| L(s) = 1 | + (1.16 + 0.806i)2-s + (−1.33 + 1.10i)3-s + (0.699 + 1.87i)4-s + (0.297 + 0.0525i)5-s + (−2.44 + 0.207i)6-s + (−0.312 − 0.371i)7-s + (−0.697 + 2.74i)8-s + (0.560 − 2.94i)9-s + (0.303 + 0.301i)10-s + (−0.00722 − 0.0410i)11-s + (−3.00 − 1.72i)12-s + (4.05 − 1.47i)13-s + (−0.0627 − 0.683i)14-s + (−0.455 + 0.258i)15-s + (−3.02 + 2.62i)16-s + (−2.14 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (0.821 + 0.570i)2-s + (−0.770 + 0.637i)3-s + (0.349 + 0.936i)4-s + (0.133 + 0.0234i)5-s + (−0.996 + 0.0847i)6-s + (−0.117 − 0.140i)7-s + (−0.246 + 0.969i)8-s + (0.186 − 0.982i)9-s + (0.0960 + 0.0952i)10-s + (−0.00217 − 0.0123i)11-s + (−0.866 − 0.498i)12-s + (1.12 − 0.409i)13-s + (−0.0167 − 0.182i)14-s + (−0.117 + 0.0668i)15-s + (−0.755 + 0.655i)16-s + (−0.520 − 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.959196 + 0.843805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.959196 + 0.843805i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 - 0.806i)T \) |
| 3 | \( 1 + (1.33 - 1.10i)T \) |
| good | 5 | \( 1 + (-0.297 - 0.0525i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.312 + 0.371i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.00722 + 0.0410i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.05 + 1.47i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.14 + 1.23i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.55 + 3.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.03 + 4.22i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.40 - 6.60i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.30 + 2.75i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.36 - 5.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.16 - 5.94i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.38 - 0.420i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.51 - 4.62i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 9.90iT - 53T^{2} \) |
| 59 | \( 1 + (-0.755 + 4.28i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.77 - 4.84i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.80 - 10.4i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.68 + 2.92i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.315 - 0.546i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.03 - 2.84i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.41 - 1.96i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.00 + 5.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.578 + 3.28i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.91889030921397797681899766874, −13.05809509420188541039182698810, −11.82857808834824447818367626364, −11.11188826635572012552265354338, −9.790139031208613670671339931798, −8.389365954070436905076011359322, −6.85648010131122803570393214581, −5.87290838660293172238946875906, −4.74495448416761549136118456979, −3.41320696488269155926977226928,
1.73833606489049682407962373563, 3.85260262705354219408419612325, 5.52753959624336696368829056822, 6.24260687608330153624466967651, 7.67007654159158228870412918816, 9.511104476940570314877563954677, 10.69243639868887833208960191218, 11.65034581495903656853409740662, 12.27870351247559735514269521520, 13.59830540246505227208415892132
