050042, 4.21881920925903999976686063069, 5.83494172895362497856173496813, 7.00332258945355786122062542556, 7.45757036000761819743043897171, 9.205343227374405316456305698600, 9.644422403971298346313980519030, 10.86185039401310161998674962333, 11.25706219156803627864221505223, 13.44277325064472241494314841429
