L(s) = 1 | + (−1.20 + 0.439i)2-s + (0.500 − 0.419i)4-s + (−0.642 + 0.766i)5-s + (0.223 − 0.386i)8-s + (0.439 − 1.20i)10-s + (−0.213 + 1.20i)16-s + (0.642 + 1.76i)17-s + (−0.173 + 0.984i)19-s + 0.652i·20-s + (−0.642 − 0.766i)23-s + (−0.173 − 0.984i)25-s + (−1.70 + 0.984i)31-s + (−0.196 − 1.11i)32-s + (−1.55 − 1.85i)34-s + (−0.223 − 1.26i)38-s + ⋯ |
L(s) = 1 | + (−1.20 + 0.439i)2-s + (0.500 − 0.419i)4-s + (−0.642 + 0.766i)5-s + (0.223 − 0.386i)8-s + (0.439 − 1.20i)10-s + (−0.213 + 1.20i)16-s + (0.642 + 1.76i)17-s + (−0.173 + 0.984i)19-s + 0.652i·20-s + (−0.642 − 0.766i)23-s + (−0.173 − 0.984i)25-s + (−1.70 + 0.984i)31-s + (−0.196 − 1.11i)32-s + (−1.55 − 1.85i)34-s + (−0.223 − 1.26i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3568392599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3568392599\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.642 - 0.766i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
good | 2 | \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.642 - 1.76i)T + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.524 + 0.439i)T + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.766 - 0.642i)T^{2} \) |
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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
−10.51651262266626580927030374128, −9.917127207083053661727799834724, −8.831919183761697183655205736494, −8.083377054527619127392585143313, −7.61617575909176315098451239138, −6.61404559689627751773058512930, −5.90855075548525541528393027297, −4.21627728059452987189738723846, −3.42749585225663157951832239598, −1.69367753717295034609660843948,
0.54721432964599381798924581044, 2.02403356141327046263579741277, 3.43194313689422837073917706549, 4.76933945986132359492714720898, 5.47885630800582923062054540128, 7.15756646837573963870427366172, 7.68199587312803895347554861306, 8.566727721317085276744414658819, 9.334946251207563605357273489441, 9.734617961629327672429326547403