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 | \( 1 + 17.3iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.97517590636691842209829807427, −10.29233125531613752785678339070, −9.502844452221826321801328251494, −8.670676711468156846644875228916, −7.87254139132288101596382382545, −6.36841696237561543730530495532, −5.57355300128677886401587579543, −4.38082759710359553075574900837, −3.24354565953934307254864852773, −2.00381032521859041150670228900,
1.39876642209692133624301680685, 2.31965705615484027027562375473, 4.14129297242995947238260935568, 5.10412713105915544256972825094, 6.56208256735920456303519445966, 7.20488959036581092647778589963, 8.178879946103446912780320300217, 9.033770817009944580319910657090, 9.920592196395164269045865155912, 11.06751089746579069021160570858