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In this paper, we provide Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet L-functions. The details are as follows. Let  L ( s, χ ) be the Dirichlet  L -function and  G ( χ ) be the Gauss sum associated with a primitive Dirichlet character  χ  (mod  q ). We definef  (  s  ,  χ  )  :  =   q  s   L  (  s  ,  χ  )  +       i    −  κ  (  χ  )    G  (  χ  )  L  (  s  ,   χ  ¯   ), whereχ  ¯is the complex conjugate of  χ  and  κ ( χ ) := (1 –  χ (−1))/2. Then, we prove that  f  ( s ,  χ ) satisfies Riemann’s functional equation in Hamburger’s theorem if  χ  is even. In addition, we show that  f  ( σ ,  χ ) ≠ 0 for all  σ  ≥ 1. Moreover, we prove that  f ( σ ,  χ ) ≠ 0 for all 1/2 ≤  σ  &lt; 1 if and only if  L ( σ ,  χ ) ≠ 0 for all 1/2 ≤  σ  &lt; 1. When  χ  is real, all zeros of  f  ( s ,  χ ) withℜ  (  s  )  &gt;  0are on the line  σ  = 1/2 if and only if the generalized Riemann hypothesis for  L ( s ,  χ ) is true. However,  f  ( s ,  χ ) has infinitely many zeros off the critical line  σ  = 1/2 if  χ  is non-real.

Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L -Functions