| L(s) = 1 | + (0.841 − 0.540i)2-s + (−3.08 + 0.906i)3-s + (0.415 − 0.909i)4-s + (−2.36 − 2.72i)5-s + (−2.10 + 2.43i)6-s + (−3.01 + 1.93i)7-s + (−0.142 − 0.989i)8-s + (6.19 − 3.97i)9-s + (−3.46 − 1.01i)10-s + (−1.83 − 2.11i)11-s + (−0.458 + 3.18i)12-s + (−0.254 + 1.77i)13-s + (−1.48 + 3.25i)14-s + (9.77 + 6.28i)15-s + (−0.654 − 0.755i)16-s + (−0.531 − 1.16i)17-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (−1.78 + 0.523i)3-s + (0.207 − 0.454i)4-s + (−1.05 − 1.22i)5-s + (−0.860 + 0.993i)6-s + (−1.13 + 0.731i)7-s + (−0.0503 − 0.349i)8-s + (2.06 − 1.32i)9-s + (−1.09 − 0.321i)10-s + (−0.552 − 0.637i)11-s + (−0.132 + 0.919i)12-s + (−0.0706 + 0.491i)13-s + (−0.397 + 0.870i)14-s + (2.52 + 1.62i)15-s + (−0.163 − 0.188i)16-s + (−0.128 − 0.282i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0398986 - 0.284409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0398986 - 0.284409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (-5.96 - 5.60i)T \) |
| good | 3 | \( 1 + (3.08 - 0.906i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (2.36 + 2.72i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (3.01 - 1.93i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (1.83 + 2.11i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.254 - 1.77i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (0.531 + 1.16i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.450 + 0.289i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 1.43i)T + (19.3 - 12.4i)T^{2} \) |
| 29 | \( 1 + 3.22T + 29T^{2} \) |
| 31 | \( 1 + (0.862 + 5.99i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 + (0.848 + 1.85i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.29 + 7.21i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (5.71 - 1.67i)T + (39.5 - 25.4i)T^{2} \) |
| 53 | \( 1 + (-3.06 + 6.71i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.0868 + 0.603i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.73 + 4.30i)T + (-8.68 - 60.3i)T^{2} \) |
| 71 | \( 1 + (1.31 - 2.87i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (7.99 - 9.22i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.539 - 3.74i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.0897 - 0.103i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.73 - 1.39i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−12.62703851139855930374217722616, −11.74997346254888873710478205243, −11.21362812503844501579783727407, −9.932776548658744664384311989312, −8.859054170050007066521268284824, −6.89279063942102249637906227872, −5.67680250739579146449939046267, −4.92744426304369685379813327995, −3.74866806518002372359429496291, −0.29286095046830098085423504683,
3.42205500173818841433591911993, 4.87788454570792339468672389659, 6.28027598636412541158699406622, 7.02014783066982767660441491286, 7.56250253166908316331606039340, 10.24940344136472535316366199433, 10.81806448469688268262270591684, 11.76326526593823983947704030930, 12.68194420133584704745420529217, 13.31275345773853208901261946174
