| L(s) = 1 | + (−1.41 − 0.0781i)2-s + (1.98 + 0.220i)4-s + (0.479 + 0.174i)5-s + (−1.63 − 0.287i)7-s + (−2.78 − 0.467i)8-s + (−0.663 − 0.283i)10-s + (0.576 + 1.58i)11-s + (−3.81 − 4.54i)13-s + (2.28 + 0.533i)14-s + (3.90 + 0.877i)16-s + (−5.08 + 2.93i)17-s + (−1.91 + 3.32i)19-s + (0.914 + 0.452i)20-s + (−0.690 − 2.28i)22-s + (−0.765 − 4.34i)23-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0552i)2-s + (0.993 + 0.110i)4-s + (0.214 + 0.0780i)5-s + (−0.616 − 0.108i)7-s + (−0.986 − 0.165i)8-s + (−0.209 − 0.0897i)10-s + (0.173 + 0.477i)11-s + (−1.05 − 1.26i)13-s + (0.609 + 0.142i)14-s + (0.975 + 0.219i)16-s + (−1.23 + 0.712i)17-s + (−0.440 + 0.762i)19-s + (0.204 + 0.101i)20-s + (−0.147 − 0.486i)22-s + (−0.159 − 0.905i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00720230 - 0.0919894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00720230 - 0.0919894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0781i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.479 - 0.174i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.63 + 0.287i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.576 - 1.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.81 + 4.54i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.08 - 2.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.765 + 4.34i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.748 + 0.628i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.88 + 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.88 - 3.39i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.01 - 2.40i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.826 + 0.300i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0117 + 0.0666i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.83T + 53T^{2} \) |
| 59 | \( 1 + (-0.521 + 1.43i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (10.6 + 1.88i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 5.92i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (8.21 + 14.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.650 + 1.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.66 - 4.36i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.44 + 1.72i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (11.2 + 6.52i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 5.06i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.27587894417079526457987949330, −9.402407983233056898593485947801, −8.382982194363419211517542884293, −7.70131542762958086841420517622, −6.59953917516759561785683383993, −6.03116453248682143003461140066, −4.50101515128966548320871810023, −3.04582381292065602411963281104, −1.99845696844882610470993098398, −0.06223511608293130215160348830,
1.89402104358406954668986026839, 2.98148409262156312209112675946, 4.53013801960585639268035951384, 5.85892263570790399939043058477, 6.79002465003423570022661446604, 7.33700185439802495009974575711, 8.645399429889696320230450775192, 9.307884967650786191644199459174, 9.768287743847489245917412745829, 10.93862585717783817272330726940
