A lot of guitar players freeze up completely when they come across thirteenth chords in a song chart. I'd like to give you some very easy ways to finger 13ths so that you can jump to them quickly when called upon.
The actual spelling of a G 13th is, G B D F A C E. This is a dominant function with a flat 7th scale degree also known as Myxolydian Mode. When you look at the spelling of the notes You can think of this as a G7 (G B D F) with an A minor triad (A C E) on top (at the same time).
Of course, you only have 6 strings on most guitars, and you need some practical shapes that are hassle free. In this example, I'm omitting the B, D and F notes. These are the third, fifth and seventh. By taking three notes out, you can break down something very complex into alot simpler format (G, A, C, E).
When you think about it, the G, A, C and E notes are really what make this a 13th sound. These notes are the Root (G) and the upper extended notes which are the 9th, 11th, and 13th (In this case A, C, E). If you included the B, D, and F notes, you might as well just be playing a vanilla G7 shape.
You can try adding back in the third or seventh in addition to the simplified shape. Of course, you can get different sounds by experimenting with interchanging or deleting any of the other notes as well.
A way to really simplify the 13th concept is to think of it as a root note with a triad on top of it. The triad, in this case, is built on the note one whole step above the root. For example,in a G 13th, you have a G note as the root with an A minor triad on top. So the notes are G, A, C, E. A cool thing to notice is that this abbreviated 13th shape (G A C E) is really an A minor 7th in 3rd inversion.
One of the common alterations of this shape you will see is the 13th#11. All you need to do is raise the 11th a half step which will be G A C# E. Notice the main difference is you have a root note (G) with a Major triad on top (A C# E) instead of a Minor triad.
I hope this helps you get a hold of the concepts. Drop me a line if you have any questions. See ya next time!