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We show that there is no spectral value in the set $\\mathbb{S}\\cup \\mathbb{D}_{c_0}$, $c_0=0.2078750206\\ldots$, where $\\mathbb{S}$ is the sector $\\{ 0<|z|<0.6,{\\rm arg}(z)\\in [\\pi /4 ,7\\pi /4 ]\\}$. There is a single spectral value in the set $\\mathbb{S}\\cup \\mathbb{D}_{0.31}$ which equals $0.309249\\ldots$. 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