| L(s) = 1 | + (−0.0733 − 1.41i)2-s + (1.70 − 0.317i)3-s + (−1.98 + 0.207i)4-s + (−0.0828 − 0.0944i)5-s + (−0.572 − 2.38i)6-s + (−4.69 − 0.617i)7-s + (0.438 + 2.79i)8-s + (2.79 − 1.08i)9-s + (−0.127 + 0.123i)10-s + (−0.634 − 1.28i)11-s + (−3.32 + 0.983i)12-s + (−1.77 − 5.22i)13-s + (−0.528 + 6.67i)14-s + (−0.170 − 0.134i)15-s + (3.91 − 0.823i)16-s + (−4.39 − 4.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.0518 − 0.998i)2-s + (0.983 − 0.183i)3-s + (−0.994 + 0.103i)4-s + (−0.0370 − 0.0422i)5-s + (−0.233 − 0.972i)6-s + (−1.77 − 0.233i)7-s + (0.154 + 0.987i)8-s + (0.932 − 0.360i)9-s + (−0.0402 + 0.0391i)10-s + (−0.191 − 0.387i)11-s + (−0.958 + 0.284i)12-s + (−0.492 − 1.45i)13-s + (−0.141 + 1.78i)14-s + (−0.0441 − 0.0347i)15-s + (0.978 − 0.205i)16-s + (−1.06 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0563690 + 0.965336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0563690 + 0.965336i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0733 + 1.41i)T \) |
| 3 | \( 1 + (-1.70 + 0.317i)T \) |
| good | 5 | \( 1 + (0.0828 + 0.0944i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (4.69 + 0.617i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (0.634 + 1.28i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.77 + 5.22i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.39 + 4.39i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.571 - 2.87i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.378 - 2.87i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (5.33 - 0.349i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (0.790 - 0.456i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 9.21i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.604 + 4.58i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-6.33 + 3.12i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-6.50 + 1.74i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.975 + 1.46i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (0.686 + 0.783i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.533 + 8.13i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-12.2 - 6.05i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (2.69 + 6.51i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.00719 - 0.0173i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.64 - 6.15i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-5.76 - 5.05i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (17.3 - 7.18i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-3.83 - 2.21i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.03552448561160037847899538730, −9.615807045846184662333201975925, −8.753370175673753572411445731168, −7.82388449916992525951478620394, −6.81813209841842911079897530070, −5.50395457372471464041970944865, −4.04646180530110405684025538342, −3.17066797325940212663989381601, −2.49858907252580381390055178687, −0.47474947009904226511289787951,
2.36206269207498837153820145941, 3.75289558117269262051879038821, 4.47121532682386877445653338078, 5.98565615696181981407042277389, 6.87933093437864776666292140610, 7.38523812219544001317728207816, 8.797101977396007967139972736017, 9.216047827227821188330359550269, 9.774387882988634293310464901421, 10.86682076774273413876550146344
