| L(s) = 1 | + (10.1 + 57.7i)3-s + (−349. + 293. i)5-s + (−759. + 1.31e3i)7-s + (−1.17e3 + 429. i)9-s + (−706. − 1.22e3i)11-s + (−102. + 582. i)13-s + (−2.05e4 − 1.72e4i)15-s + (6.19e3 + 2.25e3i)17-s + (2.79e4 + 1.07e4i)19-s + (−8.37e4 − 3.04e4i)21-s + (3.17e4 + 2.66e4i)23-s + (2.26e4 − 1.28e5i)25-s + (2.73e4 + 4.73e4i)27-s + (1.51e5 − 5.49e4i)29-s + (−5.49e4 + 9.51e4i)31-s + ⋯ |
| L(s) = 1 | + (0.217 + 1.23i)3-s + (−1.25 + 1.05i)5-s + (−0.836 + 1.44i)7-s + (−0.538 + 0.196i)9-s + (−0.159 − 0.277i)11-s + (−0.0129 + 0.0734i)13-s + (−1.57 − 1.31i)15-s + (0.305 + 0.111i)17-s + (0.933 + 0.357i)19-s + (−1.97 − 0.718i)21-s + (0.543 + 0.456i)23-s + (0.290 − 1.64i)25-s + (0.267 + 0.463i)27-s + (1.14 − 0.418i)29-s + (−0.331 + 0.573i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.470295 - 0.713338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.470295 - 0.713338i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.79e4 - 1.07e4i)T \) |
| good | 3 | \( 1 + (-10.1 - 57.7i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (349. - 293. i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (759. - 1.31e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (706. + 1.22e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (102. - 582. i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-6.19e3 - 2.25e3i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-3.17e4 - 2.66e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.51e5 + 5.49e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (5.49e4 - 9.51e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.69e4 + 2.09e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-1.64e5 + 1.37e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (1.22e6 - 4.45e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (2.47e5 + 2.07e5i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-2.15e6 - 7.82e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (7.55e5 + 6.33e5i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-8.70e5 + 3.16e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-2.65e6 + 2.22e6i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-3.79e5 - 2.15e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (9.13e5 + 5.17e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (4.92e6 - 8.52e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.87e6 - 1.06e7i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-7.33e6 - 2.66e6i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−14.22493181653617747514641614888, −12.41442879402230903358582655618, −11.54133953953281339543911912134, −10.43402169956395102649801552435, −9.430432837977797085334808271112, −8.301244981092520042270462835718, −6.80055380701853192341546300814, −5.26926909804118897082628742943, −3.60138395903402304641056914457, −2.97307068143173164788450157159,
0.33361875895005415546675152398, 1.13916666435797020049834766987, 3.37788475546608625636553246209, 4.72734495299530096930230307583, 6.85942380859926850992562381628, 7.50027081692352175321268600885, 8.494392333751237133751607350274, 10.03707147842382237016148485255, 11.55325368586780355347448885700, 12.60105012157059371106741318556
