| L(s) = 1 | − 4·4-s − 11·7-s + 22·13-s + 52·19-s − 25·25-s + 44·28-s + 13·31-s − 146·37-s − 83·43-s + 49·49-s − 88·52-s + 121·61-s + 64·64-s + 109·67-s − 194·73-s − 208·76-s + 142·79-s − 242·91-s − 2·97-s + 100·100-s + 37·103-s + 142·109-s − 121·121-s − 52·124-s + 127-s + 131-s − 572·133-s + ⋯ |
| L(s) = 1 | − 4-s − 1.57·7-s + 1.69·13-s + 2.73·19-s − 25-s + 11/7·28-s + 0.419·31-s − 3.94·37-s − 1.93·43-s + 49-s − 1.69·52-s + 1.98·61-s + 64-s + 1.62·67-s − 2.65·73-s − 2.73·76-s + 1.79·79-s − 2.65·91-s − 0.0206·97-s + 100-s + 0.359·103-s + 1.30·109-s − 121-s − 0.419·124-s + 0.00787·127-s + 0.00763·131-s − 4.30·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.007554461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007554461\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 - 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−11.97443947961758062277571903882, −11.81665279896007131272708491773, −11.26435442618341913003505498507, −10.52023409708469375586464216427, −9.886575073449185070821132025944, −9.883017161406670449205910830109, −9.274289860066989745310240649863, −8.613350997680044264048085333605, −8.560869071073504338394998332409, −7.71208101889481213688889481271, −6.86082296424511453787695535562, −6.81021228481683706497037587427, −5.92536824276041341363662873232, −5.38328873934200320089991809104, −5.01109181839463085790046077550, −3.89206492771762718266429759829, −3.44616630578991031180337498466, −3.22215371327991025871540503024, −1.69634415217055647026360914582, −0.55682468455908070927384990622,
0.55682468455908070927384990622, 1.69634415217055647026360914582, 3.22215371327991025871540503024, 3.44616630578991031180337498466, 3.89206492771762718266429759829, 5.01109181839463085790046077550, 5.38328873934200320089991809104, 5.92536824276041341363662873232, 6.81021228481683706497037587427, 6.86082296424511453787695535562, 7.71208101889481213688889481271, 8.560869071073504338394998332409, 8.613350997680044264048085333605, 9.274289860066989745310240649863, 9.883017161406670449205910830109, 9.886575073449185070821132025944, 10.52023409708469375586464216427, 11.26435442618341913003505498507, 11.81665279896007131272708491773, 11.97443947961758062277571903882
