| L(s) = 1 | − 9·2-s − 49·4-s − 180·5-s + 700·7-s + 459·8-s + 1.62e3·10-s − 1.08e4·11-s − 5.48e3·13-s − 6.30e3·14-s − 3.77e3·16-s − 1.64e4·17-s + 1.60e4·19-s + 8.82e3·20-s + 9.80e4·22-s − 2.43e4·23-s − 1.31e5·25-s + 4.93e4·26-s − 3.43e4·28-s + 1.43e5·29-s − 3.87e4·31-s + 1.78e5·32-s + 1.47e5·34-s − 1.26e5·35-s + 4.55e5·37-s − 1.44e5·38-s − 8.26e4·40-s − 7.31e5·41-s + ⋯ |
| L(s) = 1 | − 0.795·2-s − 0.382·4-s − 0.643·5-s + 0.771·7-s + 0.316·8-s + 0.512·10-s − 2.46·11-s − 0.691·13-s − 0.613·14-s − 0.230·16-s − 0.810·17-s + 0.535·19-s + 0.246·20-s + 1.96·22-s − 0.417·23-s − 1.68·25-s + 0.550·26-s − 0.295·28-s + 1.09·29-s − 0.233·31-s + 0.961·32-s + 0.644·34-s − 0.496·35-s + 1.47·37-s − 0.426·38-s − 0.204·40-s − 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 9 T + 65 p T^{2} + 9 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 36 p T + 32753 p T^{2} + 36 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 100 p T + 584961 T^{2} - 100 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 90 p^{2} T + 68388367 T^{2} + 90 p^{9} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5480 T + 57188634 T^{2} + 5480 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 16416 T + 79007650 T^{2} + 16416 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16024 T + 1645526562 T^{2} - 16024 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 24372 T + 2905606450 T^{2} + 24372 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 143280 T + 39622155718 T^{2} - 143280 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 38708 T + 26496803553 T^{2} + 38708 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 455620 T + 173526750366 T^{2} - 455620 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 731880 T + 472932732862 T^{2} + 731880 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1088840 T + 751093452114 T^{2} + 1088840 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1561500 T + 1424242194466 T^{2} + 1561500 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2610468 T + 3933113324245 T^{2} + 2610468 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1731960 T + 4329510506038 T^{2} + 1731960 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 620192 T + 6303036139818 T^{2} + 620192 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 346600 T + 12137122138146 T^{2} - 346600 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4242240 T + 14648438075182 T^{2} - 4242240 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3145190 T + 18855097518219 T^{2} + 3145190 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10110616 T + 57627345888222 T^{2} - 10110616 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 644202 T + 52577539308895 T^{2} + 644202 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6021000 T + 94843763242558 T^{2} + 6021000 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4098670 T + 5732921354451 T^{2} - 4098670 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.44300699008772398870322230485, −15.19694017290412953747874130964, −14.16771241367756159284030634661, −13.46290917077449186145970445196, −13.10210044178011672654975543935, −12.12825516654550966983914809766, −11.47269640361738180135143110430, −10.89419681588924115655180323773, −9.823570931174348196955616277347, −9.802433375458030670966204227072, −8.397291883269100581771662600694, −8.035162359291653759836583903955, −7.69797392899539787341016412379, −6.39014394685925507515243771147, −4.96483972803010063458016663411, −4.77369401669797021759338215721, −3.14313834901834165140929803545, −1.96325339036313430748040369469, 0, 0,
1.96325339036313430748040369469, 3.14313834901834165140929803545, 4.77369401669797021759338215721, 4.96483972803010063458016663411, 6.39014394685925507515243771147, 7.69797392899539787341016412379, 8.035162359291653759836583903955, 8.397291883269100581771662600694, 9.802433375458030670966204227072, 9.823570931174348196955616277347, 10.89419681588924115655180323773, 11.47269640361738180135143110430, 12.12825516654550966983914809766, 13.10210044178011672654975543935, 13.46290917077449186145970445196, 14.16771241367756159284030634661, 15.19694017290412953747874130964, 15.44300699008772398870322230485
