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Argument principle calculations of zeroes using extended Riemann Siegel function analogues are investigatedfor the Davenport-Heilbronn function and another 5-periodic Dirichlet series. Both these 5-periodic Dirichletseries functions have previously been reported as possessing functional equations and not being expressibleas Euler products. Informative counts of the non-trivial zeroes (including off the critical line) are obtained for variouss = σ + i ∗ t lines in the positive quadrant of thedoi:10.6084/m9.figshare.5721085.v1 fatcat:baudvjwt6zepzgz3qsu5ih5oke