c:=200   (*   c is actually going to Infinity  *)

a:=1.789
t:=2.867

zt2a:=Zeta[2*a]

zr2:=Sum[Sum[Cos[t*Log[(k-r)/k]]/(((k-r)^a)*(k^a)) , {r,0,k-1}] , {k,0,c}]
zr3:=Sum[Sum[Cos[t*Log[(k-r)/k]]/(((k-r)^a)*(k^a)) , {r,1,k-1}] , {k,1,c}]

result:=zr2-zr3

N[zt2a,14]
N[result,14]

1.1182587797427
1.1182583290692

    Source: geocities.com/timeparadox