LINQ to SQL - Left Outer Join with multiple join conditions

this works too, ...if you have multiple column joins

from p in context.Periods
join f in context.Facts 
on new {
    id = p.periodid,
    p.otherid
} equals new {
    f.id,
    f.otherid
} into fg
from fgi in fg.DefaultIfEmpty()
where p.companyid == 100
select f.value

You need to introduce your join condition before calling DefaultIfEmpty(). I would just use extension method syntax:

from p in context.Periods
join f in context.Facts on p.id equals f.periodid into fg
from fgi in fg.Where(f => f.otherid == 17).DefaultIfEmpty()
where p.companyid == 100
select f.value

Or you could use a subquery:

from p in context.Periods
join f in context.Facts on p.id equals f.periodid into fg
from fgi in (from f in fg
             where f.otherid == 17
             select f).DefaultIfEmpty()
where p.companyid == 100
select f.value