| L(s) = 1 | + (68.7 + 42.8i)3-s + (−357. + 206. i)5-s + (−24.5 − 2.40e3i)7-s + (2.88e3 + 5.89e3i)9-s + (−7.84e3 − 4.52e3i)11-s + 3.10e4·13-s + (−3.34e4 − 1.14e3i)15-s + (9.04e4 + 5.22e4i)17-s + (1.22e5 + 2.12e5i)19-s + (1.01e5 − 1.66e5i)21-s + (−3.64e5 + 2.10e5i)23-s + (−1.09e5 + 1.90e5i)25-s + (−5.42e4 + 5.28e5i)27-s − 1.62e5i·29-s + (3.58e5 − 6.21e5i)31-s + ⋯ |
| L(s) = 1 | + (0.848 + 0.529i)3-s + (−0.572 + 0.330i)5-s + (−0.0102 − 0.999i)7-s + (0.439 + 0.898i)9-s + (−0.535 − 0.309i)11-s + 1.08·13-s + (−0.660 − 0.0225i)15-s + (1.08 + 0.625i)17-s + (0.942 + 1.63i)19-s + (0.520 − 0.853i)21-s + (−1.30 + 0.752i)23-s + (−0.281 + 0.487i)25-s + (−0.102 + 0.994i)27-s − 0.229i·29-s + (0.388 − 0.673i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.68795 + 1.54680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68795 + 1.54680i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-68.7 - 42.8i)T \) |
| 7 | \( 1 + (24.5 + 2.40e3i)T \) |
| good | 5 | \( 1 + (357. - 206. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.84e3 + 4.52e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 3.10e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-9.04e4 - 5.22e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.22e5 - 2.12e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.64e5 - 2.10e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.62e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-3.58e5 + 6.21e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (9.38e5 + 1.62e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 5.16e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 9.82e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (6.43e6 - 3.71e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-7.45e6 - 4.30e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.43e7 - 8.26e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-8.43e6 - 1.46e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (5.74e5 - 9.95e5i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.64e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.34e7 + 4.05e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.49e6 - 2.58e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.80e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-9.59e7 + 5.54e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 2.73e6T + 7.83e15T^{2} \) |
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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
−13.20545201088190411482228568882, −11.63903290630632084203803676264, −10.48610476274004224026714315173, −9.771862312548659341092455910329, −8.039405657819669132865439860798, −7.70989204725375137087437799949, −5.79163826847365913695823546799, −3.93987385244819062079018689220, −3.40121615141674504801410697182, −1.39337318382279177279757625245,
0.66402942726287711299955889370, 2.28712603141275804034129436627, 3.50896270154853986425454436112, 5.19354417783834983626695544583, 6.75586097218080336336469901897, 8.052979883979749282496897117105, 8.736542022665616956139844208078, 9.926005152028007786970434904951, 11.66734818866657994187509968710, 12.35594218358978968419554687783
