The standard grips this chord is usually shown with.
G13 is G, B, D, F, A and E, a dominant thirteenth. The shape this page opens on is written 302001, lowest string first.

Also written Gdom13 or G dominant 13. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 8 common shapes for G13 on the guitar, and 55 derived voicings in 5 families.
In which keys. Every note of G13 belongs to C major (the V) and A minor (the ♭VII).
The common shapes, written lowest string first: 302001, 322231, 333455, 353453, x-10-9-10-10-12, x-10-9-10-12-12, 10-10-10-10-12-12, 15-12-14-12-12-13.
Shell. Root, third and seventh: the guide tones fix the quality, so the fifth is dropped. That is what leaves a string free, so pick a tension to put on top. G13 has 3x345x, x-10-x-10-12-12.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. G13 has xx2001, 1220xx, xx3453, x8998x, x-x-14-12-12-13, 13-14-14-12-x-x, x2305x, 3734xx.
Drop 3. The third voice from the top lowered an octave. The dropped voice becomes the bass, which is why a string is skipped. G13 has x2x051, 1x200x, x7x463, 3x345x, 7x596x, x8x987, 12-x-9-10-8-x, x-10-x-10-12-12.
Rootless. The root left to the bass player. Frees a finger for a tension. G13 has xx2201, xx3455, x5345x, x-8-9-9-10-x, x-x-12-10-12-12, x-x-14-14-12-13, x2x251, x5x451.
Open string. An open string left ringing against a fretted grip. Because it costs no finger, it sounds a note the closed shape has none left for -- a tension on top, or a pedal root underneath. Either a familiar grip with open strings added, or a shape so sparse the open strings carry the chord. These belong to their key rather than to a shape, so D, A, G and E have many and Eb a handful. G13 has x03453, 070465, 00300x, x53000, 0x0067, 7x0060.
Tensions. A shell of G13, root, third and seventh, leaves a string free, and the tones that belong over it are the #11 (C#).
The common shapes and the derived voicings come from different places, and both are explained in where the chords and the theory come from.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.