There's a trivial case, where the "sum" is only of the first prime itself: 2^2 = 4, which is technically palindromic.

However the sum of the squares of the first seven primes is also palindromic:

pr squ

 2   4
 3   9
 5  25
 7  49
11 121
13 169
17 289
   ---
   666 sum of squares

The sum has also come to be called the Number of the Beast from the New Testament book Revelation, or Apocalypse, with various interpretations of what that means. Most commonly it's taken as the Hebrew-alphabet numbering equivalent to the name Nero.     

DefDbl A-Z

Dim crlf$

Function mform$(x, t$)

  a$ = Format$(x, t$)

  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$

  mform$ = a$

End Function

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 Form1.Visible = True

  

  p = 1

  Do

    p = nxtprm(p)

    p2 = p * p

    tot = tot + p2

    If isPalin(tot) Then

      Text1.Text = Text1.Text & p & Str(tot) & crlf

      p1 = 1

      Do

        p1 = nxtprm(p1)

        Text1.Text = Text1.Text & p1 & Str(p1 * p1) & crlf

      Loop Until p1 = p

    End If

    

    DoEvents

  Loop

  

  

  End Sub

 

 Function prmdiv(num)

 Dim n, dv, q

 If num = 1 Then prmdiv = 1: Exit Function

 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0

 If limit <> Int(limit) Then limit = Int(limit + 1)

 dv = 2: GoSub DivideIt

 dv = 3: GoSub DivideIt

 dv = 5: GoSub DivideIt

 dv = 7

 Do Until dv > limit

   GoSub DivideIt: dv = dv + 4 '11

   GoSub DivideIt: dv = dv + 2 '13

   GoSub DivideIt: dv = dv + 4 '17

   GoSub DivideIt: dv = dv + 2 '19

   GoSub DivideIt: dv = dv + 4 '23

   GoSub DivideIt: dv = dv + 6 '29

   GoSub DivideIt: dv = dv + 2 '31

   GoSub DivideIt: dv = dv + 6 '37

 Loop

 If n > 1 Then prmdiv = n

 Exit Function

DivideIt:

 Do

  q = Int(n / dv)

  If q * dv = n And n > 0 Then

    prmdiv = dv: Exit Function

   Else

    Exit Do

  End If

 Loop

 Return

End Function

Function nxtprm(x)

  Dim n

  n = x + 1

  While prmdiv(n) < n Or n < 2

    n = n + 1

  Wend

  nxtprm = n

End Function

Function isPalin(n)

 s$ = LTrim(Str(n))

 

 good = 1

 For i = 1 To Len(s$) / 2

   If Mid$(s$, i, 1) <> Mid$(s$, Len(s$) + 1 - i, 1) Then good = 0: Exit For

 Next

 

 isPalin = good

End Function