| L(s) = 1 | + (−0.686 − 0.727i)2-s + (−1.70 + 0.322i)3-s + (−0.0581 + 0.998i)4-s + (0.269 − 0.0314i)5-s + (1.40 + 1.01i)6-s + (1.70 + 0.854i)7-s + (0.766 − 0.642i)8-s + (2.79 − 1.09i)9-s + (−0.207 − 0.174i)10-s + (0.493 + 1.14i)11-s + (−0.223 − 1.71i)12-s + (6.47 + 1.53i)13-s + (−0.546 − 1.82i)14-s + (−0.448 + 0.140i)15-s + (−0.993 − 0.116i)16-s + (0.874 − 4.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.514i)2-s + (−0.982 + 0.186i)3-s + (−0.0290 + 0.499i)4-s + (0.120 − 0.0140i)5-s + (0.572 + 0.414i)6-s + (0.643 + 0.323i)7-s + (0.270 − 0.227i)8-s + (0.930 − 0.366i)9-s + (−0.0657 − 0.0551i)10-s + (0.148 + 0.344i)11-s + (−0.0644 − 0.495i)12-s + (1.79 + 0.425i)13-s + (−0.145 − 0.487i)14-s + (−0.115 + 0.0363i)15-s + (−0.248 − 0.0290i)16-s + (0.212 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.769065 - 0.0397462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769065 - 0.0397462i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.686 + 0.727i)T \) |
| 3 | \( 1 + (1.70 - 0.322i)T \) |
| good | 5 | \( 1 + (-0.269 + 0.0314i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-1.70 - 0.854i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.493 - 1.14i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (-6.47 - 1.53i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.874 + 4.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.683 - 3.87i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.82 + 0.916i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (1.75 - 5.85i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (1.57 + 1.03i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-7.10 + 2.58i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (0.0646 - 0.0685i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.298 + 0.401i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (8.22 - 5.40i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-5.41 + 9.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.219 - 0.509i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.127 + 2.18i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (3.40 + 11.3i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-3.56 - 2.98i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.592 + 0.497i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.82 + 10.4i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-5.24 - 5.55i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (8.83 - 7.41i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.13 - 0.366i)T + (94.3 + 22.3i)T^{2} \) |
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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
−12.54897820627034914269010313626, −11.47690098782845637359603066118, −11.15062819752522736182120803436, −9.921769202686689398838119227374, −9.000355856457051815665158077425, −7.71722948474398924286535123408, −6.39315242113628954888058383697, −5.17862785995804399061583480099, −3.78604629933488377345627119790, −1.50660155169308263672462866736,
1.27302096723230869400929266977, 4.15359594269836394976969549568, 5.65520859516271516585732285044, 6.34539890343323777011408350699, 7.66992925503058213053652485296, 8.575740866810735792447645700918, 9.998939390360626549034844354229, 10.98921058164457559509642690813, 11.48424936104702450474847755017, 13.04753630150730591347880216801
