| L(s) = 1 | + (−22.1 − 12.7i)2-s + (−80.9 − 3.05i)3-s + (199. + 344. i)4-s + (570. − 329. i)5-s + (1.75e3 + 1.10e3i)6-s + (−1.51e3 + 2.61e3i)7-s − 3.63e3i·8-s + (6.54e3 + 494. i)9-s − 1.68e4·10-s + (−8.01e3 − 4.62e3i)11-s + (−1.50e4 − 2.85e4i)12-s + (2.17e4 + 3.77e4i)13-s + (6.70e4 − 3.86e4i)14-s + (−4.71e4 + 2.49e4i)15-s + (4.44e3 − 7.70e3i)16-s + 1.07e5i·17-s + ⋯ |
| L(s) = 1 | + (−1.38 − 0.799i)2-s + (−0.999 − 0.0376i)3-s + (0.777 + 1.34i)4-s + (0.912 − 0.526i)5-s + (1.35 + 0.850i)6-s + (−0.629 + 1.09i)7-s − 0.888i·8-s + (0.997 + 0.0753i)9-s − 1.68·10-s + (−0.547 − 0.316i)11-s + (−0.726 − 1.37i)12-s + (0.762 + 1.32i)13-s + (1.74 − 1.00i)14-s + (−0.931 + 0.492i)15-s + (0.0678 − 0.117i)16-s + 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.387578 + 0.163240i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.387578 + 0.163240i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (80.9 + 3.05i)T \) |
| good | 2 | \( 1 + (22.1 + 12.7i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-570. + 329. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.51e3 - 2.61e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (8.01e3 + 4.62e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-2.17e4 - 3.77e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.07e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.23e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (1.15e5 - 6.69e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (2.19e5 + 1.26e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-3.93e5 - 6.82e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.55e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.91e6 - 1.10e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (2.77e5 - 4.79e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-2.05e6 - 1.18e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 7.83e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.48e6 + 2.59e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.77e6 - 8.26e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (4.31e6 + 7.47e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.31e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-4.50e6 + 7.80e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (3.31e7 + 1.91e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 1.41e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-7.52e7 + 1.30e8i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−19.15769037614425050572245773122, −18.31342217779406176492453852242, −17.14878932406913325732578774527, −16.09433998175246507014094515150, −12.93639979836971049995693979203, −11.56932617623686265557173171285, −10.06560762510905738783193777304, −8.819168163032880300650681293837, −5.99259420850941214303848521549, −1.69526611868712884942989270072,
0.53681807599824510589392014368, 6.04845802692798669774395671846, 7.36718518045089871773723745946, 9.891960330351807970546288923020, 10.59577347419126559789928603604, 13.31919606956957855104958359725, 15.61798045533313297295241252963, 16.73629872267008683638573636928, 17.79933411224223872195822985924, 18.48336511862362186309696302061
