For what purpose are you doing the analysis? Is this theory homework where you have to give chord symbols which represent all the played notes and nothing but the played notes as accurately as possible? Remember that chord symbols don't describe any voicing, which might be very important for getting a specific sound. Or are you making an approximation for accompaniment purposes or for making arrangements?
If it's for accompaniment or re-arranging, I would label the chords G13 and G11 or Dm/G. That's not theoretically accurate, but for example on the guitar (and it's a good to think about a guitar reduction for a blues), those chords might be played with pretty much exactly with the notes you have in the picture.
In theory, a G13 chord has seven notes: G - B - D - F - A - C - E, but the guitar only has six strings, and in practice the chord is approximated as 3x3455 (G - F - B - E - A) or even just 3x345x (G - F - B - E), which is what you have.
In theory, a G11 chord has six notes: G - B - D - F - A - C, but in practice it's used as a sus4 chord G - D - F - A - C and written as an alias for something like F6/G. The notes you have are G - D - F - A which corresponds to Dm/G. Would the added C note ruin the feeling?
If it's for arrangement purposes, here's a cool sounding variation (in double time compared to the original version):
The function of the first chord is clearly a dominant V of C, because after this bluesy chord lick there's a C chord which provides a resolution. In blues, dominant-seventh chords are used as "normal" chords without resolving the tension, but your example goes to a C chord.