| L(s) = 1 | − 9i·3-s + 107. i·5-s − 150.·7-s − 81·9-s + 221. i·11-s − 504. i·13-s + 963.·15-s − 1.98e3·17-s + 2.36e3i·19-s + 1.35e3i·21-s − 1.14e3·23-s − 8.33e3·25-s + 729i·27-s − 3.90e3i·29-s + 9.00e3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.91i·5-s − 1.15·7-s − 0.333·9-s + 0.552i·11-s − 0.827i·13-s + 1.10·15-s − 1.66·17-s + 1.50i·19-s + 0.668i·21-s − 0.450·23-s − 2.66·25-s + 0.192i·27-s − 0.861i·29-s + 1.68·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4020031129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4020031129\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| good | 5 | \( 1 - 107. iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 150.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 221. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 504. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.36e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.90e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 9.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 448. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 9.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.62e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.74e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.59e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.25e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.53e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.76e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.90e4T + 8.58e9T^{2} \) |
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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
−10.23607162745263788338420641642, −9.734475676281308139755918393783, −8.169039353725848014252889411136, −7.29463970010889956776932332973, −6.43836863613147706861833456361, −6.03737071417665381162968603094, −4.00211331323592398680696226755, −2.94365877847732722086851195070, −2.18661829118034553055665285950, −0.12344515772578131591357926600,
0.866086013749546980511020047501, 2.54292538526697547812994678260, 4.11963370129667092854287094304, 4.67695896332574989178222437483, 5.83067812173904112059939038781, 6.83319798835758804484285859976, 8.404604870015701939031959599314, 9.126533197479148921054982156093, 9.387804166328965867100037368014, 10.71582636050022759368290210899
