| L(s) = 1 | + (−5.61 − 4.71i)3-s + (101. − 37.0i)5-s + (618. + 1.07e3i)7-s + (−370. − 2.10e3i)9-s + (−17.5 + 30.4i)11-s + (−6.62e3 + 5.56e3i)13-s + (−745. − 271. i)15-s + (−406. + 2.30e3i)17-s + (2.92e4 − 6.31e3i)19-s + (1.57e3 − 8.92e3i)21-s + (7.93e4 + 2.88e4i)23-s + (−5.08e4 + 4.26e4i)25-s + (−1.58e4 + 2.74e4i)27-s + (2.89e4 + 1.64e5i)29-s + (1.75e4 + 3.04e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.120 − 0.100i)3-s + (0.364 − 0.132i)5-s + (0.681 + 1.18i)7-s + (−0.169 − 0.960i)9-s + (−0.00397 + 0.00689i)11-s + (−0.836 + 0.702i)13-s + (−0.0570 − 0.0207i)15-s + (−0.0200 + 0.113i)17-s + (0.977 − 0.211i)19-s + (0.0370 − 0.210i)21-s + (1.35 + 0.494i)23-s + (−0.650 + 0.546i)25-s + (−0.154 + 0.268i)27-s + (0.220 + 1.24i)29-s + (0.105 + 0.183i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.75306 + 0.841590i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75306 + 0.841590i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.92e4 + 6.31e3i)T \) |
| good | 3 | \( 1 + (5.61 + 4.71i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-101. + 37.0i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-618. - 1.07e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (17.5 - 30.4i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.62e3 - 5.56e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (406. - 2.30e3i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-7.93e4 - 2.88e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-2.89e4 - 1.64e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.75e4 - 3.04e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.72e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.39e5 - 4.52e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (3.86e5 - 1.40e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-7.94e4 - 4.50e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-5.12e5 - 1.86e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (1.09e5 - 6.18e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.67e6 - 6.10e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (4.20e5 + 2.38e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (3.36e6 - 1.22e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (5.06e6 + 4.24e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-3.17e6 - 2.66e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (3.91e6 + 6.77e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-9.44e6 + 7.92e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (1.16e6 - 6.60e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−13.09973146974714580169172581938, −11.99861816325918620271186428655, −11.34517064866026104355485689703, −9.541566821836931362092435612890, −8.930701194149950803026036627456, −7.36284557889693399375282007616, −5.95428552547385533816657073988, −4.86915518584281262867557434849, −2.88907531202148048033287160540, −1.34268685302273531119569495719,
0.74649014000737257595968519283, 2.49023148836841745475829943394, 4.37618569503262851287547040664, 5.50512722263479474396811306557, 7.24872552850299339848749837050, 8.063266449711949339864013319159, 9.819648450942452525242009998234, 10.62843226970583809860716514067, 11.62183092663382312558262137521, 13.15636812441674943850979111289
