| L(s) = 1 | + (1.87 − 0.330i)2-s + (0.583 − 1.63i)3-s + (1.52 − 0.553i)4-s + (−0.848 + 0.308i)5-s + (0.555 − 3.24i)6-s + (2.48 + 0.894i)7-s + (−0.629 + 0.363i)8-s + (−2.31 − 1.90i)9-s + (−1.48 + 0.858i)10-s + (−1.19 + 3.28i)11-s + (−0.0147 − 2.80i)12-s + (−1.35 − 3.72i)13-s + (4.95 + 0.853i)14-s + (0.00822 + 1.56i)15-s + (−3.53 + 2.96i)16-s + (−0.233 − 0.405i)17-s + ⋯ |
| L(s) = 1 | + (1.32 − 0.233i)2-s + (0.337 − 0.941i)3-s + (0.760 − 0.276i)4-s + (−0.379 + 0.138i)5-s + (0.226 − 1.32i)6-s + (0.941 + 0.338i)7-s + (−0.222 + 0.128i)8-s + (−0.772 − 0.634i)9-s + (−0.470 + 0.271i)10-s + (−0.360 + 0.990i)11-s + (−0.00425 − 0.808i)12-s + (−0.375 − 1.03i)13-s + (1.32 + 0.228i)14-s + (0.00212 + 0.403i)15-s + (−0.884 + 0.742i)16-s + (−0.0567 − 0.0982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04254 - 0.914994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04254 - 0.914994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.583 + 1.63i)T \) |
| 7 | \( 1 + (-2.48 - 0.894i)T \) |
| good | 2 | \( 1 + (-1.87 + 0.330i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.848 - 0.308i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.19 - 3.28i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.35 + 3.72i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.14 - 1.81i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.44 - 0.607i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.79 - 4.92i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.38 + 6.54i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 - 9.85T + 37T^{2} \) |
| 41 | \( 1 + (-2.71 + 0.987i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.54 + 8.76i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.13 + 1.14i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (8.75 + 5.05i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.48 + 2.08i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.0209 + 0.0574i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.78 + 10.0i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.19 + 1.26i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 8.37iT - 73T^{2} \) |
| 79 | \( 1 + (-1.64 - 9.30i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (10.6 + 3.86i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (1.51 - 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.02 + 0.886i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.62681037422796846333179126269, −11.81530023803919797338574623561, −11.06423370733848590682802038159, −9.366786377171804740490226717998, −8.002334980662612255616928127362, −7.33211332509488762400074144965, −5.79170612284559118556525396294, −4.90978622787150760789711869180, −3.39103301457937071945087216692, −2.13405859673227140884382404607,
2.92999914974445558358520277675, 4.21830620834330837790633898637, 4.82724702482613196075096145106, 5.96511048829373325979799988787, 7.53423317503839545008139098489, 8.656385184019373709389890451378, 9.718379670183925820220762445982, 11.24324381381499326617283158355, 11.52216542018660756740909721675, 12.98377722528098393933359484660
