| L(s) = 1 | + (0.596 + 1.28i)2-s + (0.791 + 1.54i)3-s + (−1.28 + 1.52i)4-s + (0.236 − 1.18i)5-s + (−1.50 + 1.93i)6-s + (1.67 + 0.692i)7-s + (−2.73 − 0.738i)8-s + (−1.74 + 2.43i)9-s + (1.66 − 0.405i)10-s + (0.195 + 0.292i)11-s + (−3.37 − 0.773i)12-s + (1.51 − 0.302i)13-s + (0.109 + 2.55i)14-s + (2.01 − 0.575i)15-s + (−0.681 − 3.94i)16-s + (−5.35 − 5.35i)17-s + ⋯ |
| L(s) = 1 | + (0.421 + 0.906i)2-s + (0.456 + 0.889i)3-s + (−0.644 + 0.764i)4-s + (0.105 − 0.530i)5-s + (−0.613 + 0.789i)6-s + (0.632 + 0.261i)7-s + (−0.965 − 0.261i)8-s + (−0.582 + 0.812i)9-s + (0.525 − 0.128i)10-s + (0.0590 + 0.0883i)11-s + (−0.974 − 0.223i)12-s + (0.421 − 0.0838i)13-s + (0.0292 + 0.683i)14-s + (0.520 − 0.148i)15-s + (−0.170 − 0.985i)16-s + (−1.29 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849951 + 1.33470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849951 + 1.33470i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.596 - 1.28i)T \) |
| 3 | \( 1 + (-0.791 - 1.54i)T \) |
| good | 5 | \( 1 + (-0.236 + 1.18i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-1.67 - 0.692i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.195 - 0.292i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.51 + 0.302i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (5.35 + 5.35i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.449 + 0.0895i)T + (17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-5.61 + 2.32i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (2.38 + 1.59i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + (1.66 - 8.34i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.93 - 4.66i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (1.78 + 2.67i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.866i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.141 + 0.0944i)T + (20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (8.97 + 1.78i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (11.0 + 7.39i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (2.80 - 4.19i)T + (-25.6 - 61.8i)T^{2} \) |
| 71 | \( 1 + (-2.24 + 5.41i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (6.08 + 14.6i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-5.00 + 5.00i)T - 79iT^{2} \) |
| 83 | \( 1 + (-1.64 - 8.26i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (4.01 - 9.68i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 0.511iT - 97T^{2} \) |
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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
−13.31488881317779207422370153556, −11.93705449734940757943761729955, −10.94077070410159251593432101521, −9.420077355225537170646030159852, −8.815927608593261015379463183548, −7.948458098962720203742017936811, −6.56150018159381236958657191531, −5.01465822191069229132087594183, −4.62820967241345295582557904058, −2.96449777513754953463166991760,
1.54368967724063531225195280967, 2.89871349168007790937850654588, 4.26619552546333218920755393537, 5.90443584524594786051086355042, 6.95993405506390460196103291172, 8.367179046261461463875424215042, 9.180901189654804134874753368467, 10.71022444933934316439945286453, 11.17697003096851622783598981666, 12.35394411540524837274900516852
