| L(s) = 1 | − 4·4-s − 2·7-s − 23·13-s − 74·19-s − 25·25-s + 8·28-s − 59·31-s + 52·37-s + 61·43-s + 49·49-s + 92·52-s + 121·61-s + 64·64-s + 109·67-s − 194·73-s + 296·76-s − 11·79-s + 46·91-s + 169·97-s + 100·100-s + 37·103-s + 286·109-s − 121·121-s + 236·124-s + 127-s + 131-s + 148·133-s + ⋯ |
| L(s) = 1 | − 4-s − 2/7·7-s − 1.76·13-s − 3.89·19-s − 25-s + 2/7·28-s − 1.90·31-s + 1.40·37-s + 1.41·43-s + 49-s + 1.76·52-s + 1.98·61-s + 64-s + 1.62·67-s − 2.65·73-s + 3.89·76-s − 0.139·79-s + 0.505·91-s + 1.74·97-s + 100-s + 0.359·103-s + 2.62·109-s − 121-s + 1.90·124-s + 0.00787·127-s + 0.00763·131-s + 1.11·133-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3037562968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3037562968\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
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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
−12.37992966960291085887811145869, −11.52898023645220013378679846282, −11.25851360790636429507443813956, −10.40730560727753524919974254103, −10.34410570439859903141531630213, −9.631302808049334591067923245851, −9.242955165758210258881934141469, −8.641497375764232328008268811742, −8.508544345603712462454948089836, −7.52546184942337447572848596000, −7.35596499364999224576952405450, −6.45918777368354100007570043504, −6.09607942887488467497314158862, −5.33604224059679194868611566965, −4.71218828786393916689322823150, −4.06950849096600244420587296207, −3.94633202720989909307320011233, −2.32575741368796304430358997092, −2.27733626257811772901905472395, −0.27786641153951727928793548843,
0.27786641153951727928793548843, 2.27733626257811772901905472395, 2.32575741368796304430358997092, 3.94633202720989909307320011233, 4.06950849096600244420587296207, 4.71218828786393916689322823150, 5.33604224059679194868611566965, 6.09607942887488467497314158862, 6.45918777368354100007570043504, 7.35596499364999224576952405450, 7.52546184942337447572848596000, 8.508544345603712462454948089836, 8.641497375764232328008268811742, 9.242955165758210258881934141469, 9.631302808049334591067923245851, 10.34410570439859903141531630213, 10.40730560727753524919974254103, 11.25851360790636429507443813956, 11.52898023645220013378679846282, 12.37992966960291085887811145869
