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The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T \ }$ zeros in the strip with $|t| 0 < t < T$. See section 5.3 of Iwaniec and Kowalski's "Analytic Number Theory," in particular Theorem 5.8.

It should be possible, if it hasn't been done already, to show that a positive proportion of these zeros are on the critical line using Selberg's method. Hafner extended Selberg's method to various families of degree 2 $L$-functions in a series of papers in the 1980s.

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The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T \ }$ zeros in the strip with $|t| < T$. See section 5.3 of Iwaniec & and Kowalski's "Analytic Number Theory," in particular Theorem 5.8.

It should be possible, if it hasn't been done already, to show that a positive proportion of these zeros are on the critical line using Selberg's method. Hafner extended Selberg's method to various families of degree 2 $L$-functions in a series of papers in the 1980s.

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The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T \ }$ zeros in the strip with $|t| < T$.

See section 5.3 of Iwaniec & Kowalski's "Analytic Number Theory," in particular Theorem 5.8.