L(s) = 1 | + (−1.64 + 2.05i)2-s + (−1.09 − 4.79i)4-s + (−3.33 − 1.60i)5-s + (−2.39 + 1.11i)7-s + (6.91 + 3.33i)8-s + (8.78 − 4.22i)10-s + (1.47 − 1.84i)11-s + (−0.990 + 1.24i)13-s + (1.63 − 6.76i)14-s + (−9.33 + 4.49i)16-s + (0.263 − 1.15i)17-s + 4.37·19-s + (−4.05 + 17.7i)20-s + (1.38 + 6.05i)22-s + (1.93 + 8.46i)23-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.45i)2-s + (−0.547 − 2.39i)4-s + (−1.49 − 0.719i)5-s + (−0.906 + 0.422i)7-s + (2.44 + 1.17i)8-s + (2.77 − 1.33i)10-s + (0.443 − 0.556i)11-s + (−0.274 + 0.344i)13-s + (0.436 − 1.80i)14-s + (−2.33 + 1.12i)16-s + (0.0637 − 0.279i)17-s + 1.00·19-s + (−0.907 + 3.97i)20-s + (0.294 + 1.29i)22-s + (0.402 + 1.76i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0714 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0714 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.290656 + 0.312212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.290656 + 0.312212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.39 - 1.11i)T \) |
good | 2 | \( 1 + (1.64 - 2.05i)T + (-0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (3.33 + 1.60i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-1.47 + 1.84i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.990 - 1.24i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.263 + 1.15i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 + (-1.93 - 8.46i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.443 + 1.94i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 + (0.0740 - 0.324i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-3.96 - 1.91i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.93 - 1.41i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-4.93 + 6.19i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.847 - 3.71i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (-8.45 + 4.07i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.964 - 4.22i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 + (-1.49 - 6.53i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (3.58 + 4.48i)T + (-16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + (7.79 + 9.77i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.76 - 4.71i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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
−11.36577850635572178966850839855, −9.923667253170402098256051241371, −9.204788592570739813652382825238, −8.605918752383840324657995703309, −7.63870903761533645663066291184, −7.09426690486323139660062624224, −5.93212158254015427951734066474, −4.99972729314427577481396994649, −3.59239425586031594528221039320, −0.832437725305347766130269558699,
0.62400265287847216045786820859, 2.68829622736624329010455266159, 3.50929447645130682004534485699, 4.34541411447014350778482561991, 6.79636779965383694553119816148, 7.46523375735057524492024427736, 8.318936081306068435555771635722, 9.306931879053653187620537350569, 10.21642917617522832723559942068, 10.76308949906306576096944631559