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Abstract We consider the one-parameter generalization S q 4 of 4-sphere with a conical singularity due to identification τ = τ +2πq in one isometric angle. We compute the value of the spectral zeta-function at zero ζ ^ q = ζ 0 q $$ \widehat{\zeta}(q)=\zeta \left(0;q\right) $$ that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on S q 4 . While the value of the conformal anomaly a-coefficient is proportional to ζ ^ 1 $$ \widehat{\zeta}(1) $$ , we argue that in general the second c ∼ C T anomaly coefficient is related to a particular combination of the second and first derivatives of ζ ^ q $$ \widehat{\zeta}(q) $$ at q = 1. The universality of this relation for C T is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the “r = −1” value of the one-parameter Ansatz suggested in arXiv:1309.0785 . Like the sums of a s and c s coefficients, the regularized sum of ζ ^ s q $$ {\widehat{\zeta}}_s(q) $$ over the whole tower of conformal higher spins s = 1, 2,… is found to vanish, implying UV finiteness on S q 4 and thus also the vanishing of the associated Rényi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2, 0) supersymmetric extension.