| L(s) = 1 | + (3.95 + 8.21i)2-s + (−41.0 − 60.1i)3-s + (107. − 135. i)4-s + (−599. + 645. i)5-s + (332. − 575. i)6-s + (1.65e3 − 953. i)7-s + (3.81e3 + 869. i)8-s + (457. − 1.16e3i)9-s + (−7.67e3 − 2.36e3i)10-s + (−5.02e3 − 6.29e3i)11-s + (−1.25e4 − 941. i)12-s + (−3.16e4 + 9.77e3i)13-s + (1.43e4 + 9.78e3i)14-s + (6.34e4 + 9.56e3i)15-s + (−1.92e3 − 8.41e3i)16-s + (−9.29e4 + 8.62e4i)17-s + ⋯ |
| L(s) = 1 | + (0.247 + 0.513i)2-s + (−0.506 − 0.743i)3-s + (0.421 − 0.528i)4-s + (−0.958 + 1.03i)5-s + (0.256 − 0.443i)6-s + (0.687 − 0.396i)7-s + (0.930 + 0.212i)8-s + (0.0697 − 0.177i)9-s + (−0.767 − 0.236i)10-s + (−0.343 − 0.430i)11-s + (−0.605 − 0.0454i)12-s + (−1.10 + 0.342i)13-s + (0.373 + 0.254i)14-s + (1.25 + 0.188i)15-s + (−0.0293 − 0.128i)16-s + (−1.11 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0619i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.998 + 0.0619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.00375916 - 0.121304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00375916 - 0.121304i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.33e6 - 7.53e5i)T \) |
| good | 2 | \( 1 + (-3.95 - 8.21i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (41.0 + 60.1i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (599. - 645. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-1.65e3 + 953. i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (5.02e3 + 6.29e3i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (3.16e4 - 9.77e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (9.29e4 - 8.62e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (1.28e5 - 5.02e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (3.26e5 - 4.92e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-4.38e5 + 6.42e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (5.23e4 - 6.98e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (1.60e6 + 9.25e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-3.77e6 + 1.81e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (1.13e6 - 1.42e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-1.31e7 - 4.05e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (2.93e6 + 1.28e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (1.51e7 - 1.13e6i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (8.71e6 + 2.21e7i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (2.26e6 - 1.50e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (9.53e6 + 3.09e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (-2.11e7 - 3.65e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (4.11e7 - 2.80e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-8.42e6 - 1.23e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (6.23e7 + 7.81e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.94569494097252203938207608238, −12.29462461908054788699660900438, −11.22555351385336167062579225478, −10.45072940237140685495834313302, −7.943871169206701292304392044979, −7.03188577760715636440688135680, −6.13098296821634224296119615892, −4.26691786601662243504424016671, −1.96795568563303118386450444024, −0.04159372905385878900953595305,
2.25706471603598543158077723404, 4.36983025297122585109432321073, 4.88802897961590505182296657559, 7.41600421521519738471748836289, 8.540678620053270953250837951003, 10.28166408198716813745943950787, 11.47457168821996112758085639998, 12.11617397161768784800533475142, 13.18727152613129060302639043019, 15.14391790818205417872306338489
