L(s) = 1 | + (0.880 − 2.24i)2-s + (−0.697 + 1.58i)3-s + (−2.79 − 2.58i)4-s + (−0.193 + 0.0931i)5-s + (2.94 + 2.96i)6-s + (−2.55 − 0.680i)7-s + (−3.92 + 1.88i)8-s + (−2.02 − 2.21i)9-s + (0.0386 + 0.515i)10-s + (−2.29 − 2.88i)11-s + (6.05 − 2.61i)12-s + (1.02 − 2.62i)13-s + (−3.77 + 5.13i)14-s + (−0.0127 − 0.371i)15-s + (0.215 + 2.87i)16-s + (−4.84 + 4.49i)17-s + ⋯ |
L(s) = 1 | + (0.622 − 1.58i)2-s + (−0.402 + 0.915i)3-s + (−1.39 − 1.29i)4-s + (−0.0864 + 0.0416i)5-s + (1.20 + 1.20i)6-s + (−0.966 − 0.257i)7-s + (−1.38 + 0.668i)8-s + (−0.675 − 0.737i)9-s + (0.0122 + 0.163i)10-s + (−0.692 − 0.868i)11-s + (1.74 − 0.755i)12-s + (0.285 − 0.727i)13-s + (−1.00 + 1.37i)14-s + (−0.00329 − 0.0959i)15-s + (0.0538 + 0.718i)16-s + (−1.17 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.181951 + 0.617820i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.181951 + 0.617820i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.697 - 1.58i)T \) |
| 7 | \( 1 + (2.55 + 0.680i)T \) |
good | 2 | \( 1 + (-0.880 + 2.24i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.193 - 0.0931i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (2.29 + 2.88i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 2.62i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (4.84 - 4.49i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.18 - 3.78i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 + 5.82i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (4.39 + 1.35i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-3.77 + 6.53i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.89 - 1.50i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-2.47 - 1.68i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (2.32 - 1.58i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-4.35 + 11.0i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-8.19 + 2.52i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.335 - 0.228i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (1.92 - 1.78i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 12.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (6.79 + 1.02i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-6.35 - 11.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.74 + 7.00i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.422 - 1.07i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-2.51 + 4.35i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.63612166962713538540459678101, −10.21245516170214566559841610736, −9.269279406061263510554321229458, −8.245525754100112349267993192151, −6.27390728778498238239097202010, −5.55115680636415266574660963123, −4.20838197160669604991904574623, −3.62921209759183876770315039804, −2.55613636556161161927042019058, −0.32896310686168311910551818440,
2.54005247900338231766103602471, 4.32456626015214210432766083427, 5.27446374823681020951490013119, 6.23062775578365467876432093366, 6.98048308798406694120610564943, 7.43081929785698077426036664588, 8.638425898354811930991295046742, 9.435452049426435943184615368434, 10.96301446813289480907893283743, 12.01643125791634477238920862452