| L(s) = 1 | + (−8.67 + 5.00i)2-s + (−77.8 + 134. i)4-s + (−514. − 297. i)5-s + (−2.20e3 − 3.81e3i)7-s − 4.12e3i·8-s + 5.95e3·10-s + (3.01e3 − 1.73e3i)11-s + (9.13e3 − 1.58e4i)13-s + (3.82e4 + 2.20e4i)14-s + (718. + 1.24e3i)16-s + 8.66e4i·17-s + 7.61e4·19-s + (8.01e4 − 4.62e4i)20-s + (−1.74e4 + 3.01e4i)22-s + (−4.53e5 − 2.62e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.542 + 0.312i)2-s + (−0.304 + 0.526i)4-s + (−0.823 − 0.475i)5-s + (−0.918 − 1.59i)7-s − 1.00i·8-s + 0.595·10-s + (0.205 − 0.118i)11-s + (0.319 − 0.553i)13-s + (0.995 + 0.574i)14-s + (0.0109 + 0.0189i)16-s + 1.03i·17-s + 0.584·19-s + (0.500 − 0.289i)20-s + (−0.0743 + 0.128i)22-s + (−1.62 − 0.936i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0724898 + 0.113786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0724898 + 0.113786i\) |
| \(L(5)\) |
|
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 | \( 1 + (8.67 - 5.00i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (514. + 297. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (2.20e3 + 3.81e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-3.01e3 + 1.73e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-9.13e3 + 1.58e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 8.66e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.61e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (4.53e5 + 2.62e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (6.71e5 - 3.87e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-4.62e5 + 8.01e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 2.37e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + (4.46e5 + 2.57e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.97e6 + 3.42e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (4.61e6 - 2.66e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.30e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.08e7 - 6.27e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (5.94e6 + 1.02e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (6.76e6 - 1.17e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.33e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.24e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.39e7 - 4.14e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-5.06e7 + 2.92e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 4.62e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-4.41e7 - 7.64e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−13.01103787382105915079495852675, −12.17081537496250485635288052774, −10.60842260266518473148679372689, −9.669682057048455284639041865232, −8.283056335483127407000524625325, −7.61226133911685740775198562854, −6.40528629099009539280213828464, −4.16015483841246667015561374814, −3.60485824958111117144604010202, −0.78711726411121222150365078203,
0.07396847349291641888153980588, 1.99853827246527097687815315187, 3.41530714784474549732574487225, 5.26540731223938062345009562641, 6.44473106115450968573842536884, 8.041389837697426796993935423559, 9.255853613008745015397663573656, 9.835960318222104822025074733586, 11.52245567421398629812675872924, 11.84420465356935089703250473003
