L(s) = 1 | + (0.866 − 1.5i)3-s + (1.5 − 0.866i)5-s + (1.73 − 2i)7-s + (0.866 + 0.5i)11-s + 3.46i·13-s − 3i·15-s + (−1.5 − 0.866i)17-s + (−2.59 − 4.5i)19-s + (−1.50 − 4.33i)21-s + (−0.866 + 0.5i)23-s + (−1 + 1.73i)25-s + 5.19·27-s − 4·29-s + (0.866 − 1.5i)31-s + (1.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s + (0.670 − 0.387i)5-s + (0.654 − 0.755i)7-s + (0.261 + 0.150i)11-s + 0.960i·13-s − 0.774i·15-s + (−0.363 − 0.210i)17-s + (−0.596 − 1.03i)19-s + (−0.327 − 0.944i)21-s + (−0.180 + 0.104i)23-s + (−0.200 + 0.346i)25-s + 1.00·27-s − 0.742·29-s + (0.155 − 0.269i)31-s + (0.261 − 0.150i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63779 - 1.01609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63779 - 1.01609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 3 | \( 1 + (-0.866 + 1.5i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.59 + 4.5i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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
−11.06751089746579069021160570858, −9.920592196395164269045865155912, −9.033770817009944580319910657090, −8.178879946103446912780320300217, −7.20488959036581092647778589963, −6.56208256735920456303519445966, −5.10412713105915544256972825094, −4.14129297242995947238260935568, −2.31965705615484027027562375473, −1.39876642209692133624301680685,
2.00381032521859041150670228900, 3.24354565953934307254864852773, 4.38082759710359553075574900837, 5.57355300128677886401587579543, 6.36841696237561543730530495532, 7.87254139132288101596382382545, 8.670676711468156846644875228916, 9.502844452221826321801328251494, 10.29233125531613752785678339070, 10.97517590636691842209829807427