| L(s) = 1 | + (−0.195 − 1.72i)3-s + (1.34 + 3.70i)5-s + (−0.00177 − 0.0100i)7-s + (−2.92 + 0.673i)9-s + (−0.343 + 0.944i)11-s + (2.57 − 3.06i)13-s + (6.11 − 3.04i)15-s + (−0.948 + 1.64i)17-s + (3.76 − 2.17i)19-s + (−0.0169 + 0.00502i)21-s + (−1.19 + 6.79i)23-s + (−8.06 + 6.77i)25-s + (1.73 + 4.89i)27-s + (3.82 + 4.56i)29-s + (−1.38 + 7.85i)31-s + ⋯ |
| L(s) = 1 | + (−0.112 − 0.993i)3-s + (0.602 + 1.65i)5-s + (−0.000670 − 0.00380i)7-s + (−0.974 + 0.224i)9-s + (−0.103 + 0.284i)11-s + (0.713 − 0.850i)13-s + (1.57 − 0.786i)15-s + (−0.230 + 0.398i)17-s + (0.863 − 0.498i)19-s + (−0.00370 + 0.00109i)21-s + (−0.249 + 1.41i)23-s + (−1.61 + 1.35i)25-s + (0.333 + 0.942i)27-s + (0.710 + 0.847i)29-s + (−0.248 + 1.41i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50353 + 0.483887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50353 + 0.483887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.195 + 1.72i)T \) |
| good | 5 | \( 1 + (-1.34 - 3.70i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.00177 + 0.0100i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.343 - 0.944i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.57 + 3.06i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.948 - 1.64i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.76 + 2.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.19 - 6.79i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.82 - 4.56i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.38 - 7.85i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-6.65 - 3.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.44 - 3.73i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.42 + 3.91i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.37 + 7.79i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 2.25iT - 53T^{2} \) |
| 59 | \( 1 + (2.24 + 6.17i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.106 - 0.0188i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.47 - 2.94i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.442 - 0.766i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.26 + 12.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.46 - 2.06i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.76 - 11.6i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (1.45 + 2.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.0 + 4.37i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.46255007002618983077961436995, −9.510844411685945625642996646450, −8.370904046475625818122042909844, −7.42458502570017526093243613142, −6.88980707632490872678555932252, −6.05975481912910836771752210926, −5.33955156844187355618496183537, −3.40446814355597309638880454380, −2.73888585388735692113566276743, −1.48354077649000370521955187374,
0.845443340418916952269826724371, 2.47238625078927634464097093303, 4.15166810711939344055641278059, 4.53287695234133088653240052229, 5.69350506781201417828740697922, 6.13872513907800192396940652204, 7.87103608561002419880860400827, 8.677914088116237984868455301878, 9.298582457603579541495800134733, 9.801054070780531496184683478739
