| L(s) = 1 | − 9i·3-s − 34.2·5-s + 120.·7-s − 81·9-s + (257. + 308. i)11-s + 1.15e3i·13-s + 308. i·15-s + 1.40e3i·17-s − 903.·19-s − 1.08e3i·21-s − 5.01e3i·23-s − 1.94e3·25-s + 729i·27-s − 2.34e3i·29-s − 6.18e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.613·5-s + 0.930·7-s − 0.333·9-s + (0.640 + 0.767i)11-s + 1.90i·13-s + 0.354i·15-s + 1.17i·17-s − 0.574·19-s − 0.537i·21-s − 1.97i·23-s − 0.623·25-s + 0.192i·27-s − 0.516i·29-s − 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3119863430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3119863430\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| 11 | \( 1 + (-257. - 308. i)T \) |
| good | 5 | \( 1 + 34.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 120.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.15e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.40e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 903.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 5.01e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.34e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.18e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 6.25e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.01e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.57e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.81e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.72e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.56e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.92e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 6.13e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.75e4T + 8.58e9T^{2} \) |
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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
−10.66657255902917843319498821772, −9.400220668025122476938729992743, −8.530927098587366592799730740385, −7.81620302546479999457512293908, −6.82585487035546937806120011447, −6.13706375822026882885004125457, −4.42696649647537066558935145528, −4.17036668660540944117817177403, −2.20906106531277337300448236022, −1.52647514992529536520060976041,
0.07093648306101523060829555077, 1.30907553117068560745168930277, 3.03959380368577978750616258809, 3.77968175570850273763479444668, 5.06655836955020295974004280148, 5.61877604284911281602126366208, 7.12499301641826052273171249376, 8.024877120104585614564772827988, 8.643763195926329793507009821592, 9.719608820031074397755733583766
