  De Bruijn and Newman introduced a deformation of the Riemann zeta function
$\zeta(s)$, and found a real constant $\Lambda$ which encodes the movement of
the zeros of $\zeta(s)$ under the deformation. The Riemann hypothesis (RH) is
equivalent to $\Lambda \le 0$. Newman made the conjecture that $\Lambda \ge 0$
along with the remark that "the new conjecture is a quantitative version of the
dictum that the Riemann hypothesis, if true, is only barely so."
  Newman's conjecture is still unsolved, and previous work could only handle
the Riemann zeta function and quadratic Dirichlet $L$-functions, obtaining
lower bounds very close to zero (for example, for $\zeta(s)$ the bound is at
least $-1.14541 \cdot 10^{-11}$, and for quadratic Dirichlet $L$-functions it
is at least $-1.17 \cdot 10^{-7}$). We generalize the techniques to apply to
automorphic $L$-functions as well as function field $L$-functions. We further
determine the limit of these techniques by studying linear combinations of
$L$-functions, proving that these methods are insufficient.
  We explicitly determine the Newman constants in various function field
settings, which has strong implications for Newman's quantitative version of
RH. In particular, let $\mathcal D \in \bbZ[T]$ be a square-free polynomial of
degree 3. Let $D_p$ be the polynomial in $\bbF_p[T]$ obtained by reducing
$\mathcal D$ modulo $p$. Then the Newman constant $\Lambda_{D_p}$ equals $\log
\frac{|a_p(\mathcal D)|}{2\sqrt{p}}$; by Sato--Tate (if the curve is non-CM)
there exists a sequence of primes such that $\lim_{n \to\infty}
\Lambda_{D_{p_n}} = 0$. We end by discussing connections with random matrix
theory.
