| L(s) = 1 | − 5.65i·2-s − 32.0·4-s + 208. i·5-s + 4.19·7-s + 181. i·8-s + 1.17e3·10-s − 2.26e3i·11-s + 2.84e3·13-s − 23.7i·14-s + 1.02e3·16-s + 1.96e3i·17-s − 281.·19-s − 6.67e3i·20-s − 1.27e4·22-s + 1.67e4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.66i·5-s + 0.0122·7-s + 0.353i·8-s + 1.17·10-s − 1.69i·11-s + 1.29·13-s − 0.00865i·14-s + 0.250·16-s + 0.400i·17-s − 0.0410·19-s − 0.834i·20-s − 1.20·22-s + 1.37i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.256797430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256797430\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 208. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 4.19T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.26e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.84e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 1.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 281.T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.67e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.71e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.47e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 1.70e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.16e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.06e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.76e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.52e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.61e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.74e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.50e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.31e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 8.96e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.45e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.90e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.39e6T + 8.32e11T^{2} \) |
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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
−11.49749083927354957799627522953, −11.06876828517948214262776858413, −10.39022089915894469256152450127, −9.051558442039004527693796963604, −7.965692940750314557097277917638, −6.58877983235400973149546385791, −5.67136035591161315748859067512, −3.55102942194812949667365946150, −3.15585639334753910017239072969, −1.44217208066409831776983601739,
0.38030557152875619794465872017, 1.75135500212716955847425025014, 4.14925118757044234608576974113, 4.87354585465978095580975210731, 6.04789163722634940481002238358, 7.38360132305516723843919135754, 8.470500397925486215641505829311, 9.159884479882000299113615736262, 10.19149713579994446280865030434, 11.81389420776233214490982323127
