| L(s) = 1 | + (0.0921 + 0.522i)2-s + (−0.758 + 1.55i)3-s + (1.61 − 0.587i)4-s + (0.810 + 0.679i)5-s + (−0.883 − 0.253i)6-s + (0.939 + 0.342i)7-s + (0.986 + 1.70i)8-s + (−1.84 − 2.36i)9-s + (−0.280 + 0.486i)10-s + (−0.921 + 0.773i)11-s + (−0.309 + 2.95i)12-s + (0.0549 − 0.311i)13-s + (−0.0921 + 0.522i)14-s + (−1.67 + 0.745i)15-s + (1.83 − 1.53i)16-s + (−2.46 + 4.27i)17-s + ⋯ |
| L(s) = 1 | + (0.0651 + 0.369i)2-s + (−0.437 + 0.898i)3-s + (0.807 − 0.293i)4-s + (0.362 + 0.303i)5-s + (−0.360 − 0.103i)6-s + (0.355 + 0.129i)7-s + (0.348 + 0.604i)8-s + (−0.616 − 0.787i)9-s + (−0.0887 + 0.153i)10-s + (−0.277 + 0.233i)11-s + (−0.0894 + 0.854i)12-s + (0.0152 − 0.0865i)13-s + (−0.0246 + 0.139i)14-s + (−0.431 + 0.192i)15-s + (0.457 − 0.383i)16-s + (−0.598 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05271 + 0.813701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05271 + 0.813701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.758 - 1.55i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 2 | \( 1 + (-0.0921 - 0.522i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.810 - 0.679i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (0.921 - 0.773i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0549 + 0.311i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.46 - 4.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.145 - 0.251i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0206 - 0.00750i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.11 + 6.33i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.45 - 1.25i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.39 + 7.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.47 + 8.38i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.51 + 3.78i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.39 - 0.872i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + (-5.30 - 4.45i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.37 + 3.41i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.74 - 15.5i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.52 - 6.10i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.62 - 6.28i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.658 + 3.73i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.02 + 17.1i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (0.890 + 1.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.60 - 6.38i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.61171362702509801951834855189, −11.52207053389809747473174250531, −10.73630644650078785083817936425, −10.11282046148021516934588080678, −8.830975955643846980635811976030, −7.52533102673641094646714505147, −6.22639274728607218346018017929, −5.57080099879067353163197904461, −4.17691327288476209611233034275, −2.34384489014388720405203642667,
1.51555883502470359515268975982, 2.89477814676350522967197085091, 4.91564263503934426515731197587, 6.18630376736429872291120474800, 7.18378986704144490685302791483, 8.038293206800777831341551731049, 9.431180759578705374539751253010, 10.92165466302001537083325762755, 11.32497226950616854659116989329, 12.35782871901047134601510738533
