| L(s) = 1 | + (1.35 − 0.401i)2-s + (−1.03 − 1.38i)3-s + (1.67 − 1.08i)4-s + (1.93 + 0.705i)5-s + (−1.96 − 1.46i)6-s + (−0.744 − 0.131i)7-s + (1.83 − 2.15i)8-s + (−0.856 + 2.87i)9-s + (2.90 + 0.177i)10-s + (−0.949 − 2.60i)11-s + (−3.24 − 1.20i)12-s + (0.421 + 0.502i)13-s + (−1.06 + 0.121i)14-s + (−1.02 − 3.41i)15-s + (1.62 − 3.65i)16-s + (−3.79 + 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.958 − 0.284i)2-s + (−0.597 − 0.801i)3-s + (0.838 − 0.544i)4-s + (0.866 + 0.315i)5-s + (−0.800 − 0.598i)6-s + (−0.281 − 0.0496i)7-s + (0.649 − 0.760i)8-s + (−0.285 + 0.958i)9-s + (0.920 + 0.0562i)10-s + (−0.286 − 0.786i)11-s + (−0.937 − 0.346i)12-s + (0.116 + 0.139i)13-s + (−0.283 + 0.0323i)14-s + (−0.265 − 0.882i)15-s + (0.406 − 0.913i)16-s + (−0.919 + 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60331 - 0.995977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60331 - 0.995977i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 + 0.401i)T \) |
| 3 | \( 1 + (1.03 + 1.38i)T \) |
| good | 5 | \( 1 + (-1.93 - 0.705i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.744 + 0.131i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.949 + 2.60i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.421 - 0.502i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.79 - 2.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 0.270i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 7.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.89 - 4.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.16 - 0.733i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.70 + 1.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.31 - 5.14i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.00 - 1.09i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.89 - 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.876T + 53T^{2} \) |
| 59 | \( 1 + (-2.94 + 8.09i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (12.1 + 2.14i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 2.79i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.08 + 12.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.17 + 12.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.09 + 8.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.10 - 8.47i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.55 - 3.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 0.579i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−12.33343387137923783963896818772, −11.17793167940721678205521820811, −10.73380586487427282117096804610, −9.453082128001920867155857784745, −7.79758913029116209446599140963, −6.52882434243030239532367897146, −6.03732709871890773971453045354, −4.93157806056726409092865169640, −3.10860407485420519996600387321, −1.70359954258514379847195777350,
2.53050939657931399301769955468, 4.21992795649900299811883688425, 5.08332366681042083689839332505, 6.06548903401804312780720587787, 6.96125783066471629126946309985, 8.624822671159947132300955928580, 9.783559733688345063232748229591, 10.61478797266216093738777979099, 11.66742020394233346469996273569, 12.60265097029192644488654849790
