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Let 
$f$ be a modular form that is non-ordinary at 
$p$ . Loeffler has recently constructed four two-variable $p$ -adic 
$L$ -functions associated with $f$ . In the case where 
${{a}_{p}}\,=\,0$ , he showed that, as in the one-variable case, Pollack’s plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where 
${{a}_{p}}\ne 0$ using Sprung’s logarithmic matrix.
