L(s) = 1 | + (0.809 − 2.06i)2-s + (1.12 − 1.31i)3-s + (−2.12 − 1.97i)4-s + (0.273 − 0.131i)5-s + (−1.80 − 3.38i)6-s + (2.25 − 1.38i)7-s + (−1.80 + 0.869i)8-s + (−0.476 − 2.96i)9-s + (−0.0501 − 0.669i)10-s + (2.70 + 3.39i)11-s + (−4.99 + 0.587i)12-s + (−0.884 + 2.25i)13-s + (−1.02 − 5.76i)14-s + (0.133 − 0.507i)15-s + (−0.102 − 1.36i)16-s + (−0.427 + 0.396i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 1.45i)2-s + (0.648 − 0.761i)3-s + (−1.06 − 0.988i)4-s + (0.122 − 0.0588i)5-s + (−0.738 − 1.38i)6-s + (0.852 − 0.522i)7-s + (−0.638 + 0.307i)8-s + (−0.158 − 0.987i)9-s + (−0.0158 − 0.211i)10-s + (0.816 + 1.02i)11-s + (−1.44 + 0.169i)12-s + (−0.245 + 0.624i)13-s + (−0.274 − 1.54i)14-s + (0.0344 − 0.131i)15-s + (−0.0255 − 0.341i)16-s + (−0.103 + 0.0962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.557701 - 2.35589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.557701 - 2.35589i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.12 + 1.31i)T \) |
| 7 | \( 1 + (-2.25 + 1.38i)T \) |
good | 2 | \( 1 + (-0.809 + 2.06i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.273 + 0.131i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.70 - 3.39i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.884 - 2.25i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (0.427 - 0.396i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.54 - 6.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 - 7.60i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (7.33 + 2.26i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-3.13 + 5.43i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.26 - 2.85i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (6.84 + 4.66i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-7.64 + 5.21i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (2.33 - 5.94i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-5.06 + 1.56i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (6.52 - 4.44i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-0.637 + 0.591i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.88 + 6.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.974 + 4.26i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.06 - 0.763i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (1.66 + 2.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.997 + 2.54i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (1.95 + 4.99i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-1.87 + 3.24i)T + (-48.5 - 84.0i)T^{2} \) |
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
−11.07223072905074038708902849474, −9.791173978808540625100885246012, −9.380145317545811989275271454509, −7.925076209729733969299217719632, −7.23739851610496216722409351924, −5.81177606291655388734140196991, −4.26299576271833427151817553681, −3.76740418537176751398673139269, −2.02516152089676341696806320741, −1.58191823930907928618256846671,
2.55802457525224572255599130894, 4.08183690027553256749427702032, 4.84598449408894714348851160234, 5.78843807843673512369680732196, 6.73485799822824841987161861203, 8.028236081843652003036910979766, 8.484147388951068712453853654817, 9.251624881395066220486519572575, 10.63953007902845030607238272587, 11.35731325349949801881270317447