| 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 | \( 1 + (-14.5 + 2.55i)T + (91.1 - 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.72782176639392746729505753878, −12.00931297162052718351910105323, −10.50762017424586084353974482866, −9.422078589720078081359003515473, −8.228352880009328846003913396475, −7.84267183576469791706348658329, −6.48975508753989288984267554583, −5.11927155393591232900286323589, −3.17674697030904953857823489889, −2.30222797095000372415086318635,
2.12215285300381282511656673305, 3.34862435027317260679713836122, 4.88292153820271451847752433299, 6.58415489858408926407252366312, 7.29924559421858115762824651727, 8.525593773459567715502415520573, 9.952836705718468247638983160329, 10.40647498631566529849348186544, 11.17203306311286477485808565376, 12.76110322068616249917619672982
