| L(s) = 1 | + (9.72 + 16.8i)2-s + (−125 + 216. i)4-s + (194. − 336. i)5-s + (630.5 + 1.09e3i)7-s − 2.37e3·8-s + 7.55e3·10-s + (738. + 1.27e3i)11-s + (−4.79e3 + 8.29e3i)13-s + (−1.22e4 + 2.12e4i)14-s + (−7.05e3 − 1.22e4i)16-s + 2.12e4·17-s − 2.19e4·19-s + (4.86e4 + 8.41e4i)20-s + (−1.43e4 + 2.48e4i)22-s + (−4.29e4 + 7.44e4i)23-s + ⋯ |
| L(s) = 1 | + (0.859 + 1.48i)2-s + (−0.976 + 1.69i)4-s + (0.695 − 1.20i)5-s + (0.694 + 1.20i)7-s − 1.63·8-s + 2.39·10-s + (0.167 + 0.289i)11-s + (−0.604 + 1.04i)13-s + (−1.19 + 2.06i)14-s + (−0.430 − 0.746i)16-s + 1.04·17-s − 0.733·19-s + (1.35 + 2.35i)20-s + (−0.287 + 0.498i)22-s + (−0.736 + 1.27i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.577129 + 3.27306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577129 + 3.27306i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-9.72 - 16.8i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-194. + 336. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-630.5 - 1.09e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-738. - 1.27e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (4.79e3 - 8.29e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.29e4 - 7.44e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.61e4 + 2.80e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-2.54e4 + 4.40e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.05e5 + 5.29e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.57e5 + 2.73e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-2.12e5 - 3.68e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.27e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (4.82e5 - 8.34e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.48e5 - 4.31e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.68e5 - 1.15e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 9.01e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.03e6 + 5.26e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.09e6 - 7.09e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 1.30e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (3.28e6 + 5.69e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.67911534260360288706561528802, −12.55621569880521775423756821113, −11.89783211824836010148471954045, −9.501044226706750858852675369635, −8.658544031512490540623199902656, −7.56828591599157804796114168126, −5.97943077445383372939412718783, −5.30396980486927639513449787640, −4.30653724334528757122223819136, −1.87654761804361526444541861914,
0.860359570549616323972148472649, 2.31589025755912272136570685914, 3.41426895259088772398329126745, 4.72697248242845922420547192108, 6.18543675715632455128945212512, 7.77100879905559145321404492106, 9.962237130972435086528101354415, 10.46201312815115921862249547997, 11.16836237214970318445030994960, 12.45643317292475526085534407455
