L(s) = 1 | + (−0.866 + 0.5i)4-s + (0.5 − 0.866i)7-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (0.866 + 0.5i)25-s + 0.999i·28-s + (−1.36 + 0.366i)31-s + (−0.5 − 1.86i)37-s + (1.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s − 1.73·61-s + 0.999i·64-s + (−1 − i)67-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)4-s + (0.5 − 0.866i)7-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (0.866 + 0.5i)25-s + 0.999i·28-s + (−1.36 + 0.366i)31-s + (−0.5 − 1.86i)37-s + (1.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−0.499 + 0.866i)52-s − 1.73·61-s + 0.999i·64-s + (−1 − i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8971553877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8971553877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
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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
−10.52403060717540282138395632216, −9.463322661929802573631553809019, −8.789720302073274423239527270075, −7.71872498564144079265791293213, −7.43447111586758402498875781759, −5.88050261215556366916036047167, −5.04829324274203416652598710843, −3.93846544203856168203041988128, −3.31052947040954895929487667957, −1.27520525454857803587190793091,
1.39005318553599220502305882319, 2.94263895845517460930360134371, 4.28500326620099445323270361678, 5.14099150499034411808459670238, 5.85227941283572077116123002544, 6.95155218175250706796369691965, 8.119360106353005796258367289655, 9.027783125758584661260911466670, 9.248762769368219574260648237105, 10.45655616663492799850900290603