[math] \newcommand{\mathds}{\mathbb}[/math]

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Work out the formula of the basic circulant almost Hadamard matrix

[[math]] L_N=\frac{1}{\sqrt{N}} \begin{pmatrix} 1&-\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&\ldots\ldots&\cos^{-1}\frac{(N-1)\pi}{N}\\ \cos^{-1}\frac{(N-1)\pi}{N}&1&-\cos^{-1}\frac{\pi}{N}&\ldots\ldots&-\cos^{-1}\frac{(N-2)\pi}{N}\\ \vdots&\vdots&\vdots&&\vdots\\ \vdots&\vdots&\vdots&&\vdots\\ -\cos^{-1}\frac{\pi}{N}&\cos^{-1}\frac{2\pi}{N}&-\cos^{-1}\frac{3\pi}{N}&\ldots\ldots&1 \end{pmatrix} [[/math]]

at [math]N=3,5,7,9,11[/math], and compute its [math]1[/math]-norm.