| L(s) = 1 | + (−1.29 − 0.567i)2-s + (−1.12 + 0.602i)3-s + (1.35 + 1.47i)4-s + (−1.29 − 1.58i)5-s + (1.80 − 0.140i)6-s + (−1.93 + 1.29i)7-s + (−0.922 − 2.67i)8-s + (−0.758 + 1.13i)9-s + (0.784 + 2.78i)10-s + (−4.58 + 1.39i)11-s + (−2.41 − 0.840i)12-s + (−1.77 − 1.45i)13-s + (3.24 − 0.577i)14-s + (2.41 + 1.00i)15-s + (−0.322 + 3.98i)16-s + (−0.698 + 0.289i)17-s + ⋯ |
| L(s) = 1 | + (−0.915 − 0.401i)2-s + (−0.651 + 0.348i)3-s + (0.677 + 0.735i)4-s + (−0.580 − 0.707i)5-s + (0.736 − 0.0575i)6-s + (−0.732 + 0.489i)7-s + (−0.326 − 0.945i)8-s + (−0.252 + 0.378i)9-s + (0.248 + 0.881i)10-s + (−1.38 + 0.419i)11-s + (−0.697 − 0.242i)12-s + (−0.491 − 0.403i)13-s + (0.867 − 0.154i)14-s + (0.624 + 0.258i)15-s + (−0.0806 + 0.996i)16-s + (−0.169 + 0.0701i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0200761 + 0.0811680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0200761 + 0.0811680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 + 0.567i)T \) |
| good | 3 | \( 1 + (1.12 - 0.602i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (1.29 + 1.58i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.93 - 1.29i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (4.58 - 1.39i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (1.77 + 1.45i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (0.698 - 0.289i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.355 - 3.60i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (0.824 - 0.164i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-2.64 + 8.70i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-4.31 - 4.31i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.24 + 0.319i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-2.34 - 11.7i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (7.17 + 3.83i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (1.13 + 2.74i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.85 - 9.39i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (3.31 - 2.71i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.0672 - 0.125i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-3.12 - 5.84i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (6.30 + 9.44i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (4.87 + 3.25i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.87 + 11.7i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (13.3 - 1.31i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (13.5 + 2.69i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.07 - 4.07i)T + 97iT^{2} \) |
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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
−13.32335744074211760729144068043, −12.34840697983622234347436161768, −11.77842449631118821893728096149, −10.44190651148850590557739739689, −9.905348477908002946422277846618, −8.446749484721161513316955314748, −7.73834141239905790759388175941, −6.05117439073838575695675335182, −4.66578565724831013840941935545, −2.73116661081429475937941782681,
0.11664435912694360078336561450, 2.99566202736871662362920409262, 5.35519824839387940282491633753, 6.71183726849112582942047611756, 7.22076497679749874145794937309, 8.551522509600294001528306219581, 9.887065048400168638026721800232, 10.83943378153729385419970107557, 11.52844298484460107699807529436, 12.76343307490831403000124251007
