| L(s) = 1 | + (−39.9 − 33.4i)3-s + (−63.3 + 23.0i)5-s + (546. + 946. i)7-s + (91.6 + 519. i)9-s + (3.72e3 − 6.45e3i)11-s + (7.14e3 − 5.99e3i)13-s + (3.30e3 + 1.20e3i)15-s + (−5.08e3 + 2.88e4i)17-s + (−2.89e4 − 7.57e3i)19-s + (9.89e3 − 5.60e4i)21-s + (−4.28e4 − 1.55e4i)23-s + (−5.63e4 + 4.72e4i)25-s + (−4.32e4 + 7.48e4i)27-s + (−2.69e4 − 1.52e5i)29-s + (−4.99e4 − 8.65e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.853 − 0.716i)3-s + (−0.226 + 0.0824i)5-s + (0.602 + 1.04i)7-s + (0.0418 + 0.237i)9-s + (0.844 − 1.46i)11-s + (0.902 − 0.757i)13-s + (0.252 + 0.0919i)15-s + (−0.251 + 1.42i)17-s + (−0.967 − 0.253i)19-s + (0.233 − 1.32i)21-s + (−0.734 − 0.267i)23-s + (−0.721 + 0.605i)25-s + (−0.422 + 0.732i)27-s + (−0.205 − 1.16i)29-s + (−0.301 − 0.521i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.151351 - 0.699698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151351 - 0.699698i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.89e4 + 7.57e3i)T \) |
| good | 3 | \( 1 + (39.9 + 33.4i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (63.3 - 23.0i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-546. - 946. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.72e3 + 6.45e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-7.14e3 + 5.99e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (5.08e3 - 2.88e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (4.28e4 + 1.55e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (2.69e4 + 1.52e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (4.99e4 + 8.65e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (6.63e5 + 5.56e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-7.42e5 + 2.70e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-6.01e4 - 3.40e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.75e6 + 6.39e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-1.90e5 + 1.08e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (2.71e6 + 9.86e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-4.94e5 - 2.80e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (1.39e6 - 5.06e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (4.25e6 + 3.57e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-2.25e6 - 1.89e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-4.14e5 - 7.18e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-6.58e6 + 5.52e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (1.36e6 - 7.75e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−12.43318579977098985223678508802, −11.49770577989041827628164756510, −10.86842774489009272515886856267, −8.874864068312313613889755029696, −8.048096687978027039037130147555, −6.13609982730776774740819280963, −5.88755146243415445333393191963, −3.76294432269218391401578458269, −1.74123186880437803544436833981, −0.27981116763057688070325187799,
1.54845414343019438913850923003, 4.15304895648186518010435786377, 4.69412175106725577997369866737, 6.44170795574139284823906772349, 7.62697823979349814477326866924, 9.274572067285993333318918058164, 10.37803638951350630461264058853, 11.27893519071919568324390444943, 12.10426738674752361414072220026, 13.67134384752132673658531945574
