| L(s) = 1 | + (−0.0448 − 0.123i)2-s + (1.62 + 0.587i)3-s + (1.51 − 1.27i)4-s + (0.231 + 1.31i)5-s + (−0.000699 − 0.227i)6-s + (−0.772 − 2.53i)7-s + (−0.452 − 0.261i)8-s + (2.30 + 1.91i)9-s + (0.151 − 0.0873i)10-s + (−4.31 − 0.759i)11-s + (3.22 − 1.18i)12-s + (−1.38 + 3.81i)13-s + (−0.277 + 0.208i)14-s + (−0.393 + 2.27i)15-s + (0.676 − 3.83i)16-s + (1.85 + 3.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.0317 − 0.0871i)2-s + (0.940 + 0.339i)3-s + (0.759 − 0.637i)4-s + (0.103 + 0.586i)5-s + (−0.000285 − 0.0927i)6-s + (−0.292 − 0.956i)7-s + (−0.159 − 0.0923i)8-s + (0.769 + 0.638i)9-s + (0.0478 − 0.0276i)10-s + (−1.29 − 0.229i)11-s + (0.930 − 0.341i)12-s + (−0.385 + 1.05i)13-s + (−0.0740 + 0.0557i)14-s + (−0.101 + 0.587i)15-s + (0.169 − 0.959i)16-s + (0.450 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64599 - 0.0837184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64599 - 0.0837184i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.62 - 0.587i)T \) |
| 7 | \( 1 + (0.772 + 2.53i)T \) |
| good | 2 | \( 1 + (0.0448 + 0.123i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.231 - 1.31i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (4.31 + 0.759i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.38 - 3.81i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.85 - 3.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.31 + 2.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.242 - 0.289i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.57 - 4.31i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.693 + 0.826i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.172 + 0.298i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 + 1.85i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.390 - 2.21i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (9.77 + 8.19i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 9.19iT - 53T^{2} \) |
| 59 | \( 1 + (-2.24 - 12.7i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.14 + 1.36i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 - 1.96i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.30 + 1.33i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.63 - 2.67i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.48 + 1.99i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 1.46i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.59 + 9.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.5 - 2.55i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.76110322068616249917619672982, −11.17203306311286477485808565376, −10.40647498631566529849348186544, −9.952836705718468247638983160329, −8.525593773459567715502415520573, −7.29924559421858115762824651727, −6.58415489858408926407252366312, −4.88292153820271451847752433299, −3.34862435027317260679713836122, −2.12215285300381282511656673305,
2.30222797095000372415086318635, 3.17674697030904953857823489889, 5.11927155393591232900286323589, 6.48975508753989288984267554583, 7.84267183576469791706348658329, 8.228352880009328846003913396475, 9.422078589720078081359003515473, 10.50762017424586084353974482866, 12.00931297162052718351910105323, 12.72782176639392746729505753878
