| L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.16 − 1.27i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s + (−0.0725 + 1.73i)6-s + (0.781 + 4.43i)7-s + (0.500 + 0.866i)8-s + (−0.271 − 2.98i)9-s + (−0.5 + 0.866i)10-s + (0.978 + 0.356i)11-s + (−1.05 − 1.37i)12-s + (2.22 + 1.86i)13-s + (−3.44 − 2.89i)14-s + (0.660 − 1.60i)15-s + (−0.939 − 0.342i)16-s + (3.16 − 5.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.674 − 0.738i)3-s + (0.0868 − 0.492i)4-s + (0.420 − 0.152i)5-s + (−0.0296 + 0.706i)6-s + (0.295 + 1.67i)7-s + (0.176 + 0.306i)8-s + (−0.0905 − 0.995i)9-s + (−0.158 + 0.273i)10-s + (0.295 + 0.107i)11-s + (−0.305 − 0.396i)12-s + (0.617 + 0.518i)13-s + (−0.921 − 0.772i)14-s + (0.170 − 0.413i)15-s + (−0.234 − 0.0855i)16-s + (0.767 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34415 + 0.0719430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34415 + 0.0719430i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-1.16 + 1.27i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 7 | \( 1 + (-0.781 - 4.43i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.978 - 0.356i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.22 - 1.86i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.16 + 5.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.956 + 1.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.953 + 5.40i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.97 - 5.01i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.09 - 6.23i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.09 - 1.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.63 + 5.56i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.48 - 2.72i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.46 - 8.32i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + (0.00700 - 0.00255i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.63 + 14.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.46 + 7.09i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.83 - 6.64i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.26 - 7.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.91 - 2.44i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.910 + 0.763i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.75 - 3.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.30 + 0.837i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.10133282247185215649351289509, −11.05491178141582070247296138334, −9.409864833133960618645068865103, −9.042223834340056259004815314472, −8.264591536821062371709112749675, −7.05865333795224968687224472056, −6.14348032371500169327662104471, −5.08462988448090245528336912789, −2.89647940719753075649090715903, −1.66239946387196128149544617884,
1.59953290864618076217749957455, 3.48834600873749556167427231661, 4.10815392207458283201398179448, 5.82408343012118280148544053973, 7.46559352467911104662916880501, 8.063457421544810312606606345424, 9.238757628370909357315674191529, 10.20777024132735211711056598388, 10.57225249942720529912862962517, 11.49523810764583387597138846033
