L(s) = 1 | + (−0.866 + 0.5i)2-s + (−1.44 + 0.956i)3-s + (0.499 − 0.866i)4-s + (0.0938 + 0.162i)5-s + (0.772 − 1.55i)6-s + (−2.64 + 0.0638i)7-s + 0.999i·8-s + (1.16 − 2.76i)9-s + (−0.162 − 0.0938i)10-s + (0.866 + 0.5i)11-s + (0.106 + 1.72i)12-s − 4.70i·13-s + (2.25 − 1.37i)14-s + (−0.291 − 0.144i)15-s + (−0.5 − 0.866i)16-s + (−1.48 + 2.57i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.833 + 0.552i)3-s + (0.249 − 0.433i)4-s + (0.0419 + 0.0727i)5-s + (0.315 − 0.632i)6-s + (−0.999 + 0.0241i)7-s + 0.353i·8-s + (0.389 − 0.920i)9-s + (−0.0514 − 0.0296i)10-s + (0.261 + 0.150i)11-s + (0.0307 + 0.499i)12-s − 1.30i·13-s + (0.603 − 0.368i)14-s + (−0.0751 − 0.0374i)15-s + (−0.125 − 0.216i)16-s + (−0.360 + 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.608399 - 0.0954280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.608399 - 0.0954280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (1.44 - 0.956i)T \) |
| 7 | \( 1 + (2.64 - 0.0638i)T \) |
| 11 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-0.0938 - 0.162i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 + (1.48 - 2.57i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 0.929i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.05 + 1.18i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.25iT - 29T^{2} \) |
| 31 | \( 1 + (-8.78 - 5.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.73 + 4.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.187T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + (2.79 + 4.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.50 - 5.48i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.32 - 4.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 6.28i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.28 - 2.23i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (13.8 + 8.02i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.99 + 6.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + (4.29 + 7.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.00iT - 97T^{2} \) |
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
−10.50293182418019411434525549381, −10.33159571808992689989135451891, −9.337459647966211273790918053973, −8.439851545843328859545931385751, −7.14757541942327216608033718535, −6.31827951198726167727251382593, −5.59532694993044974089354255746, −4.32675173689694367020090865242, −2.93554525669192544910553461356, −0.61105991229685101048697447851,
1.18634558499619859679573545454, 2.74235216127029777850563884773, 4.25306098344425189134226133888, 5.60376769283013492634876075387, 6.80109396231120825243024516900, 7.08734709328029207782664464865, 8.530604317110885870917143272031, 9.437702414502855462452430337500, 10.16171254573950443430654365348, 11.31387383699749376807041267145