| L(s) = 1 | + 9i·3-s + 55.4·5-s + 152.·7-s − 81·9-s + (160. − 367. i)11-s + 323. i·13-s + 499. i·15-s + 1.49e3i·17-s − 2.57e3·19-s + 1.37e3i·21-s + 3.07e3i·23-s − 48.9·25-s − 729i·27-s − 2.48e3i·29-s + 7.62e3i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.992·5-s + 1.17·7-s − 0.333·9-s + (0.400 − 0.916i)11-s + 0.531i·13-s + 0.572i·15-s + 1.25i·17-s − 1.63·19-s + 0.680i·21-s + 1.21i·23-s − 0.0156·25-s − 0.192i·27-s − 0.548i·29-s + 1.42i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.698823444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.698823444\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-160. + 367. i)T \) |
| good | 5 | \( 1 - 55.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 152.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 323. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.07e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.48e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.62e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.39e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.63e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.16e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.10e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 9.33e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.86e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.79e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.62e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.54e4T + 8.58e9T^{2} \) |
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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
−10.51706652749737797430890432498, −9.290871441656595985012300120470, −8.708546920441927154054705381515, −7.81490460608938658795376502770, −6.32431457209857680417551035926, −5.75017680286810742510842983210, −4.63766396235485132358226716401, −3.72670510926136716560157626980, −2.18766177186508768560736834127, −1.32906137134895803105966819565,
0.58392576246679615987455203800, 1.88974954341511950306892544886, 2.41763248267745289067117194834, 4.29837377896810552044869818822, 5.17698337543452060083094475655, 6.19181740534041764488117071368, 7.07996813121138176399943244382, 8.022285457827111159154749925779, 8.885309212341434007309035183315, 9.827631554570774404316462839410
