| L(s) = 1 | + (−0.900 + 1.09i)2-s + (−1.22 − 1.22i)3-s + (−0.377 − 1.96i)4-s + (2.79 − 1.83i)5-s + (2.43 − 0.238i)6-s + (1.37 + 1.29i)7-s + (2.48 + 1.35i)8-s + (0.0155 + 2.99i)9-s + (−0.512 + 4.69i)10-s + (1.73 + 3.45i)11-s + (−1.93 + 2.87i)12-s + (0.329 − 0.142i)13-s + (−2.64 + 0.330i)14-s + (−5.67 − 1.15i)15-s + (−3.71 + 1.48i)16-s + (−2.24 − 2.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.636 + 0.771i)2-s + (−0.708 − 0.705i)3-s + (−0.188 − 0.981i)4-s + (1.24 − 0.821i)5-s + (0.995 − 0.0974i)6-s + (0.519 + 0.489i)7-s + (0.877 + 0.479i)8-s + (0.00518 + 0.999i)9-s + (−0.161 + 1.48i)10-s + (0.522 + 1.04i)11-s + (−0.558 + 0.829i)12-s + (0.0914 − 0.0394i)13-s + (−0.708 + 0.0883i)14-s + (−1.46 − 0.298i)15-s + (−0.928 + 0.371i)16-s + (−0.545 − 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14154 + 0.175549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14154 + 0.175549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 - 1.09i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| good | 5 | \( 1 + (-2.79 + 1.83i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-1.37 - 1.29i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 3.45i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-0.329 + 0.142i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (2.24 + 2.68i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-4.69 - 3.94i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.70 - 2.87i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (3.12 - 0.364i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-1.70 - 7.18i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-10.0 + 1.76i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (4.99 + 3.72i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.0897 + 1.54i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (9.42 + 2.23i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-5.46 - 9.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.69 + 5.35i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (8.17 - 2.44i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-3.47 - 0.406i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (7.42 + 2.70i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.3 + 5.60i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.12 + 5.30i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-2.75 + 2.04i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (4.05 + 11.1i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-10.8 - 7.13i)T + (38.4 + 89.0i)T^{2} \) |
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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
−10.34890027950295249213280699082, −9.535507728265466058955012341487, −8.941179043676783124735428663974, −7.86626539341376588523214186653, −7.02577854429412527651694161881, −6.13034465929340343631373570823, −5.28963079059888122645052798649, −4.84394229088125477906471781812, −1.99808194091340847998018702204, −1.27835531105579471339714139516,
1.06947588449489728741215148441, 2.63858426165257123978792375120, 3.74059351220843850031136617283, 4.88056226363520747449076762219, 6.12193935592300341669442565413, 6.80408464019544776627345037769, 8.130081117373611427877162710057, 9.265945425827976780576208869193, 9.727122142111923622103356125909, 10.59593707319938010756221277980
