L(s) = 1 | + (0.965 + 0.258i)2-s + 3-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + i·10-s + (0.707 + 0.707i)11-s + (−0.499 + 0.866i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.707 − 0.707i)19-s + (−0.258 + 0.965i)21-s + (0.500 + 0.866i)22-s + (0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + 3-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + i·10-s + (0.707 + 0.707i)11-s + (−0.499 + 0.866i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.707 − 0.707i)19-s + (−0.258 + 0.965i)21-s + (0.500 + 0.866i)22-s + (0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.049013169\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.049013169\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.258 - 0.965i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.739630460764582039082634843997, −9.201005272717165939848912534210, −8.728116968791004562550923907290, −7.27912258054248140703298946947, −6.73517644122130343906058445132, −5.84423729050444438542932635511, −4.94506588829134604732450441286, −3.86395391679438149350426209554, −2.92911980328115114204382566273, −2.34243760690916444074598983591,
1.49231615283735433111874369824, 3.01178440414669880513342892789, 3.66618809010245038556753066462, 4.43699541897458893474468291912, 5.40431899987460415536819515466, 6.29245644542615909131855090672, 7.52018124520010807691189059643, 8.416935470053044645265099473615, 9.018396893054545109718121060230, 9.511997106809571221456765515410