| L(s) = 1 | + (6.01 − 3.47i)2-s + (−29.7 + 75.3i)3-s + (−103. + 179. i)4-s + (331. + 191. i)5-s + (82.9 + 556. i)6-s + (467. + 809. i)7-s + 3.21e3i·8-s + (−4.79e3 − 4.47e3i)9-s + 2.65e3·10-s + (4.89e3 − 2.82e3i)11-s + (−1.04e4 − 1.31e4i)12-s + (1.76e4 − 3.05e4i)13-s + (5.62e3 + 3.24e3i)14-s + (−2.42e4 + 1.92e4i)15-s + (−1.54e4 − 2.67e4i)16-s + 1.52e5i·17-s + ⋯ |
| L(s) = 1 | + (0.375 − 0.216i)2-s + (−0.366 + 0.930i)3-s + (−0.405 + 0.702i)4-s + (0.530 + 0.306i)5-s + (0.0640 + 0.429i)6-s + (0.194 + 0.337i)7-s + 0.786i·8-s + (−0.730 − 0.682i)9-s + 0.265·10-s + (0.334 − 0.192i)11-s + (−0.505 − 0.635i)12-s + (0.618 − 1.07i)13-s + (0.146 + 0.0844i)14-s + (−0.479 + 0.381i)15-s + (−0.235 − 0.407i)16-s + 1.83i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0529 - 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0529 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.07930 + 1.02363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07930 + 1.02363i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (29.7 - 75.3i)T \) |
| good | 2 | \( 1 + (-6.01 + 3.47i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (-331. - 191. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-467. - 809. i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-4.89e3 + 2.82e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.76e4 + 3.05e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.52e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.91e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.32e5 - 7.63e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-4.01e5 + 2.31e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (3.93e5 - 6.82e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.10e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (3.14e6 + 1.81e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.51e6 + 2.62e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-4.75e6 + 2.74e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.41e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-7.58e6 - 4.37e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.47e6 + 6.01e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.08e6 + 1.22e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.97e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.61e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + (1.37e6 + 2.38e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.52e7 + 2.03e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 2.90e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.58e7 + 7.93e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
| 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
−20.30295778127236931339689754033, −18.00677273737882632645879755540, −17.06150547738140974918525741998, −15.38317187300023511897913547662, −13.83270859036771247267922533071, −12.10742396971423945398687667553, −10.48392747023142718106052748452, −8.630788820797890166413663774591, −5.56153928530669329441785329668, −3.50079334468490511309708086589,
1.16720721083289990081338060380, 5.19065848468107296842545980032, 6.88614962389352937350056873446, 9.393313925242577285136124625194, 11.55178339158290585120316178168, 13.43638419500306249409761028557, 14.12946913427679148604793071793, 16.31247249728407599218954257752, 17.89607854748203375317251501054, 18.87320331611209650812824729757
