L(s) = 1 | + (−0.439 + 0.761i)2-s + (0.613 + 1.06i)4-s + (0.673 + 1.16i)5-s + (−0.5 + 0.866i)7-s − 2.83·8-s − 1.18·10-s + (0.826 − 1.43i)11-s + (1.68 + 2.91i)13-s + (−0.439 − 0.761i)14-s + (0.0209 − 0.0362i)16-s − 0.467·17-s − 3.22·19-s + (−0.826 + 1.43i)20-s + (0.726 + 1.25i)22-s + (4.47 + 7.74i)23-s + ⋯ |
L(s) = 1 | + (−0.310 + 0.538i)2-s + (0.306 + 0.531i)4-s + (0.301 + 0.521i)5-s + (−0.188 + 0.327i)7-s − 1.00·8-s − 0.374·10-s + (0.249 − 0.431i)11-s + (0.467 + 0.809i)13-s + (−0.117 − 0.203i)14-s + (0.00523 − 0.00906i)16-s − 0.113·17-s − 0.740·19-s + (−0.184 + 0.320i)20-s + (0.154 + 0.268i)22-s + (0.932 + 1.61i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.692087 + 0.824798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692087 + 0.824798i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.439 - 0.761i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.673 - 1.16i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.826 + 1.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 2.91i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 + (-4.47 - 7.74i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.13 + 5.42i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.67 + 8.10i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.573T + 53T^{2} \) |
| 59 | \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.81 - 6.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.554T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + (-0.949 + 1.64i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.89154894888064048106140319647, −11.68736117843268124713128679370, −11.08059827952534416893018254807, −9.591572592453379326014656879601, −8.790687227432763431757829816578, −7.68767236848735720792094942811, −6.61444128359880553100548282014, −5.89007315023916187007989011715, −3.90727407255354225909979301245, −2.50273273181646134525430891596,
1.18828745570387212178649372787, 2.88912242770314900545470895722, 4.70116061828204160052833276461, 5.97319602943244698246172172806, 7.01614181346704908930936333758, 8.621950746548719260976662824586, 9.356180252745843499348996705443, 10.56958287162613011098782157462, 10.92188029914600836593950433077, 12.46414059082505661558898573096