| L(s) = 1 | + (0.470 + 0.631i)2-s + (−1.10 − 1.32i)3-s + (0.395 − 1.32i)4-s + (2.32 − 1.53i)5-s + (0.318 − 1.32i)6-s + (−3.41 + 3.62i)7-s + (2.50 − 0.910i)8-s + (−0.537 + 2.95i)9-s + (2.06 + 0.750i)10-s + (1.55 − 0.779i)11-s + (−2.19 + 0.940i)12-s + (0.859 + 1.99i)13-s + (−3.89 − 0.455i)14-s + (−4.61 − 1.39i)15-s + (−0.554 − 0.364i)16-s + (−3.39 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (0.332 + 0.446i)2-s + (−0.640 − 0.767i)3-s + (0.197 − 0.660i)4-s + (1.04 − 0.684i)5-s + (0.129 − 0.541i)6-s + (−1.29 + 1.36i)7-s + (0.884 − 0.321i)8-s + (−0.179 + 0.983i)9-s + (0.651 + 0.237i)10-s + (0.468 − 0.235i)11-s + (−0.634 + 0.271i)12-s + (0.238 + 0.552i)13-s + (−1.04 − 0.121i)14-s + (−1.19 − 0.360i)15-s + (−0.138 − 0.0911i)16-s + (−0.823 + 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00987 - 0.224772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00987 - 0.224772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.10 + 1.32i)T \) |
| good | 2 | \( 1 + (-0.470 - 0.631i)T + (-0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (-2.32 + 1.53i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (3.41 - 3.62i)T + (-0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.55 + 0.779i)T + (6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-0.859 - 1.99i)T + (-8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (3.39 - 2.84i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.63 + 1.37i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.465 - 0.493i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-5.71 + 0.668i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (5.72 - 1.35i)T + (27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-0.131 - 0.747i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-0.0737 + 0.0990i)T + (-11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (0.148 - 2.54i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-1.65 - 0.391i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (5.02 + 8.69i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.3 + 5.21i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (2.27 + 7.58i)T + (-50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (-0.462 - 0.0540i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 4.17i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.01 - 0.732i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 5.62i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (1.97 + 2.65i)T + (-23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (4.72 - 1.71i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.27 - 4.12i)T + (38.4 + 89.0i)T^{2} \) |
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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
−14.08490272624711029381037393186, −13.14176340449461180238428012468, −12.51696827023224799632617551314, −11.14900048331360980102139326239, −9.732673311041363893839305098217, −8.780739740836086362857167110068, −6.53118194856908351262457286282, −6.19117784704273531809153657737, −5.12758723213643698044715777103, −1.96162584811106564359673526296,
3.09878609196109257559462121005, 4.28971000591971014807319959503, 6.23413646773448018757056195562, 7.10657487481799078710613704057, 9.324518027983059370893149376167, 10.38288920989507422787735279449, 10.86886455369434261241571860001, 12.33233462941760843870831494148, 13.33717688149255042428254495813, 14.12820440540439237880256385490
