| L(s) = 1 | + (−2.49 − 0.530i)2-s + (−0.0429 + 0.408i)3-s + (4.12 + 1.83i)4-s + (−2.29 − 2.54i)5-s + (0.323 − 0.996i)6-s + (−5.18 − 3.76i)8-s + (2.76 + 0.588i)9-s + (4.37 + 7.57i)10-s + (2.88 − 1.63i)11-s + (−0.926 + 1.60i)12-s + (0.672 + 2.06i)13-s + (1.13 − 0.827i)15-s + (4.90 + 5.45i)16-s + (−4.41 + 0.939i)17-s + (−6.60 − 2.93i)18-s + (2.21 − 0.985i)19-s + ⋯ |
| L(s) = 1 | + (−1.76 − 0.375i)2-s + (−0.0247 + 0.235i)3-s + (2.06 + 0.917i)4-s + (−1.02 − 1.13i)5-s + (0.132 − 0.406i)6-s + (−1.83 − 1.33i)8-s + (0.923 + 0.196i)9-s + (1.38 + 2.39i)10-s + (0.870 − 0.492i)11-s + (−0.267 + 0.463i)12-s + (0.186 + 0.573i)13-s + (0.294 − 0.213i)15-s + (1.22 + 1.36i)16-s + (−1.07 + 0.227i)17-s + (−1.55 − 0.692i)18-s + (0.507 − 0.226i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331337 - 0.371961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331337 - 0.371961i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-2.88 + 1.63i)T \) |
| good | 2 | \( 1 + (2.49 + 0.530i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.0429 - 0.408i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (2.29 + 2.54i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-0.672 - 2.06i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (4.41 - 0.939i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-2.21 + 0.985i)T + (12.7 - 14.1i)T^{2} \) |
| 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 | \( 1 + (-5.37 + 5.97i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.524 + 4.98i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (2.12 + 1.54i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + (-4.60 + 2.05i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-8.96 + 9.95i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (7.00 + 3.11i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (9.59 + 10.6i)T + (-6.37 + 60.6i)T^{2} \) |
| 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 | \( 1 + (-5.41 - 2.40i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (9.54 + 2.02i)T + (72.1 + 32.1i)T^{2} \) |
| 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
−10.46341374545129884740503634106, −9.412654234381366943476231056728, −8.960510961462903157027572290177, −8.192780498934770541678017456415, −7.39384638346424239471846617302, −6.45411523662697673558346439875, −4.63474174361386639913219268518, −3.71473217661762790714965528045, −1.83656527681490446726412222832, −0.60805404380629348412282481757,
1.24877156517280267065591936977, 2.82858529009226480842079301203, 4.26460831251248677237369573002, 6.28274874421804853750518555919, 6.97733568736650566941586000515, 7.39905684175996708260680517061, 8.327756932509480321529227558118, 9.226111106460211999125272899942, 10.22391932931700592475156468613, 10.66872768781298416132401514284
