Bull. London Math. Soc. 49 (2017) 926–936 C ❡ 2017 London Mathematical Society doi:10.1112/blms.12080 Zeros of L-functions attached to modular forms of half-integral weight Jaban Meher, Sudhir Pujahari and Srinivas Kotyada Abstract We show that there are inﬁnitely many zeros on the critical line for the L-functions attached to certain modular forms of half-integral weight. 1. Introduction For any integer k  1, let S k+ 1 2 (4) denote the space of cusp forms of weight k + 1 2 on the congruence subgroup Γ 0 (4). Let f be a cusp form of half-integral weight k + 1 2 on the group Γ 0 (4) with Fourier coeﬃcients a(n)(n  1). The L-series attached to f is given by L(s, f )= ∞  n=1 a(n) n s . L(s, f ) converges for ℜ(s) suﬃciently large, where ℜ(s) is the real part of s. Similar to the case of L-series attached to an integral weight cusp form, the L-series L(s, f ) can be analytically continued to the whole complex plane C and it satisﬁes certain functional equation. For more details, see Section 2 of this paper. But unlike the integral weight cusp forms, the L-function attached to a half-integral weight cusp form does not have an Euler product. This contributes to the diﬃculty in studying certain properties of L-functions attached to half-integral weight cusp forms. For example, while studying the zeros of the L-function attached to an integral weight Hecke cusp form, one deduces that, in certain right half-plane the L-function does not vanish since it has an Euler product in that region, and then by the functional equation one rules out the possibility of having zeros of the L-function in certain left half-plane except for zeros on the real axis (contributed by the gamma function) which are called trivial zeros. Thus the non-trivial zeros lie in the remaining vertical strip of the complex plane. This vertical strip is called the critical strip. However, since we do not have Euler products in the case of L-functions attached to half-integral weight cusp forms, we cannot guarantee the location of the zeros of these L-functions. Thus vanishing and non-vanishing properties of L-functions attached to half-integral weight cusp forms at particular points in the complex plane are bit mysterious. In the case of any integral weight Hecke eigenforms of level 1, Kohnen [7] proved a non-vanishing result at an arbitrary point s 0 which lies in the critical strip but not on the critical line using certain kernel functions. Following the method of Kohnen, Ramakrishnan and Shankhadhar [13] proved an analogous result for cusp forms of half-integral weight of arbitrary level. Recently, Choie and Kohnen [2] have investigated the non-vanishing properties of L-functions attached to certain half-integral weight cusp forms on Γ 0 (4) on the real line. Also very recently, Kohnen and Raji [9] have proved a non-vanishing result in certain half planes for L-functions attached to modular forms of half-integral weight in the Kohnen plus space. In this article we concentrate on the zeros of L-functions attached to half-integral weight Received 2 March 2017; revised 24 June 2017; published online 23 August 2017. 2010 Mathematics Subject Classiﬁcation 11F37, 11F66, 11M41 (primary).