| L(s) = 1 | + (−2.18 + 2.05i)3-s + (−2.05 − 1.18i)5-s + (−4.05 − 7.02i)7-s + (0.558 − 8.98i)9-s + (17.6 − 10.1i)11-s + (−3.05 + 5.29i)13-s + (6.94 − 1.63i)15-s − 17.9i·17-s − 9.11·19-s + (23.3 + 7.02i)21-s + (−29.0 − 16.7i)23-s + (−9.67 − 16.7i)25-s + (17.2 + 20.7i)27-s + (14.4 − 8.31i)29-s + (−11.1 + 19.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.728 + 0.684i)3-s + (−0.411 − 0.237i)5-s + (−0.579 − 1.00i)7-s + (0.0620 − 0.998i)9-s + (1.60 − 0.924i)11-s + (−0.235 + 0.407i)13-s + (0.462 − 0.108i)15-s − 1.05i·17-s − 0.479·19-s + (1.11 + 0.334i)21-s + (−1.26 − 0.729i)23-s + (−0.387 − 0.670i)25-s + (0.638 + 0.769i)27-s + (0.496 − 0.286i)29-s + (−0.360 + 0.624i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.568305 - 0.507743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568305 - 0.507743i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.18 - 2.05i)T \) |
| good | 5 | \( 1 + (2.05 + 1.18i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.05 + 7.02i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.05 - 5.29i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 17.9iT - 289T^{2} \) |
| 19 | \( 1 + 9.11T + 361T^{2} \) |
| 23 | \( 1 + (29.0 + 16.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-14.4 + 8.31i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11.1 - 19.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-29.9 - 17.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-33.1 + 19.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-2.96 - 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (3.14 - 5.45i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (42.2 + 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-33.1 + 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.3 + 69.9i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.26334760966394651482480026545, −11.67158242049134946075826658422, −10.60717258771064680202038720861, −9.665867658030573891784873539167, −8.654328719676497497899593775973, −6.97045443707983262170098715538, −6.14078357180180067412212514364, −4.45635333501527627130821515314, −3.67560794279252365039610890123, −0.55238639663421763721218855534,
1.92157986650593612286107693687, 3.94914786953374667369238860954, 5.66898602710482737840393133874, 6.53934082316317018217289894828, 7.60465809793590322586196620587, 8.933471970740367358970774834565, 10.09888819913574581995941859312, 11.36433871993597297799655450351, 12.25953643838186336360235939480, 12.60886187896907820310796944689
