L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + 10-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)28-s + (−0.309 − 0.951i)29-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + 10-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)28-s + (−0.309 − 0.951i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2226903527 + 0.2331210515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2226903527 + 0.2331210515i\) |
\(L(1)\) |
\(\approx\) |
\(0.5402889161 - 0.1049431347i\) |
\(L(1)\) |
\(\approx\) |
\(0.5402889161 - 0.1049431347i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.029261283663249348286565594708, −23.12481420982784714672609321082, −22.09891749914273475439819524856, −20.948995359374800521906805496381, −19.96244983832791804940616776981, −19.26028736737591999046241890767, −18.61096446153805024246088378168, −17.58481910092343770955876923886, −16.66540492261968432388082378066, −15.87984737123267850209429581511, −15.2875281218309030317666584352, −14.41499270220519875424465790489, −13.0203225409362487434169799339, −12.05631758693992850930152297218, −11.24617243037123119238116909147, −10.04849009531831687852780599006, −9.10266486161088682418819431980, −8.380825735668847032234239764445, −7.5981233344700724491634846317, −6.400029171502507417039845495094, −5.50130262239441297690314126755, −4.43848061330561706471024408916, −2.89476993869930541992513924271, −1.48131236301462595999791301850, −0.146288949971700986694101042142,
0.93496123109748676620155463053, 2.45267091942875487683375125626, 3.60216914739325012409938914147, 4.18987495900871680610237151023, 6.131356948146316251259980411415, 7.37415525336967663815879235946, 7.70058130369774872649031251796, 8.90757044907378004454701111658, 10.24079944930170167393835592408, 10.45571268292710913608585785915, 11.73808849001022986261018691763, 12.25548735459031904349615557494, 13.53514354671027410824216660665, 14.45692335891177715927281974634, 15.77432186231560262932703968330, 16.383922174997393735922199749437, 17.30254105373785727572291216114, 18.27989674270615709476128159665, 19.08826882538807942648821490015, 19.682646943873311107945174031522, 20.549925658447721188024540892709, 21.31090818982285494407518077518, 22.639163352496920218129040704194, 22.972486261881411450311647891395, 24.163468356066990954034219800128