| L(s) = 1 | + (19.4 − 23.2i)2-s + (−115. − 652. i)4-s + (197. − 542. i)5-s + (379. − 2.15e3i)7-s + (−1.06e4 − 6.15e3i)8-s + (−8.74e3 − 1.51e4i)10-s + (3.14e3 + 8.63e3i)11-s + (−2.15e4 + 1.80e4i)13-s + (−4.25e4 − 5.07e4i)14-s + (−1.91e5 + 6.96e4i)16-s + (1.18e5 − 6.86e4i)17-s + (9.47e4 − 1.64e5i)19-s + (−3.76e5 − 6.63e4i)20-s + (2.61e5 + 9.52e4i)22-s + (−2.06e5 + 3.64e4i)23-s + ⋯ |
| L(s) = 1 | + (1.21 − 1.45i)2-s + (−0.449 − 2.54i)4-s + (0.315 − 0.867i)5-s + (0.158 − 0.896i)7-s + (−2.60 − 1.50i)8-s + (−0.874 − 1.51i)10-s + (0.214 + 0.589i)11-s + (−0.754 + 0.633i)13-s + (−1.10 − 1.32i)14-s + (−2.91 + 1.06i)16-s + (1.42 − 0.822i)17-s + (0.727 − 1.25i)19-s + (−2.35 − 0.414i)20-s + (1.11 + 0.406i)22-s + (−0.739 + 0.130i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.36867 + 3.30985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36867 + 3.30985i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-19.4 + 23.2i)T + (-44.4 - 252. i)T^{2} \) |
| 5 | \( 1 + (-197. + 542. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-379. + 2.15e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-3.14e3 - 8.63e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (2.15e4 - 1.80e4i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-1.18e5 + 6.86e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.47e4 + 1.64e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.06e5 - 3.64e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (5.59e5 - 6.66e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-2.01e5 - 1.14e6i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (1.07e5 + 1.86e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-8.39e5 - 1.00e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-1.24e6 + 4.52e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (3.65e6 + 6.45e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 2.09e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.17e6 + 1.14e7i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-4.25e6 + 2.41e7i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-8.20e6 + 6.88e6i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (8.38e6 - 4.84e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-9.79e5 + 1.69e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.87e7 + 1.56e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-6.79e5 + 8.09e5i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (1.03e8 + 5.96e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.40e8 + 5.11e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−12.17275821965303080735306955076, −11.25388958630255604249312156407, −9.983017231520414986559904167166, −9.329073502385749063674351347500, −7.10526077470444585739525404674, −5.25164798947972003864593813825, −4.64433563853971428724015947957, −3.28008913234175000672333321688, −1.72310837247735969649791846583, −0.73169136046461978916422434556,
2.68865539220535841456527731113, 3.85165423392964409389328780960, 5.68749804004074500174626863805, 5.91816555274751111580991976571, 7.48307641823128658272357930944, 8.266098779992328556506266818508, 9.954124651686346393827845320288, 11.78001515880860787772136655590, 12.53710749084476885565882039562, 13.77525107481611601632030765714
