L(s) = 1 | + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (−2 + 3.46i)5-s + (2 − 1.73i)7-s + 3·8-s + (1.99 + 3.46i)10-s + (1 + 1.73i)11-s + 13-s + (−0.499 − 2.59i)14-s + (0.500 − 0.866i)16-s + (−3 − 5.19i)17-s + (−2 + 3.46i)19-s − 4·20-s + 1.99·22-s + (3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.894 + 1.54i)5-s + (0.755 − 0.654i)7-s + 1.06·8-s + (0.632 + 1.09i)10-s + (0.301 + 0.522i)11-s + 0.277·13-s + (−0.133 − 0.694i)14-s + (0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.458 + 0.794i)19-s − 0.894·20-s + 0.426·22-s + (0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43149 + 0.182420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43149 + 0.182420i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 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 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 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 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−12.34746182005289072543014347442, −11.40145777942347560492962968641, −11.02764885332401663289194776595, −10.14691296986573405702794469107, −8.300079537147651787454175107691, −7.31358002864404554212208615440, −6.77741518484778729195316987935, −4.52109332697467195722041791113, −3.63529057435228304430447279733, −2.36587211598253029337752680266,
1.48624052642395739323500039129, 4.14387406148145574160586639230, 5.06267473736550086588148582320, 5.98028871689103106929561727479, 7.46298497246863215298517608625, 8.491108875433920823632139901996, 9.097171371402099595716051490550, 10.91163410613688514849341903148, 11.52135417211737529592403696610, 12.66325444120093442589088370226