L(s) = 1 | + (0.149 + 0.988i)2-s + (0.680 + 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.0747 + 0.997i)9-s + (0.294 + 0.955i)10-s + (0.997 − 0.0747i)11-s + (−0.866 − 0.5i)12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)15-s + (0.826 − 0.563i)16-s + (−0.866 + 0.5i)17-s + (−0.997 + 0.0747i)18-s + (−0.680 + 0.733i)19-s + ⋯ |
L(s) = 1 | + (0.149 + 0.988i)2-s + (0.680 + 0.733i)3-s + (−0.955 + 0.294i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.0747 + 0.997i)9-s + (0.294 + 0.955i)10-s + (0.997 − 0.0747i)11-s + (−0.866 − 0.5i)12-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)15-s + (0.826 − 0.563i)16-s + (−0.866 + 0.5i)17-s + (−0.997 + 0.0747i)18-s + (−0.680 + 0.733i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6450913809 + 2.700882777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6450913809 + 2.700882777i\) |
\(L(1)\) |
\(\approx\) |
\(1.069937682 + 1.203457904i\) |
\(L(1)\) |
\(\approx\) |
\(1.069937682 + 1.203457904i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.149 + 0.988i)T \) |
| 3 | \( 1 + (0.680 + 0.733i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 11 | \( 1 + (0.997 - 0.0747i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.680 + 0.733i)T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 31 | \( 1 + (-0.930 - 0.365i)T \) |
| 37 | \( 1 + (0.997 + 0.0747i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.781 - 0.623i)T \) |
| 47 | \( 1 + (0.563 + 0.826i)T \) |
| 53 | \( 1 + (0.365 - 0.930i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.294 - 0.955i)T \) |
| 67 | \( 1 + (-0.826 - 0.563i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.149 - 0.988i)T \) |
| 79 | \( 1 + (-0.997 - 0.0747i)T \) |
| 83 | \( 1 + (-0.222 + 0.974i)T \) |
| 89 | \( 1 + (0.149 + 0.988i)T \) |
| 97 | \( 1 + (-0.974 - 0.222i)T \) |
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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
−26.16378582130755519929155421149, −25.27825394676612254325227102618, −24.40721114108351929174284626894, −23.23191927135720293266683842501, −22.258073431474202135053971066084, −21.34077264938531174053643425897, −20.36063931640999981457814318363, −19.75508452458418880047042019618, −18.54506747652545181137901853455, −18.005594574179009870409185013347, −17.03641494725316422147471293874, −15.06236803762134702603467208076, −14.1999646734083766013058387668, −13.366159150271532780613500245901, −12.757510265288386565015943995, −11.484853625398333460959619832305, −10.42975817995473871927551773862, −9.09021353971210151187933509831, −8.74515367270167758343205745344, −6.90149790210482947473222719410, −5.885726457873775223082858418185, −4.276100927509120323391304741616, −2.95203139866434991511851923358, −2.001301581147698971778056728715, −0.88260303353937249420991484008,
1.71950773953182297509789730676, 3.54376386640105995502311239483, 4.46966128203417944510124565892, 5.7592650651146416683670156231, 6.652059921066350551474233225158, 8.20650638254814431805081334514, 9.05869752889049114864640780989, 9.69848222696895073334799226500, 11.05455763294338529382328971730, 12.90448338202204535423382729023, 13.69105044830659964093240342749, 14.48742792100552244092111236450, 15.30480776150218581808649588244, 16.495497301870292191961340531676, 17.04521728386443927870737171991, 18.211048659347984324540930385161, 19.341663796400842737120711652542, 20.5991100541234060889804209967, 21.638423266025996964868831592004, 22.04708739848037737730353006593, 23.31383194471887759735651255724, 24.4985464420930898253028739664, 25.36908890955111770492200464805, 25.75072658875368438118918443738, 26.81128772628936095179296189076