The mass spectra and wave functions for the doubly heavy baryons are computed under the picture that the two heavy quarks inside a doubly heavy baryon, such as two $c$-quarks in $Xi_{cc}$, combine into a heavy `diquark core in color anti-triplet firstly, then the diquark core turns into a color-less doubly heavy baryon via combining the light $q$-quark inside the baryon. Namely both of the combinations, the two heavy quarks inside the baryon into a diquark core in color anti-triplet and the heavy diquark core with the light quark into the baryon, are depicted by relativistic Bethe-Salpeter equations (BSEs) with an accordingly QCD inspired kernel respectively, although in the paper only the heavy diquark cores with the quantum numbers $J^P=1^+$ are considered. Since the `second combination is of the heavy diquark core and the light quark, so the structure effect of the diquark core to the relevant kernel of the BSE is specially considered in terms of the diquark-core wave functions. The mass spectra and wave functions for the `low-laying doubly heavy baryons in the flavors $(ccq)$, $(bcq)$ and $(bbq)$ and in the quantum numbers $J^P=frac{1}{2}^+$, $J^P=frac{3}{2}^+$, achieved by solving the equations under the so-called instantaneous approximation, are presented properly and some comparisons with the others results under different approaches in the literature are made.
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