| L(s) = 1 | + (3.11 − 3.90i)2-s + (2.22 + 2.78i)3-s + (1.56 + 6.86i)4-s + (−1.98 − 0.953i)5-s + 17.8·6-s + 187.·7-s + (175. + 84.6i)8-s + (51.2 − 224. i)9-s + (−9.89 + 4.76i)10-s + (−44.1 + 193. i)11-s + (−15.6 + 19.6i)12-s + (−21.4 − 10.3i)13-s + (584. − 732. i)14-s + (−1.74 − 7.64i)15-s + (674. − 324. i)16-s + (−2.74 + 1.32i)17-s + ⋯ |
| L(s) = 1 | + (0.550 − 0.690i)2-s + (0.142 + 0.178i)3-s + (0.0489 + 0.214i)4-s + (−0.0354 − 0.0170i)5-s + 0.202·6-s + 1.44·7-s + (0.970 + 0.467i)8-s + (0.210 − 0.923i)9-s + (−0.0312 + 0.0150i)10-s + (−0.110 + 0.482i)11-s + (−0.0314 + 0.0393i)12-s + (−0.0352 − 0.0169i)13-s + (0.796 − 0.998i)14-s + (−0.00200 − 0.00877i)15-s + (0.658 − 0.317i)16-s + (−0.00230 + 0.00111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.55222 - 0.511290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55222 - 0.511290i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.03e4 + 6.30e3i)T \) |
| good | 2 | \( 1 + (-3.11 + 3.90i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (-2.22 - 2.78i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (1.98 + 0.953i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 - 187.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (44.1 - 193. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (21.4 + 10.3i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (2.74 - 1.32i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (137. + 604. i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (702. - 3.07e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-1.57e3 + 1.97e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (5.17e3 - 6.48e3i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 + 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.65e3 + 5.84e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (2.29e3 + 1.00e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-9.15e3 + 4.40e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-2.09e4 + 1.00e4i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (3.86e3 + 4.84e3i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-5.78e3 - 2.53e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (1.11e4 + 4.89e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (6.31e4 + 3.04e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + 5.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-5.10e4 - 6.40e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-3.67e3 - 4.60e3i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (6.47e3 - 2.83e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−14.65908345487112928681981724847, −13.67002906544338578714140264079, −12.26784271178981371847337911366, −11.58167519112781085253328441396, −10.31500972388529954876829694846, −8.610738905899714518719549268961, −7.31185717589690274182801202265, −5.02260010348877826636337237736, −3.73529664035081009823964987518, −1.82308834183047882635416530970,
1.71826592482323601813357322770, 4.51869792245307990930040450308, 5.63601856883599625503613693410, 7.33237874432277231994682174873, 8.334668820557957419387206099010, 10.37818663713670571223171605310, 11.35633608592370139513374097439, 13.09276988040944217999307207640, 14.13476606523909613856565274377, 14.77395278937667532280074047778
