LINQ to SQL - Left Outer Join with multiple join conditions

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

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

I know it's "a bit late" but just in case if anybody needs to do this in LINQ Method syntax (which is why I found this post initially), this would be how to do that:

var results = context.Periods
    .GroupJoin(
        context.Facts,
        period => period.id,
        fk => fk.periodid,
        (period, fact) => fact.Where(f => f.otherid == 17)
                              .Select(fact.Value)
                              .DefaultIfEmpty()
    )
    .Where(period.companyid==100)
    .SelectMany(fact=>fact).ToList();

Another valid option is to spread the joins across multiple LINQ clauses, as follows:

public static IEnumerable<Announcementboard> GetSiteContent(string pageName, DateTime date)
{
    IEnumerable<Announcementboard> content = null;
    IEnumerable<Announcementboard> addMoreContent = null;
        try
        {
            content = from c in DB.Announcementboards
              // Can be displayed beginning on this date
              where c.Displayondate > date.AddDays(-1)
              // Doesn't Expire or Expires at future date
              && (c.Displaythrudate == null || c.Displaythrudate > date)
              // Content is NOT draft, and IS published
              && c.Isdraft == "N" && c.Publishedon != null
              orderby c.Sortorder ascending, c.Heading ascending
              select c;

            // Get the content specific to page names
            if (!string.IsNullOrEmpty(pageName))
            {
              addMoreContent = from c in content
                  join p in DB.Announceonpages on c.Announcementid equals p.Announcementid
                  join s in DB.Apppagenames on p.Apppagenameid equals s.Apppagenameid
                  where s.Apppageref.ToLower() == pageName.ToLower()
                  select c;
            }

            // Add the specified content using UNION
            content = content.Union(addMoreContent);

            // Exclude the duplicates using DISTINCT
            content = content.Distinct();

            return content;
        }
    catch (MyLovelyException ex)
    {
        // Add your exception handling here
        throw ex;
    }
}

Can be written using composite join key. Also if there is need to select properties from both left and right sides the LINQ can be written as

var result = context.Periods
    .Where(p => p.companyid == 100)
    .GroupJoin(
        context.Facts,
        p => new {p.id, otherid = 17},
        f => new {id = f.periodid, f.otherid},
        (p, f) => new {p, f})
    .SelectMany(
        pf => pf.f.DefaultIfEmpty(),
        (pf, f) => new MyJoinEntity
        {
            Id = pf.p.id,
            Value = f.value,
            // and so on...
        });

Whilst my response below does not directly answer the question, I believe it offers an alternative to the core issue that a read might find valuable.

I end up at this thread and others looking for the EF equivalent of a simple self join SQL I had written. I included Entity Framework in my project to make my DB interactions easier, but having to use "GroupJoin" , "SelectMany" and "DefaultIfEmpty" is like having to translate to another language.

Additionally I work with engineer who are excellent at SQL but have limited C# skills. So I want a solution that they could read.

The solution that worked for me was:

context.Database.SqlQuery<Class>

This allows executing of SQL commands with result return in a typed object. So long as the column names returned match the property names of the given Class. For example:

 public class MeasurementEvent
{
    public int ID { get; set; }
    public string JobAssemID { get; set; }
    public DateTime? InspDate { get; set; }


}

var list = context.Database.SqlQuery<MeasurementEvent>(@"
                Select op.umeMeasurementEventID as ID, op.umeJobID+'.'+Cast(op.umeAssemblyID as varchar) as JobAssemID ,  insp.umeCreatedDate as InspDate 
                from uMeasurementEvents as op 
                    left JOIN   uMeasurementEvents as insp on op.umeJobID = insp.umeJobID and op.umeAssemblyID = insp.umeAssemblyID and insp.umeInstanceId = 1 and insp.umeIsInspector = 1
                  where  op.umeInstanceId = 1  and op.umeIsInspector = 0")
            .ToList();

It seems to me there is value in considering some rewrites to your SQL code before attempting to translate it.

Personally, I'd write such a query as a union (although I'd avoid nulls entirely!):

SELECT f.value
  FROM period as p JOIN facts AS f ON p.id = f.periodid
WHERE p.companyid = 100
      AND f.otherid = 17
UNION
SELECT NULL AS value
  FROM period as p
WHERE p.companyid = 100
      AND NOT EXISTS ( 
                      SELECT * 
                        FROM facts AS f
                       WHERE p.id = f.periodid
                             AND f.otherid = 17
                     );

So I guess I agree with the spirit of @MAbraham1's answer (though their code seems to be unrelated to the question).

However, it seems the query is expressly designed to produce a single column result comprising duplicate rows -- indeed duplicate nulls! It's hard not to come to the conclusion that this approach is flawed.