  Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four
corresponding zeta functions in $s$ are defined by \[\zeta_F(s) \,:=\,
\sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad \zeta_F^*(s)
\,:=\,\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{F_n^s}\,,\quad \zeta_L(s) \,:=\,
\sum_{n=1}^{\infty} \frac{1}{L_n^s}\,,\quad \zeta_L^*(s) \,:=\,
\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{L_n^s} \,.\] For positive integers $s$
the transcendence of these values is known as well as algebraic independence or
dependence results for sets of these numbers. In this paper, we investigate
linear forms in the above zeta functions and determine the dimension of linear
spaces spanned by such linear forms. In particular, it is established that for
any positive integer $m$ the solutions of \[\sum_{s=1}^m \big( \,t_s\zeta_F(2s)
+ u_s\zeta_F^*(2s) + v_s\zeta_L(2s) + w_s\zeta_L^*(2s) \,\big) \,=\, 0 \] with
$t_{s},u_{s},v_{s},w_{s} \in \mathbb{Q}$ $(1\leq s\leq m)$ form a
$\mathbb{Q}$-vector space of dimension $m$. This proves a conjecture from the
Ph.D. thesis of M. Stein, who, in 2012, was inspired by the relation
$-2\zeta_F(2)+\zeta_F^*(2)+5\zeta_L^*(2)=0$. All the results are also true for
zeta functions in $s$ where the Fibonacci and Lucas numbers are replaced by
numbers from sequences satisfying a second order recurrence formula.
