| L(s) = 1 | + (8.42 + 14.5i)2-s + (−7.74 + 46.1i)3-s + (−77.8 + 134. i)4-s + (−239. + 413. i)5-s + (−738. + 275. i)6-s + (−171.5 − 297. i)7-s − 467.·8-s + (−2.06e3 − 714. i)9-s − 8.05e3·10-s + (3.72e3 + 6.45e3i)11-s + (−5.61e3 − 4.63e3i)12-s + (5.03e3 − 8.72e3i)13-s + (2.88e3 − 5.00e3i)14-s + (−1.72e4 − 1.42e4i)15-s + (6.03e3 + 1.04e4i)16-s + 3.59e3·17-s + ⋯ |
| L(s) = 1 | + (0.744 + 1.28i)2-s + (−0.165 + 0.986i)3-s + (−0.608 + 1.05i)4-s + (−0.855 + 1.48i)5-s + (−1.39 + 0.520i)6-s + (−0.188 − 0.327i)7-s − 0.322·8-s + (−0.945 − 0.326i)9-s − 2.54·10-s + (0.844 + 1.46i)11-s + (−0.938 − 0.774i)12-s + (0.636 − 1.10i)13-s + (0.281 − 0.487i)14-s + (−1.31 − 1.08i)15-s + (0.368 + 0.637i)16-s + 0.177·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.24736 - 1.46246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24736 - 1.46246i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.74 - 46.1i)T \) |
| 7 | \( 1 + (171.5 + 297. i)T \) |
| good | 2 | \( 1 + (-8.42 - 14.5i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (239. - 413. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-3.72e3 - 6.45e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.03e3 + 8.72e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 3.59e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.10e4 - 7.10e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.17e4 - 2.04e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-8.13e4 + 1.40e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.98e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.16e5 + 2.01e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.83e5 + 4.91e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-2.58e5 - 4.47e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 8.89e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (7.91e5 - 1.37e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (6.63e5 + 1.14e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.04e6 - 3.53e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 2.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-4.01e6 - 6.95e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.44e6 + 2.50e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 2.96e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.63e6 - 2.82e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.70000186519367456942184928302, −13.80423858309824188501461370378, −12.07249826597750461192293943815, −10.84986338005258066164917288614, −9.843968409373144213514078509609, −7.88613125038514239445831640760, −6.98100867006250586048144838472, −5.82520062591746039855394579683, −4.25139900951174705139294249146, −3.41370728299643859441811378459,
0.60260249058226886645683524577, 1.53213822784396635262681790047, 3.39310897971547671604669763617, 4.69027830169018431413242429769, 6.18694451917035306845775326691, 8.166172949553270206501120403384, 9.054811063991160473773022710369, 11.14649055344862276126221962047, 11.88736439631267706541083337176, 12.38436984610783824300938697760
