| L(s) = 1 | + (−0.747 − 1.56i)3-s + (−0.738 + 4.19i)5-s + (−2.50 + 2.10i)7-s + (−1.88 + 2.33i)9-s + (0.0777 + 0.441i)11-s + (2.92 − 1.06i)13-s + (7.09 − 1.97i)15-s + (−1.84 + 3.19i)17-s + (−1.19 − 2.06i)19-s + (5.15 + 2.34i)21-s + (5.22 + 4.38i)23-s + (−12.3 − 4.48i)25-s + (5.05 + 1.19i)27-s + (0.616 + 0.224i)29-s + (−3.54 − 2.97i)31-s + ⋯ |
| L(s) = 1 | + (−0.431 − 0.902i)3-s + (−0.330 + 1.87i)5-s + (−0.946 + 0.793i)7-s + (−0.627 + 0.778i)9-s + (0.0234 + 0.133i)11-s + (0.810 − 0.294i)13-s + (1.83 − 0.510i)15-s + (−0.446 + 0.773i)17-s + (−0.273 − 0.474i)19-s + (1.12 + 0.510i)21-s + (1.09 + 0.915i)23-s + (−2.46 − 0.896i)25-s + (0.973 + 0.229i)27-s + (0.114 + 0.0416i)29-s + (−0.635 − 0.533i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000691 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000691 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.519409 + 0.519768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519409 + 0.519768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.747 + 1.56i)T \) |
| good | 5 | \( 1 + (0.738 - 4.19i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.50 - 2.10i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0777 - 0.441i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.92 + 1.06i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.84 - 3.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.19 + 2.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.22 - 4.38i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.616 - 0.224i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.54 + 2.97i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.459 - 0.795i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.96 + 1.07i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.522 - 2.96i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.99 - 5.86i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (-0.976 + 5.53i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.68 + 8.12i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.22 + 2.99i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.81 - 8.34i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.57 - 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.69 - 1.34i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (9.79 + 3.56i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (3.11 + 5.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.477 + 2.70i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.64976623572182666125463604248, −11.28927855513225560556096246058, −11.07074185801315650009586751160, −9.803660347741980718530558539586, −8.404800008948028071690637325519, −7.20119739095668770406572038540, −6.52455213782896265788111354474, −5.76600259626709056588787141929, −3.47574293214940609252650748661, −2.42431453151081065313826946219,
0.65344102378856691914585958666, 3.67562528980882096970221107684, 4.52452291386513207730113467583, 5.54486400563873482146287717416, 6.85171372452034336844736770923, 8.531550733848026098787601231357, 9.082997126274476024563492979533, 10.04186896308087513794001315161, 11.11043036707553034793275696339, 12.10719341266879982197910765291
