| L(s) = 1 | + (1.34 − 0.775i)2-s + (−44.1 − 67.9i)3-s + (−126. + 219. i)4-s + (−604. − 349. i)5-s + (−111. − 56.9i)6-s + (−1.12e3 − 1.94e3i)7-s + 790. i·8-s + (−2.66e3 + 5.99e3i)9-s − 1.08e3·10-s + (1.88e4 − 1.08e4i)11-s + (2.05e4 − 1.07e3i)12-s + (−9.20e3 + 1.59e4i)13-s + (−3.02e3 − 1.74e3i)14-s + (2.97e3 + 5.64e4i)15-s + (−3.18e4 − 5.51e4i)16-s − 5.65e4i·17-s + ⋯ |
| L(s) = 1 | + (0.0839 − 0.0484i)2-s + (−0.544 − 0.838i)3-s + (−0.495 + 0.857i)4-s + (−0.967 − 0.558i)5-s + (−0.0863 − 0.0439i)6-s + (−0.468 − 0.811i)7-s + 0.192i·8-s + (−0.406 + 0.913i)9-s − 0.108·10-s + (1.28 − 0.742i)11-s + (0.989 − 0.0520i)12-s + (−0.322 + 0.558i)13-s + (−0.0786 − 0.0453i)14-s + (0.0587 + 1.11i)15-s + (−0.485 − 0.841i)16-s − 0.676i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0569948 - 0.404852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0569948 - 0.404852i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (44.1 + 67.9i)T \) |
| good | 2 | \( 1 + (-1.34 + 0.775i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (604. + 349. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.12e3 + 1.94e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.88e4 + 1.08e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (9.20e3 - 1.59e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 5.65e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.12e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.43e4 - 8.31e3i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (2.94e5 - 1.70e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.29e4 - 1.43e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.11e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (3.60e6 + 2.08e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.28e6 - 2.23e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-8.10e6 + 4.67e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + 4.75e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (8.39e6 + 4.84e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.04e6 + 5.27e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.02e7 - 1.77e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.08e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 9.02e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.67e7 - 2.89e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (5.09e7 - 2.94e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 8.65e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.68e7 + 8.12e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−18.92100249743989561271407551382, −17.06184714829564696604477284828, −16.54290211647330127592341812288, −13.86865905076271324912283410378, −12.56000524145047644427543794850, −11.49774621494844381802356828779, −8.602020683839552045489335433552, −7.01584354746971139933710875418, −4.11866726565879182457521211816, −0.30478727359902946164962514141,
4.17227290223665537786876960297, 6.20971333674209743702952450390, 9.189456624125193648901034867659, 10.67490965304052345682325926865, 12.25139192342945363042276329920, 14.94652705139606633003598277171, 15.23446361470023058955530260091, 17.19825473198927906313369498742, 18.85295136686562556724721549750, 19.88221859347148920978542310640
