Matthew begins his gospel with a genealogy of Jesus, right before then introducing the birth narrative. Luke, however, waits until after the story is somewhat underway before presenting his genealogy of Jesus. What is Luke's purpose in giving the genealogy after Jesus' baptism by John rather than as part of his birth narrative?
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The reason Luke provides the genealogy of Jesus after Jesus' baptism, at which time the Father calls Jesus "My beloved son" (Luke 3:22), is because those very words from the Father are the basis for Luke's later assertion that Jesus is "the son of God" (Luke 3:38).
In his commentary on Luke 3:23, John Lightfoot wrote,
Lightfoot, John. A Commentary of the New Testament from the Talmud and Hebraica.
One explanation that some scholars (though not a majority) think explains this, is that there was an earlier version of Luke which began at verse 3:1 (or which skipped straight from the preface 1:1-1:4 to 3:1). The evidence is somewhat circumstantial, but Luke 3:1 looks like a beginning and there's some differences in language and themes between the first two chapters and the rest. Finally, it's known that Marcion's gospel looked a lot like Luke and definitely started at 3:1. The location of the genealogy is much less weird if it was in its current location before a birth narrative was added. This argument is briefly sketched in this blog post of Bart Ehrman's, though the idea is not his originally.
See my answer to Jesus as Adam in the Gospels?
Luke presents Jesus as a new Adam.
This is beyond a doubt Luke’s purpose in the placement and arrangement of Jesus’ genealogy. Unlike Matthew who places his genealogy at the outset of his gospel, Luke places it immedietly after Jesus’ adult baptism and immediately prior to his three temptations. The genealogy is thus bookend by the issue of Jesus’ sonship. In the baptism God declares Jesus to be His “beloved Son” and in the temptations Satan challenges Jesus, “if you are the Son of God…”
Also instead of beginning with Abraham and working forward to Jesus, as Matthew does in the genealogy he records (Matthew 1:1-16), Luke genealogy begins with Jesus and works backwards to Adam (Luke 3:23-38). The net effect makes his genealogy a list of sons rather than a list of father and points to Adam as also God's son. Luke is making an implicit comparison between these two men. Only Jesus and Adam are said to be God's son.
To combine the answers above, Luke positions the genealogy of Jesus between his baptism and the temptation to build on several themes: 1. Jesus as the Son of God, and 2. Jesus as the fulfillment of Adam's sonship, 3. Jesus as the fulfillment of Israel sonship.
It might be helpful if visually we can see how Luke has arranged the text:
First Luke begins with God's own testimony that Jesus is his Son. Next, Luke organizes his genealogy in order of son to father instead of father to son in order to be able to end with the statement of Jesus again being "son of God." Finally, by placing "the son of Adam, the son of God" back to back with the temptation narrative, Luke links for his readers the temptation of Jesus with the temptation of Adam. Jesus thus shows himself to be not only the son of God, like Adam, but also the true Son of God who is able to resist temptation.
This furthermore plays off of the idea of the wilderness temptation of Israel, also called God's son. Thus by placing the genealogy here, Luke is able to show Jesus as not only fulfilling the role of Israel as God's son, but also the role of Adam and humanity.