| L(s) = 1 | + 2·4-s + 4·7-s + 7·13-s − 2·19-s + 5·25-s + 8·28-s − 11·31-s − 20·37-s − 5·43-s + 7·49-s + 14·52-s + 61-s − 8·64-s − 5·67-s − 14·73-s − 4·76-s + 13·79-s + 28·91-s − 5·97-s + 10·100-s + 13·103-s − 38·109-s + 11·121-s − 22·124-s + 127-s + 131-s − 8·133-s + ⋯ |
| L(s) = 1 | + 4-s + 1.51·7-s + 1.94·13-s − 0.458·19-s + 25-s + 1.51·28-s − 1.97·31-s − 3.28·37-s − 0.762·43-s + 49-s + 1.94·52-s + 0.128·61-s − 64-s − 0.610·67-s − 1.63·73-s − 0.458·76-s + 1.46·79-s + 2.93·91-s − 0.507·97-s + 100-s + 1.28·103-s − 3.63·109-s + 121-s − 1.97·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163989186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163989186\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.97176877688928411767925748799, −11.96324524829119598033844710065, −11.24056823535085208879544823194, −10.96096379134801597032705417011, −10.56037986866456853977201504282, −10.41906286223794603794670020728, −9.088340175494034891205275362919, −9.032681696819044858699813405982, −8.305379320250581603205108826742, −8.123564695784635754761888175223, −7.20965693379886886712998282992, −6.94654208274578141535972637128, −6.39529797301599778835698702081, −5.61645945366736046919660453916, −5.25867214836061762996343017701, −4.47899069305727190389317624933, −3.72171311063105546675373390782, −3.10820885202341447541150209339, −1.83508433814506093189173428511, −1.61416762214392151984078775814,
1.61416762214392151984078775814, 1.83508433814506093189173428511, 3.10820885202341447541150209339, 3.72171311063105546675373390782, 4.47899069305727190389317624933, 5.25867214836061762996343017701, 5.61645945366736046919660453916, 6.39529797301599778835698702081, 6.94654208274578141535972637128, 7.20965693379886886712998282992, 8.123564695784635754761888175223, 8.305379320250581603205108826742, 9.032681696819044858699813405982, 9.088340175494034891205275362919, 10.41906286223794603794670020728, 10.56037986866456853977201504282, 10.96096379134801597032705417011, 11.24056823535085208879544823194, 11.96324524829119598033844710065, 11.97176877688928411767925748799
