| 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
−10.66872768781298416132401514284, −10.22391932931700592475156468613, −9.226111106460211999125272899942, −8.327756932509480321529227558118, −7.39905684175996708260680517061, −6.97733568736650566941586000515, −6.28274874421804853750518555919, −4.26460831251248677237369573002, −2.82858529009226480842079301203, −1.24877156517280267065591936977,
0.60805404380629348412282481757, 1.83656527681490446726412222832, 3.71473217661762790714965528045, 4.63474174361386639913219268518, 6.45411523662697673558346439875, 7.39384638346424239471846617302, 8.192780498934770541678017456415, 8.960510961462903157027572290177, 9.412654234381366943476231056728, 10.46341374545129884740503634106
