L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−132. + 76.7i)5-s + (−336. + 582. i)7-s − 181. i·8-s + 868.·10-s + (2.21e3 + 1.27e3i)11-s + (885. + 1.53e3i)13-s + (3.29e3 − 1.90e3i)14-s + (−512. + 886. i)16-s + 3.97e3i·17-s − 7.76e3·19-s + (−4.25e3 − 2.45e3i)20-s + (−7.24e3 − 1.25e4i)22-s + (2.26e3 − 1.30e3i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.06 + 0.614i)5-s + (−0.979 + 1.69i)7-s − 0.353i·8-s + 0.868·10-s + (1.66 + 0.961i)11-s + (0.402 + 0.697i)13-s + (1.20 − 0.692i)14-s + (−0.125 + 0.216i)16-s + 0.809i·17-s − 1.13·19-s + (−0.531 − 0.307i)20-s + (−0.679 − 1.17i)22-s + (0.186 − 0.107i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7733886340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7733886340\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (132. - 76.7i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (336. - 582. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-2.21e3 - 1.27e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-885. - 1.53e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 3.97e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.76e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-2.26e3 + 1.30e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.25e4 - 7.23e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-2.22e4 - 3.84e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.95e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (1.41e4 - 8.15e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (2.76e4 - 4.78e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.03e5 - 5.96e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 5.93e3iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-6.43e3 + 3.71e3i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.72e4 - 4.72e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.11e5 - 1.93e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 7.24e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.17e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (8.64e4 - 1.49e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.26e4 + 7.33e3i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 8.00e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-8.39e4 + 1.45e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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
−12.11178074301917587591024502869, −11.45218202011484169777886014117, −10.19329418386973567251185907495, −9.049029143138030244730118012747, −8.544535601729318052999936709369, −6.89543601190936714985204049269, −6.32978437469452089316587136933, −4.20053769388336073935848969369, −3.11481225569883797997364361747, −1.75422484896071429779228433891,
0.39541696214694971454449522988, 0.858829791280845570190690621125, 3.51727341195596417855306126526, 4.29342394411152670400902565185, 6.23264429642926248802631093230, 7.07107814908178845288083877149, 8.133593610475344373053087741321, 9.041177520575314434163942839872, 10.16904218364551561799428375233, 11.12601017949002621685482598792