| L(s) = 1 | + (−0.322 + 1.41i)2-s + (−1.49 − 0.871i)3-s + (−0.0960 − 0.0462i)4-s + (−1.42 − 3.62i)5-s + (1.71 − 1.83i)6-s + (1.71 − 2.01i)7-s + (−1.71 + 2.14i)8-s + (1.47 + 2.60i)9-s + (5.58 − 0.842i)10-s + (−4.66 + 1.43i)11-s + (0.103 + 0.152i)12-s + (−3.94 + 1.21i)13-s + (2.29 + 3.07i)14-s + (−1.03 + 6.66i)15-s + (−2.61 − 3.28i)16-s + (−0.361 + 4.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.228 + 1.00i)2-s + (−0.864 − 0.503i)3-s + (−0.0480 − 0.0231i)4-s + (−0.636 − 1.62i)5-s + (0.700 − 0.749i)6-s + (0.648 − 0.761i)7-s + (−0.605 + 0.759i)8-s + (0.493 + 0.869i)9-s + (1.76 − 0.266i)10-s + (−1.40 + 0.433i)11-s + (0.0298 + 0.0441i)12-s + (−1.09 + 0.337i)13-s + (0.613 + 0.822i)14-s + (−0.266 + 1.72i)15-s + (−0.654 − 0.821i)16-s + (−0.0877 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00488911 - 0.0415159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00488911 - 0.0415159i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.49 + 0.871i)T \) |
| 7 | \( 1 + (-1.71 + 2.01i)T \) |
| good | 2 | \( 1 + (0.322 - 1.41i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (1.42 + 3.62i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (4.66 - 1.43i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (3.94 - 1.21i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (0.361 - 4.82i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-1.88 - 3.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.23 - 1.52i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.300 + 4.00i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 + (-0.460 - 0.314i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-5.09 - 0.767i)T + (39.1 + 12.0i)T^{2} \) |
| 43 | \( 1 + (0.856 - 0.129i)T + (41.0 - 12.6i)T^{2} \) |
| 47 | \( 1 + (-1.52 + 6.69i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (11.6 - 7.97i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (6.29 + 7.89i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (1.24 - 0.600i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + (-6.63 - 3.19i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (3.40 + 1.04i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 - 2.68T + 79T^{2} \) |
| 83 | \( 1 + (16.1 + 4.98i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.20 + 2.04i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (4.85 - 8.41i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.79476511485707992537796740236, −10.82524647393260754232412458919, −9.726582131087977720446864918534, −8.286356905992354556495699674006, −7.80673050205013809449176241223, −7.33364457671702759649214520598, −5.89147711645052462887634152415, −5.09629379854054809580182627362, −4.37971336544273666033115470314, −1.80534903352093922112119063086,
0.02927467044329722117179858245, 2.56234632820821857099090382596, 3.11539318569761854522434418806, 4.74750274853709478420815708633, 5.78862112428981153507742093832, 6.94825697534502178438891869347, 7.72207754230177943502998806604, 9.315863581739202795244679493860, 10.15402168028226870906782310848, 10.92043311275135393953479382146
