L(s) = 1 | + (−18.8 − 32.5i)3-s + (−125. − 218. i)5-s + 965.·7-s + (385. − 667. i)9-s + 2.97e3·11-s + (4.04e3 − 7.00e3i)13-s + (−4.73e3 + 8.20e3i)15-s + (568. + 984. i)17-s + (−2.84e4 − 9.14e3i)19-s + (−1.81e4 − 3.14e4i)21-s + (−8.58e3 + 1.48e4i)23-s + (7.34e3 − 1.27e4i)25-s − 1.11e5·27-s + (−7.85e3 + 1.36e4i)29-s + 1.40e4·31-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.696i)3-s + (−0.450 − 0.780i)5-s + 1.06·7-s + (0.176 − 0.305i)9-s + 0.674·11-s + (0.510 − 0.883i)13-s + (−0.362 + 0.627i)15-s + (0.0280 + 0.0485i)17-s + (−0.952 − 0.306i)19-s + (−0.427 − 0.741i)21-s + (−0.147 + 0.254i)23-s + (0.0940 − 0.162i)25-s − 1.08·27-s + (−0.0598 + 0.103i)29-s + 0.0848·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.372907 - 1.39242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.372907 - 1.39242i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.84e4 + 9.14e3i)T \) |
good | 3 | \( 1 + (18.8 + 32.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (125. + 218. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 - 965.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.04e3 + 7.00e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-568. - 984. i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (8.58e3 - 1.48e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.85e3 - 1.36e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.40e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.52e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.48e4 + 1.29e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.60e4 - 2.78e4i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-4.68e5 + 8.12e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.88e5 + 5.00e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (5.70e5 + 9.87e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.40e6 - 2.43e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.94e5 + 3.37e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-5.23e5 - 9.06e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.51e6 - 2.63e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.11e6 - 3.66e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 7.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.29e6 + 7.43e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-5.22e6 - 9.05e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.43574522132605331007919738431, −11.78341767838270408671398344113, −10.62153007375633588450049796107, −8.887702390728003581259056855581, −8.015055576187408699743443923487, −6.71382829990246738588401951706, −5.31280084178304524206172772773, −3.96389174961082236990002551153, −1.62562291220833284582129282941, −0.54265577092672549044799011206,
1.77213032276138049892730929376, 3.81246330084680483479841383851, 4.82458823340403366974353398327, 6.43617528225095933230950843501, 7.73821844256831330098020824190, 9.044711510559557856015230331437, 10.56461271632152454757664293561, 11.10578549659315372327656074131, 12.07644252587503946465378888671, 13.80055932704168781586968386355