Require powershell command to retrieve a specific list and that particular list properties

This should do it

$spWebApp = Get-SPWebApplication "http://myapplication.sharepoint.com/"

function getListProperties ($spweb)
{
    foreach ($spList in $spweb.Lists | Where {$_.Title -eq "Project Center"})
    {
        Write-Host "List found in: $($spweb.Url)"
        $spList.PSObject.Properties | Select-Object Name,Value
    }

    foreach ( $subweb in $spweb.Webs)
    {
        getListProperties $subweb
    }

    $spweb.Dispose()
}


foreach ($spSite in $spWebApp.Sites)
{
    Write-Host "Looking in $($spSite.Url)..."
    $rootWeb = $spSite.RootWeb
    getListProperties $rootWeb
}

Don't forget to replace the spweb application URL with your SharePoint URL.