| L(s) = 1 | + (−5.89 − 3.40i)2-s + (1.92 − 15.4i)3-s + (7.14 + 12.3i)4-s + (−7.45 + 12.9i)5-s + (−63.9 + 84.5i)6-s + (−125. + 34.0i)7-s + 120. i·8-s + (−235. − 59.6i)9-s + (87.8 − 50.7i)10-s + (83.2 − 48.0i)11-s + (205. − 86.6i)12-s − 416. i·13-s + (852. + 225. i)14-s + (185. + 140. i)15-s + (638. − 1.10e3i)16-s + (−104. − 180. i)17-s + ⋯ |
| L(s) = 1 | + (−1.04 − 0.601i)2-s + (0.123 − 0.992i)3-s + (0.223 + 0.386i)4-s + (−0.133 + 0.231i)5-s + (−0.725 + 0.959i)6-s + (−0.964 + 0.262i)7-s + 0.665i·8-s + (−0.969 − 0.245i)9-s + (0.277 − 0.160i)10-s + (0.207 − 0.119i)11-s + (0.411 − 0.173i)12-s − 0.684i·13-s + (1.16 + 0.306i)14-s + (0.212 + 0.160i)15-s + (0.623 − 1.08i)16-s + (−0.0874 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0857181 + 0.329646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0857181 + 0.329646i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.92 + 15.4i)T \) |
| 7 | \( 1 + (125. - 34.0i)T \) |
| good | 2 | \( 1 + (5.89 + 3.40i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (7.45 - 12.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-83.2 + 48.0i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 416. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (104. + 180. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (2.29e3 + 1.32e3i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.26e3 + 1.30e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 7.22e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.72e3 + 2.14e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.83e3 - 3.18e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.01e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.95e3 + 3.38e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (3.14e4 - 1.81e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.91e3 - 6.77e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.74e4 - 1.00e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.19e4 + 3.80e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.60e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.40e4 + 2.54e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.50e4 + 4.34e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (6.83e4 - 1.18e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 2.51e4iT - 8.58e9T^{2} \) |
| 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
−16.95413191393351528037126864158, −15.09525572694064976399978294793, −13.48653709682161018674907166628, −12.21089833398228373943711031822, −10.84512981867014397477359947812, −9.332802843482526912573312005681, −8.048862431345300706449871249888, −6.28754028978951689148783112676, −2.54883702687264365959588721255, −0.32992762605310918453872945023,
3.96475087881706013984307549806, 6.48278195310453721152949706976, 8.367320996995104850708450527918, 9.461265009989644023529458881910, 10.51167670566876279644512430437, 12.57364414807406845624664667783, 14.41603235787856139878961500359, 15.93476760579914768479702408160, 16.43358250876707821447735932069, 17.47982254981742758182000352326
