L(s) = 1 | + (−0.360 − 0.208i)2-s + (−0.866 + 0.5i)3-s + (−0.913 − 1.58i)4-s + (0.414 + 0.239i)5-s + 0.416·6-s + (−2.50 + 0.863i)7-s + 1.59i·8-s + (0.499 − 0.866i)9-s + (−0.0995 − 0.172i)10-s + (−2.41 − 2.27i)11-s + (1.58 + 0.913i)12-s − 5.34·13-s + (1.08 + 0.209i)14-s − 0.478·15-s + (−1.49 + 2.58i)16-s + (−0.123 − 0.213i)17-s + ⋯ |
L(s) = 1 | + (−0.255 − 0.147i)2-s + (−0.499 + 0.288i)3-s + (−0.456 − 0.790i)4-s + (0.185 + 0.106i)5-s + 0.170·6-s + (−0.945 + 0.326i)7-s + 0.563i·8-s + (0.166 − 0.288i)9-s + (−0.0314 − 0.0545i)10-s + (−0.726 − 0.686i)11-s + (0.456 + 0.263i)12-s − 1.48·13-s + (0.289 + 0.0559i)14-s − 0.123·15-s + (−0.373 + 0.647i)16-s + (−0.0298 − 0.0517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00140121 + 0.101080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00140121 + 0.101080i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (2.50 - 0.863i)T \) |
| 11 | \( 1 + (2.41 + 2.27i)T \) |
good | 2 | \( 1 + (0.360 + 0.208i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.414 - 0.239i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 5.34T + 13T^{2} \) |
| 17 | \( 1 + (0.123 + 0.213i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.02 - 3.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.540 - 0.936i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 9.25iT - 29T^{2} \) |
| 31 | \( 1 + (1.00 - 0.580i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.23 - 3.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 - 2.69iT - 43T^{2} \) |
| 47 | \( 1 + (-6.81 - 3.93i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.242 - 0.420i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.553i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.66 + 4.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.52 + 2.64i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + (3.94 + 6.82i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (12.3 + 7.13i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + (-11.9 - 6.92i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.20iT - 97T^{2} \) |
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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
−11.63511235168770952013208312275, −10.39497720717372774360661328641, −9.980293444278516443056742170685, −9.121031779603909646672155098908, −7.81211467000199203113035781922, −6.23255505569778546391538351063, −5.62586177238445253794301444450, −4.36837805018961672780748193181, −2.53100358308475758484728296235, −0.089462001657176143203016438982,
2.68827607578542369831551896427, 4.28465978808539206438797467098, 5.43632433683827987936245663979, 7.11957898727419771301406941362, 7.32557406310664003785577561663, 8.871239174581491878260176463087, 9.729123096756549090575593191513, 10.60581072866027737382788080908, 12.02043647362298843331144334373, 12.80405681352991245786469371455