L(s) = 1 | + 0.0683·2-s + (−0.539 − 1.64i)3-s − 1.99·4-s + (1.33 + 2.30i)5-s + (−0.0368 − 0.112i)6-s − 0.273·8-s + (−2.41 + 1.77i)9-s + (0.0910 + 0.157i)10-s + (0.799 − 1.38i)11-s + (1.07 + 3.28i)12-s + (−2.62 + 4.54i)13-s + (3.07 − 3.43i)15-s + 3.97·16-s + (3.27 + 5.67i)17-s + (−0.165 + 0.121i)18-s + (−0.950 + 1.64i)19-s + ⋯ |
L(s) = 1 | + 0.0483·2-s + (−0.311 − 0.950i)3-s − 0.997·4-s + (0.595 + 1.03i)5-s + (−0.0150 − 0.0459i)6-s − 0.0965·8-s + (−0.805 + 0.592i)9-s + (0.0287 + 0.0498i)10-s + (0.241 − 0.417i)11-s + (0.310 + 0.948i)12-s + (−0.728 + 1.26i)13-s + (0.794 − 0.887i)15-s + 0.992·16-s + (0.793 + 1.37i)17-s + (−0.0389 + 0.0286i)18-s + (−0.218 + 0.377i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.805611 + 0.444023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.805611 + 0.444023i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.539 + 1.64i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.0683T + 2T^{2} \) |
| 5 | \( 1 + (-1.33 - 2.30i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.799 + 1.38i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.27 - 5.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.950 - 1.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 + (2.11 - 3.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 + (4.44 + 7.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.09 + 1.90i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.813T + 79T^{2} \) |
| 83 | \( 1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.235 - 0.407i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \) |
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
−11.37356331510097691929170628857, −10.29665588981368186102840124577, −9.569562458315441369522186760993, −8.423562235239927939774819805710, −7.58158063112828346780940198102, −6.39254203154300839198978803635, −5.90974109215165717625268264998, −4.52051484424678774028110464524, −3.09991472158846961963283331705, −1.62646251060612844466334832767,
0.64256605923952240127117853126, 3.05358426302652539366719604375, 4.44552127577185007856518633336, 5.14912573714231146072031805163, 5.65212183534813691551426763076, 7.40775539113715336429282120735, 8.722668091555946463569117790161, 9.196803409292925822283372653271, 9.953630347855809537761529175037, 10.64557008054040942899039942331