| L(s) = 1 | + (1.19 − 2.31i)2-s + (−2.77 − 3.89i)4-s + (0.904 − 3.72i)5-s + (2.50 + 0.866i)7-s + (−7.17 + 1.03i)8-s + (−7.55 − 6.54i)10-s + (−0.0853 + 1.79i)11-s + (−1.08 − 3.13i)13-s + (4.99 − 4.76i)14-s + (−3.04 + 8.79i)16-s + (0.919 + 2.01i)17-s + (5.33 + 2.43i)19-s + (−17.0 + 6.81i)20-s + (4.04 + 2.33i)22-s + (−0.537 + 4.76i)23-s + ⋯ |
| L(s) = 1 | + (0.843 − 1.63i)2-s + (−1.38 − 1.94i)4-s + (0.404 − 1.66i)5-s + (0.946 + 0.327i)7-s + (−2.53 + 0.364i)8-s + (−2.38 − 2.06i)10-s + (−0.0257 + 0.540i)11-s + (−0.300 − 0.868i)13-s + (1.33 − 1.27i)14-s + (−0.760 + 2.19i)16-s + (0.222 + 0.488i)17-s + (1.22 + 0.559i)19-s + (−3.80 + 1.52i)20-s + (0.862 + 0.498i)22-s + (−0.112 + 0.993i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.239607 + 2.35672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239607 + 2.35672i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (0.537 - 4.76i)T \) |
| good | 2 | \( 1 + (-1.19 + 2.31i)T + (-1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.904 + 3.72i)T + (-4.44 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-2.50 - 0.866i)T + (5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (0.0853 - 1.79i)T + (-10.9 - 1.04i)T^{2} \) |
| 13 | \( 1 + (1.08 + 3.13i)T + (-10.2 + 8.03i)T^{2} \) |
| 17 | \( 1 + (-0.919 - 2.01i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.33 - 2.43i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.362 + 0.258i)T + (9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-3.86 - 1.54i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (2.15 - 7.35i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.19 + 0.532i)T + (36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (-0.297 - 0.743i)T + (-31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (2.64 - 1.52i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.12 + 1.30i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-10.9 + 3.79i)T + (46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 3.35i)T + (-14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (2.17 - 0.103i)T + (66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (8.51 + 13.2i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.67 + 3.67i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.836 + 4.34i)T + (-73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-3.00 - 12.3i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (0.218 - 1.51i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-9.85 + 10.3i)T + (-4.61 - 96.8i)T^{2} \) |
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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
−10.07488449266005194183014676050, −9.732259583772431176035661601741, −8.683938148813237084022381696779, −7.87802317311333476379394952483, −5.74083008398253517387275443936, −5.14463971606002416030802946111, −4.61268689708571178343463629316, −3.36127669884863987698244780485, −1.88232328432097179854002692566, −1.15031742439410146674489417847,
2.64804258541275577505659871193, 3.78306629622989041582929562009, 4.87193006097833663202815082786, 5.78384078561760152503457474717, 6.73363976754130150184563460439, 7.20300738123237274339383441619, 7.973236886226836510347506781155, 9.051050380840497570643129869992, 10.17036055387685283907055673528, 11.24763597870999225134316474618
