| L(s) = 1 | + (0.873 + 2.40i)2-s + (−1.61 + 0.637i)3-s + (−3.46 + 2.91i)4-s + (0.802 − 0.673i)5-s + (−2.93 − 3.30i)6-s + (−2.47 + 0.941i)7-s + (−5.59 − 3.23i)8-s + (2.18 − 2.05i)9-s + (2.31 + 1.33i)10-s + (−2.64 + 3.15i)11-s + (3.73 − 6.90i)12-s + (2.92 + 3.48i)13-s + (−4.42 − 5.11i)14-s + (−0.862 + 1.59i)15-s + (1.29 − 7.33i)16-s + (1.74 − 3.01i)17-s + ⋯ |
| L(s) = 1 | + (0.617 + 1.69i)2-s + (−0.929 + 0.368i)3-s + (−1.73 + 1.45i)4-s + (0.358 − 0.301i)5-s + (−1.19 − 1.35i)6-s + (−0.934 + 0.355i)7-s + (−1.97 − 1.14i)8-s + (0.728 − 0.684i)9-s + (0.732 + 0.423i)10-s + (−0.797 + 0.949i)11-s + (1.07 − 1.99i)12-s + (0.812 + 0.967i)13-s + (−1.18 − 1.36i)14-s + (−0.222 + 0.411i)15-s + (0.323 − 1.83i)16-s + (0.422 − 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.205838 - 0.925878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.205838 - 0.925878i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.61 - 0.637i)T \) |
| 7 | \( 1 + (2.47 - 0.941i)T \) |
| good | 2 | \( 1 + (-0.873 - 2.40i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.802 + 0.673i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (2.64 - 3.15i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.92 - 3.48i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.74 + 3.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.80 + 2.77i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.07 - 5.70i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.57 - 1.87i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.681 + 0.812i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + (-0.164 + 0.138i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-9.06 + 3.29i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.674 + 0.565i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.554 + 0.320i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 10.5i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 1.55i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.6 - 3.86i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.48 + 5.47i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 + (-5.09 + 1.85i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.87 + 6.60i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.18 - 3.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.53 + 9.72i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−13.23317959569873549118777132139, −12.54361122957769259278108170173, −11.42137855210521041957496399709, −9.654937145840953379527924901650, −9.246270856356103886670545079763, −7.51670190735233851810908254394, −6.75566670041193463827742889632, −5.66469787381174889956692165605, −5.09819097373067904332528813999, −3.75635886203007979883192867488,
0.820124321125029821765027257881, 2.74351241362424646419876194469, 3.92716919181257650188677822319, 5.55424712061918595075419999097, 6.14583233211653740314904403159, 8.031321376243232915480508324562, 9.751089048046865358752642166089, 10.52835312063294822739667636609, 10.88283225329249020060861080036, 12.12754841328838692341568753020
