| L(s) = 1 | + (0.841 + 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.415 + 0.909i)6-s + (−2.29 + 2.64i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (−2.35 + 1.51i)11-s + (−0.841 + 0.540i)12-s + (0.0353 + 0.0407i)13-s + (−3.36 + 0.986i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (1.10 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.169 + 0.371i)6-s + (−0.866 + 1.00i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (−0.711 + 0.457i)11-s + (−0.242 + 0.156i)12-s + (0.00979 + 0.0113i)13-s + (−0.898 + 0.263i)14-s + (0.0367 − 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.267 − 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0705162 + 1.16720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0705162 + 1.16720i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (3.56 + 3.20i)T \) |
| good | 7 | \( 1 + (2.29 - 2.64i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.35 - 1.51i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.0353 - 0.0407i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 2.41i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.969 + 2.12i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.79 - 8.31i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.07 - 7.50i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.72 + 1.09i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-10.3 - 3.03i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.39 - 9.67i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + (-8.43 + 9.73i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.72 - 3.14i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.713 - 4.96i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (5.39 + 3.46i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.0178 - 0.0114i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.60 - 12.2i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.03 - 4.66i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-9.07 + 2.66i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.08 + 7.57i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (8.12 + 2.38i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.96193573412277333144979407126, −9.926355825167101534529835173342, −9.123124474720080130555199089515, −8.320229539453080541726223171034, −7.29900937149517609516558777262, −6.32413363821590061577195268821, −5.34951788620929516733839007723, −4.61371127894532297104972092649, −3.37238927889235438401987420576, −2.54521670668619674301554148959,
0.47356397613499002236703220823, 2.23261681988946454938365304074, 3.51935635817658213931216235300, 4.12555965557954800851005678846, 5.71118568775220074882798776824, 6.32750699419310203296964181201, 7.52442583539111983829863382217, 7.956047683979168600131199610039, 9.419737303857669686140130797474, 10.23477543914377640038739515149
