| L(s) = 1 | + (−1.31 − 0.514i)2-s + (−1.60 − 0.659i)3-s + (1.47 + 1.35i)4-s + (−0.503 − 3.82i)5-s + (1.77 + 1.69i)6-s + (−0.191 − 0.713i)7-s + (−1.24 − 2.54i)8-s + (2.13 + 2.11i)9-s + (−1.30 + 5.30i)10-s + (−0.526 + 0.403i)11-s + (−1.46 − 3.13i)12-s + (2.91 − 3.80i)13-s + (−0.114 + 1.03i)14-s + (−1.71 + 6.46i)15-s + (0.331 + 3.98i)16-s − 5.86·17-s + ⋯ |
| L(s) = 1 | + (−0.931 − 0.363i)2-s + (−0.924 − 0.380i)3-s + (0.735 + 0.677i)4-s + (−0.225 − 1.71i)5-s + (0.723 + 0.690i)6-s + (−0.0722 − 0.269i)7-s + (−0.439 − 0.898i)8-s + (0.710 + 0.703i)9-s + (−0.412 + 1.67i)10-s + (−0.158 + 0.121i)11-s + (−0.422 − 0.906i)12-s + (0.809 − 1.05i)13-s + (−0.0306 + 0.277i)14-s + (−0.442 + 1.66i)15-s + (0.0827 + 0.996i)16-s − 1.42·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0451184 + 0.341414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0451184 + 0.341414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 + 0.514i)T \) |
| 3 | \( 1 + (1.60 + 0.659i)T \) |
| good | 5 | \( 1 + (0.503 + 3.82i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.191 + 0.713i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.526 - 0.403i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.91 + 3.80i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + (4.81 - 1.99i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.36 - 0.901i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.10 + 0.144i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (7.73 - 4.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.210 + 0.507i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.568 + 2.12i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.978 - 1.27i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-1.31 - 0.759i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.45 - 8.34i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 0.239i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.96 + 14.9i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-8.17 + 10.6i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (1.15 + 1.15i)T + 71iT^{2} \) |
| 73 | \( 1 + (-4.58 + 4.58i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.77 + 3.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.4 + 1.37i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-8.47 + 8.47i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.26 - 3.91i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09088978798434914078507084205, −10.63303351507284411573950740393, −9.275481640424378214338232573846, −8.505450835268367230231180490271, −7.65853087630008556025607738066, −6.43631189254800042673326096279, −5.24883033279952078815425184512, −4.02341017877762301159178236189, −1.70329082302055102480191210786, −0.38178531626487840538833024714,
2.33791942670580797622214697107, 4.06920467479756021290025249309, 5.79389786186133077557211192095, 6.70697678552920371462736297554, 7.03699355756131632424562647638, 8.651068133554810343556269208592, 9.571633242610324201286034042519, 10.76776265800337906575968548162, 11.00766445105834113275617801207, 11.63955173884841221106251691542
