| L(s) = 1 | + (−38.1 + 13.8i)3-s + (91.2 − 517. i)5-s + (78.5 + 136. i)7-s + (−414. + 347. i)9-s + (3.45e3 − 5.98e3i)11-s + (−1.15e3 − 421. i)13-s + (3.70e3 + 2.09e4i)15-s + (1.50e4 + 1.26e4i)17-s + (−1.64e4 + 2.49e4i)19-s + (−4.88e3 − 4.09e3i)21-s + (−9.21e3 − 5.22e4i)23-s + (−1.86e5 − 6.77e4i)25-s + (5.53e4 − 9.58e4i)27-s + (−1.56e5 + 1.30e5i)29-s + (−1.21e5 − 2.11e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.815 + 0.296i)3-s + (0.326 − 1.85i)5-s + (0.0865 + 0.149i)7-s + (−0.189 + 0.158i)9-s + (0.783 − 1.35i)11-s + (−0.146 − 0.0531i)13-s + (0.283 + 1.60i)15-s + (0.741 + 0.622i)17-s + (−0.548 + 0.835i)19-s + (−0.115 − 0.0965i)21-s + (−0.157 − 0.895i)23-s + (−2.38 − 0.866i)25-s + (0.541 − 0.937i)27-s + (−1.18 + 0.997i)29-s + (−0.735 − 1.27i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0423i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0112220 + 0.529260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0112220 + 0.529260i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.64e4 - 2.49e4i)T \) |
| good | 3 | \( 1 + (38.1 - 13.8i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-91.2 + 517. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-78.5 - 136. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.45e3 + 5.98e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.15e3 + 421. i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-1.50e4 - 1.26e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (9.21e3 + 5.22e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (1.56e5 - 1.30e5i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.21e5 + 2.11e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.98e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (4.90e5 - 1.78e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.20e5 - 6.83e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-4.07e5 + 3.42e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-2.13e5 - 1.21e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-8.72e5 - 7.31e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.16e5 - 6.62e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.06e6 - 8.90e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (1.50e5 - 8.53e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-9.01e5 + 3.28e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (2.21e6 - 8.07e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (3.65e6 + 6.32e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.83e6 - 1.39e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (9.06e6 + 7.60e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−12.39520065094875255138449719690, −11.58442456435419498464470875586, −10.31599447512760330786550828303, −8.941443292411754495903250048690, −8.254210703470108344959311250862, −5.95706543314130478732298064399, −5.37669492605822712221068657453, −4.01516530822975016847713933240, −1.48126813684713385603485970554, −0.20015254409787906160125283765,
1.95459608814936359308164397542, 3.52348951816123644446277392705, 5.47549794962732860323212693228, 6.81899949182168584048679081014, 7.18884516052018185767313744317, 9.433323353569766261931701806420, 10.46884180926056338329125050823, 11.42117488748750278408875503818, 12.22200246652809770164751504311, 13.80403810092179646469026571139
