| L(s) = 1 | + (1.70 + 1.89i)2-s + (0.375 + 0.167i)3-s + (−0.471 + 4.48i)4-s + (3.35 + 0.712i)5-s + (0.323 + 0.996i)6-s + (−5.18 + 3.76i)8-s + (−1.89 − 2.10i)9-s + (4.37 + 7.57i)10-s + (−2.85 + 1.68i)11-s + (−0.926 + 1.60i)12-s + (0.672 − 2.06i)13-s + (1.13 + 0.827i)15-s + (−7.17 − 1.52i)16-s + (3.02 − 3.35i)17-s + (0.755 − 7.18i)18-s + (−0.253 − 2.41i)19-s + ⋯ |
| L(s) = 1 | + (1.20 + 1.34i)2-s + (0.216 + 0.0964i)3-s + (−0.235 + 2.24i)4-s + (1.49 + 0.318i)5-s + (0.132 + 0.406i)6-s + (−1.83 + 1.33i)8-s + (−0.631 − 0.701i)9-s + (1.38 + 2.39i)10-s + (−0.861 + 0.507i)11-s + (−0.267 + 0.463i)12-s + (0.186 − 0.573i)13-s + (0.294 + 0.213i)15-s + (−1.79 − 0.381i)16-s + (0.733 − 0.814i)17-s + (0.178 − 1.69i)18-s + (−0.0581 − 0.553i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48057 + 2.88464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48057 + 2.88464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (2.85 - 1.68i)T \) |
| good | 2 | \( 1 + (-1.70 - 1.89i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.375 - 0.167i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-3.35 - 0.712i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-0.672 + 2.06i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 3.35i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.253 + 2.41i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.324 - 0.561i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.01 - 0.736i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (7.86 - 1.67i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-4.57 + 2.03i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (2.12 - 1.54i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + (0.526 + 5.01i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (13.1 - 2.78i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.801 + 7.62i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-14.0 - 2.98i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (3.11 + 5.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 4.02i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.619 - 5.89i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-6.53 - 7.25i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.58 - 7.96i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 4.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.69 + 8.27i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.27426154284328600938574932003, −10.03242487493975539196311186254, −9.250407545697101211996574921274, −8.179573477642861914148129805515, −7.20523905564967064378151548695, −6.37122445588372873015363788794, −5.54011068158276381990495902041, −5.04955488162398386053768882951, −3.46263017422012154966083599669, −2.56087133745367153475752026171,
1.60221929480799382105095947977, 2.38480982146505839838610632663, 3.46100867159876642219022796093, 4.85193235832393222672879713472, 5.63428028933807891931678781175, 6.14124507611214722395772752571, 8.006239152741752417521925347462, 9.130583141715784284106673577234, 10.00980773052718145168131548792, 10.63567474636466256725434183110
