| L(s) = 1 | + (−0.798 + 1.16i)2-s + (−2.01 − 0.289i)3-s + (−0.725 − 1.86i)4-s + (−0.680 − 2.31i)5-s + (1.94 − 2.12i)6-s + (−0.800 − 0.923i)7-s + (2.75 + 0.641i)8-s + (1.10 + 0.324i)9-s + (3.24 + 1.05i)10-s + (−2.66 − 1.71i)11-s + (0.922 + 3.96i)12-s + (1.15 − 1.33i)13-s + (1.71 − 0.196i)14-s + (0.700 + 4.86i)15-s + (−2.94 + 2.70i)16-s + (−6.70 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.564 + 0.825i)2-s + (−1.16 − 0.167i)3-s + (−0.362 − 0.931i)4-s + (−0.304 − 1.03i)5-s + (0.795 − 0.866i)6-s + (−0.302 − 0.349i)7-s + (0.973 + 0.226i)8-s + (0.368 + 0.108i)9-s + (1.02 + 0.333i)10-s + (−0.804 − 0.517i)11-s + (0.266 + 1.14i)12-s + (0.321 − 0.370i)13-s + (0.459 − 0.0526i)14-s + (0.180 + 1.25i)15-s + (−0.736 + 0.675i)16-s + (−1.62 + 0.742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (9.54e-5 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (9.54e-5 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.222026 - 0.222005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222026 - 0.222005i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.798 - 1.16i)T \) |
| 23 | \( 1 + (-1.31 - 4.61i)T \) |
| good | 3 | \( 1 + (2.01 + 0.289i)T + (2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (0.680 + 2.31i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.800 + 0.923i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.66 + 1.71i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 1.33i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (6.70 - 3.06i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.65 + 5.81i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.47 + 3.23i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-3.77 + 0.543i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.727 - 2.47i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-8.68 + 2.55i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.0321 - 0.223i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 7.98iT - 47T^{2} \) |
| 53 | \( 1 + (2.04 - 1.77i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (10.9 + 9.49i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (2.87 - 0.414i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-1.08 + 0.697i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.55 - 5.53i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.05 + 2.30i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (9.02 - 10.4i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.24 - 0.366i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-12.2 - 1.76i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.02 + 10.2i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−13.56463927292780428098203120469, −12.95292733437649499830465564257, −11.41657464349886211730679881908, −10.66322548418238962226594576789, −9.188276530227911599073428178606, −8.183046292751618194566587728527, −6.81170347696397678637310576501, −5.67697545154080678343678876180, −4.68321477543183413363607573166, −0.51449673016596872418190316138,
2.76555278765542763835006623241, 4.60544353180648342226202696275, 6.34570852166504298609891124023, 7.56021417278414393967740094067, 9.141225478961172037726444264575, 10.47530179967880780769913439297, 10.97331645347486956545995252968, 11.88987141457660136757830905166, 12.82288525554956409530821530574, 14.20728829666181397816770540461
