| L(s) = 1 | + (−0.239 + 0.862i)2-s + (1.27 − 1.17i)3-s + (1.02 + 0.618i)4-s + (−0.401 − 0.0217i)5-s + (0.706 + 1.37i)6-s + (−0.709 + 4.32i)7-s + (−2.07 + 1.96i)8-s + (0.246 − 2.98i)9-s + (0.114 − 0.340i)10-s + (1.91 − 3.61i)11-s + (2.03 − 0.417i)12-s + (4.37 − 1.74i)13-s + (−3.56 − 1.64i)14-s + (−0.536 + 0.443i)15-s + (−0.0767 − 0.144i)16-s + (−3.68 + 0.604i)17-s + ⋯ |
| L(s) = 1 | + (−0.169 + 0.609i)2-s + (0.735 − 0.677i)3-s + (0.513 + 0.309i)4-s + (−0.179 − 0.00972i)5-s + (0.288 + 0.563i)6-s + (−0.268 + 1.63i)7-s + (−0.734 + 0.696i)8-s + (0.0821 − 0.996i)9-s + (0.0363 − 0.107i)10-s + (0.577 − 1.08i)11-s + (0.587 − 0.120i)12-s + (1.21 − 0.483i)13-s + (−0.951 − 0.440i)14-s + (−0.138 + 0.114i)15-s + (−0.0191 − 0.0362i)16-s + (−0.893 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33330 + 0.497579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33330 + 0.497579i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.27 + 1.17i)T \) |
| 59 | \( 1 + (3.65 - 6.75i)T \) |
| good | 2 | \( 1 + (0.239 - 0.862i)T + (-1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (0.401 + 0.0217i)T + (4.97 + 0.540i)T^{2} \) |
| 7 | \( 1 + (0.709 - 4.32i)T + (-6.63 - 2.23i)T^{2} \) |
| 11 | \( 1 + (-1.91 + 3.61i)T + (-6.17 - 9.10i)T^{2} \) |
| 13 | \( 1 + (-4.37 + 1.74i)T + (9.43 - 8.94i)T^{2} \) |
| 17 | \( 1 + (3.68 - 0.604i)T + (16.1 - 5.42i)T^{2} \) |
| 19 | \( 1 + (1.96 + 2.30i)T + (-3.07 + 18.7i)T^{2} \) |
| 23 | \( 1 + (1.95 - 1.48i)T + (6.15 - 22.1i)T^{2} \) |
| 29 | \( 1 + (-6.32 + 1.75i)T + (24.8 - 14.9i)T^{2} \) |
| 31 | \( 1 + (4.82 + 4.10i)T + (5.01 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.558 - 0.589i)T + (-2.00 - 36.9i)T^{2} \) |
| 41 | \( 1 + (1.47 - 1.94i)T + (-10.9 - 39.5i)T^{2} \) |
| 43 | \( 1 + (-5.47 + 2.90i)T + (24.1 - 35.5i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 11.4i)T + (-46.7 + 5.08i)T^{2} \) |
| 53 | \( 1 + (3.45 + 10.2i)T + (-42.1 + 32.0i)T^{2} \) |
| 61 | \( 1 + (0.368 + 0.102i)T + (52.2 + 31.4i)T^{2} \) |
| 67 | \( 1 + (-1.33 - 1.40i)T + (-3.62 + 66.9i)T^{2} \) |
| 71 | \( 1 + (3.68 - 0.199i)T + (70.5 - 7.67i)T^{2} \) |
| 73 | \( 1 + (-2.95 + 6.38i)T + (-47.2 - 55.6i)T^{2} \) |
| 79 | \( 1 + (-3.84 + 5.67i)T + (-29.2 - 73.3i)T^{2} \) |
| 83 | \( 1 + (-6.59 - 1.45i)T + (75.3 + 34.8i)T^{2} \) |
| 89 | \( 1 + (-2.26 - 8.14i)T + (-76.2 + 45.8i)T^{2} \) |
| 97 | \( 1 + (-0.983 - 2.12i)T + (-62.7 + 73.9i)T^{2} \) |
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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
−12.82124786806262982196813994957, −11.87659666720118918805993780443, −11.18307438405707306707556208605, −9.126494393024475925048046716030, −8.627064222303225194924498186793, −7.86601694658127678411446156128, −6.29307934604836505859556240359, −6.04695959996107198531015883247, −3.46901899371284126035374641386, −2.30993271280574400273122730588,
1.77968287575790505884614751261, 3.60291098328680433164540342561, 4.33372345720769828818757076525, 6.48311852461714380448422527491, 7.38333209029727011465803961273, 8.812965926947026633893385033327, 9.858900145799827446927286465370, 10.52173786664573491670827012643, 11.22407253114989461166846507679, 12.52624397235181308524993895232
