L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 − 1.73i)7-s − 0.999·8-s + (−3 − 5.19i)11-s + 5·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s − 6·22-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s + (2.5 − 4.33i)26-s + (−0.5 + 2.59i)28-s + 6·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.904 − 1.56i)11-s + 1.38·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s − 1.27·22-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + (0.490 − 0.849i)26-s + (−0.0944 + 0.490i)28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.565873 - 1.14150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.565873 - 1.14150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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
−11.08203437333440760215267638765, −10.38805947456277925055209132771, −9.285271695669495452058939774900, −8.470900719350957092343565843175, −7.14638130280981763896625698944, −6.08706715299460033565153939227, −5.09452636472162406048197616185, −3.64707567206504421166713519353, −2.91172864827382921688936916286, −0.77667562379779309584132223030,
2.30769262666340611120845357023, 3.78199205398717790541895117723, 4.87961467951190439939380869548, 6.13183919178818791994689393089, 6.65944530890540019030159135597, 8.061332521392267183355566815387, 8.645058975598650243652514653308, 9.918004128200334898047092817731, 10.57837394671706512395657219525, 12.05602328436749953487507509593