| L(s) = 1 | + (−0.246 − 2.98i)3-s − 6.63i·5-s − 0.578·7-s + (−8.87 + 1.47i)9-s − 8.68i·11-s + 17.9·13-s + (−19.8 + 1.63i)15-s − 19.0i·17-s − 32.1·19-s + (0.142 + 1.72i)21-s + 20.4i·23-s − 19.0·25-s + (6.59 + 26.1i)27-s + 22.0i·29-s − 26.2·31-s + ⋯ |
| L(s) = 1 | + (−0.0821 − 0.996i)3-s − 1.32i·5-s − 0.0825·7-s + (−0.986 + 0.163i)9-s − 0.789i·11-s + 1.37·13-s + (−1.32 + 0.109i)15-s − 1.11i·17-s − 1.69·19-s + (0.00678 + 0.0823i)21-s + 0.890i·23-s − 0.761·25-s + (0.244 + 0.969i)27-s + 0.759i·29-s − 0.846·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0491847 - 1.19488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0491847 - 1.19488i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.246 + 2.98i)T \) |
| good | 5 | \( 1 + 6.63iT - 25T^{2} \) |
| 7 | \( 1 + 0.578T + 49T^{2} \) |
| 11 | \( 1 + 8.68iT - 121T^{2} \) |
| 13 | \( 1 - 17.9T + 169T^{2} \) |
| 17 | \( 1 + 19.0iT - 289T^{2} \) |
| 19 | \( 1 + 32.1T + 361T^{2} \) |
| 23 | \( 1 - 20.4iT - 529T^{2} \) |
| 29 | \( 1 - 22.0iT - 841T^{2} \) |
| 31 | \( 1 + 26.2T + 961T^{2} \) |
| 37 | \( 1 - 53.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 88.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 63.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 33.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.60T + 5.32e3T^{2} \) |
| 79 | \( 1 - 78.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 48.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 93.3T + 9.40e3T^{2} \) |
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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
−11.09485228105336450518320216685, −9.485276593179351330100317816711, −8.542471335175719877405040193375, −8.190559517729374601835150458172, −6.78471353992484613308479978812, −5.89333116078255351090484802028, −4.92691258768584356577976278133, −3.43421964385639732640469960015, −1.71135737550822014512358310904, −0.53299343288937070833272280916,
2.32342130797280380126038348209, 3.62401126547305621980653735101, 4.39979561489589857347963658636, 6.05885012023687927796216016698, 6.51152047474506341078708112447, 7.979985098173560981301071807547, 8.895215674618557672427632151103, 9.997720239208943874464282104884, 10.83405962693651732716888866890, 10.97114731727879280910775385404
