| L(s) = 1 | + (−1.38 − 0.244i)2-s + (−0.458 + 1.67i)3-s + (−0.0163 − 0.00595i)4-s + (1.98 + 0.721i)5-s + (1.04 − 2.20i)6-s + (−2.18 + 1.49i)7-s + (2.46 + 1.42i)8-s + (−2.57 − 1.53i)9-s + (−2.57 − 1.48i)10-s + (−1.26 − 3.48i)11-s + (0.0174 − 0.0245i)12-s + (−2.27 + 6.25i)13-s + (3.39 − 1.53i)14-s + (−2.11 + 2.98i)15-s + (−3.03 − 2.54i)16-s + (−1.61 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.980 − 0.172i)2-s + (−0.264 + 0.964i)3-s + (−0.00818 − 0.00297i)4-s + (0.886 + 0.322i)5-s + (0.426 − 0.899i)6-s + (−0.826 + 0.563i)7-s + (0.869 + 0.502i)8-s + (−0.859 − 0.510i)9-s + (−0.813 − 0.469i)10-s + (−0.381 − 1.04i)11-s + (0.00503 − 0.00710i)12-s + (−0.631 + 1.73i)13-s + (0.907 − 0.409i)14-s + (−0.545 + 0.769i)15-s + (−0.759 − 0.637i)16-s + (−0.392 + 0.680i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.153882 + 0.397255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.153882 + 0.397255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 - 1.67i)T \) |
| 7 | \( 1 + (2.18 - 1.49i)T \) |
| good | 2 | \( 1 + (1.38 + 0.244i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.98 - 0.721i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.26 + 3.48i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.27 - 6.25i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.61 - 2.80i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.61 - 2.66i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.689 + 0.121i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.50 - 4.12i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.513 - 1.41i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 6.49T + 37T^{2} \) |
| 41 | \( 1 + (-10.4 - 3.81i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.29 + 7.32i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.63 - 2.05i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.442i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.77 + 4.00i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.54 - 6.98i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.82 - 10.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (3.83 - 2.21i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 3.06iT - 73T^{2} \) |
| 79 | \( 1 + (1.65 - 9.37i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 4.16i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.17 + 3.77i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.85 + 1.20i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.92194880227663347770775467342, −11.51673946208377901304120485049, −10.62685855598980932479217740796, −9.886147729679324378578067640624, −9.163041199296670826067194368814, −8.493002311808127290289564213561, −6.56104337563471079255274666902, −5.66242071433665895851426097210, −4.21198779683732818832485586620, −2.38621064467257904448767879291,
0.51952272149715405625242003550, 2.43593379331376190749460551959, 4.79832668344419113985879941256, 6.16744962866695762815975666538, 7.28903867421776247855252327966, 7.953682628464016803266528009784, 9.309930091601220907475581799136, 9.978999364125439908552765446619, 10.88628909157901642257281223156, 12.67509878557711873203004697863
