| L(s) = 1 | + (−1.99 − 0.0489i)2-s + (3.99 + 0.195i)4-s + (−3.75 + 1.36i)5-s + (−12.2 + 2.15i)7-s + (−7.97 − 0.586i)8-s + (7.57 − 2.54i)10-s + (5.05 − 13.8i)11-s + (8.77 + 7.35i)13-s + (24.5 − 3.70i)14-s + (15.9 + 1.56i)16-s + (−2.73 + 4.73i)17-s + (20.0 − 11.5i)19-s + (−15.2 + 4.72i)20-s + (−10.7 + 27.5i)22-s + (15.3 + 2.70i)23-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0244i)2-s + (0.998 + 0.0489i)4-s + (−0.751 + 0.273i)5-s + (−1.74 + 0.307i)7-s + (−0.997 − 0.0733i)8-s + (0.757 − 0.254i)10-s + (0.459 − 1.26i)11-s + (0.674 + 0.566i)13-s + (1.75 − 0.264i)14-s + (0.995 + 0.0977i)16-s + (−0.160 + 0.278i)17-s + (1.05 − 0.608i)19-s + (−0.763 + 0.236i)20-s + (−0.490 + 1.25i)22-s + (0.666 + 0.117i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.688799 - 0.0902563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.688799 - 0.0902563i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.0489i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.75 - 1.36i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (12.2 - 2.15i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-5.05 + 13.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.77 - 7.35i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.73 - 4.73i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-20.0 + 11.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-15.3 - 2.70i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-16.7 + 14.0i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-11.7 - 2.07i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (6.45 - 11.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-55.7 - 46.7i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-20.3 + 55.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-20.2 + 3.56i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 16.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (6.75 + 18.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 29.8i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-52.6 + 62.8i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (36.4 + 21.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (44.1 + 76.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.2 + 45.5i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (1.27 + 1.52i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (45.8 + 79.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-64.3 - 23.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.35690694103818191692518454875, −10.35693740429259404439424633369, −9.273055273055975229244517653211, −8.815490228457773925055268367764, −7.56943998211088796556900759685, −6.58733216952329734935154285089, −5.95200548458149694843809690123, −3.65359325228777145536733074026, −2.92543189840607120820031342606, −0.68345955508822257292515301078,
0.850617674520974600600967709479, 2.89001347678084189442867440273, 4.00298734036294593585015868934, 5.85646646614068525530534353171, 6.91913565220861300213779708400, 7.53996152054866028427287786974, 8.750971200624370255781678535132, 9.636474667247975718498160676802, 10.19196896599335704807267111857, 11.32590282819197494702420577944
