| 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
| \(L(4)\) |
\(\approx\) |
\(4.197670201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.197670201\) |
| \(L(\frac{9}{2})\) |
|
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
−16.16608444330908283915524979356, −14.95067542731888423492008013593, −14.61129536199827440290580813104, −14.36127376828542744405228850374, −13.45216182151439023345047420041, −13.37401805387148146810229708379, −12.07313384729784072575182211767, −11.89724814823951627023821017786, −11.17693146382673135756971322119, −10.06910046714440528236479689783, −9.307514850626282630634831711861, −9.079387441532127855578830158154, −7.80135456897173030435445321789, −7.05535118169219634243906319161, −5.95400044765020892187739885034, −5.35480958655317030384014312271, −4.24993938151823659642408588958, −3.78568715282118347840385501980, −2.05866821596066530584194173257, −1.03339949669651165841796199218,
1.03339949669651165841796199218, 2.05866821596066530584194173257, 3.78568715282118347840385501980, 4.24993938151823659642408588958, 5.35480958655317030384014312271, 5.95400044765020892187739885034, 7.05535118169219634243906319161, 7.80135456897173030435445321789, 9.079387441532127855578830158154, 9.307514850626282630634831711861, 10.06910046714440528236479689783, 11.17693146382673135756971322119, 11.89724814823951627023821017786, 12.07313384729784072575182211767, 13.37401805387148146810229708379, 13.45216182151439023345047420041, 14.36127376828542744405228850374, 14.61129536199827440290580813104, 14.95067542731888423492008013593, 16.16608444330908283915524979356
