| L(s) = 1 | − 7·4-s − 8·7-s + 16·13-s + 33·16-s − 38·19-s + 35·25-s + 56·28-s − 56·31-s − 56·37-s − 44·43-s − 50·49-s − 112·52-s + 88·61-s − 119·64-s + 22·67-s − 14·73-s + 266·76-s + 136·79-s − 128·91-s + 154·97-s − 245·100-s − 284·103-s − 308·109-s − 264·112-s + 182·121-s + 392·124-s + 127-s + ⋯ |
| L(s) = 1 | − 7/4·4-s − 8/7·7-s + 1.23·13-s + 2.06·16-s − 2·19-s + 7/5·25-s + 2·28-s − 1.80·31-s − 1.51·37-s − 1.02·43-s − 1.02·49-s − 2.15·52-s + 1.44·61-s − 1.85·64-s + 0.328·67-s − 0.191·73-s + 7/2·76-s + 1.72·79-s − 1.40·91-s + 1.58·97-s − 2.44·100-s − 2.75·103-s − 2.82·109-s − 2.35·112-s + 1.50·121-s + 3.16·124-s + 0.00787·127-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5053138929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5053138929\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 p T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 182 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 323 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 133 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1862 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3683 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 997 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4798 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3467 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10838 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15302 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 77 T + p^{2} T^{2} )^{2} \) |
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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
−12.44206299126176279306719509631, −11.72105439464693920321220333112, −10.90225577865438231117238712801, −10.63022711122821205794223649941, −10.25380916613010875090088803739, −9.400396772325168850754347239669, −9.361802549948283547341716553575, −8.672333404717320328734594196579, −8.452912891128604124224958869563, −7.985977100662061605231820694448, −6.88194435028054545177384372176, −6.65206773995398280644786319028, −6.00885093534105990113428260288, −5.27010059882829304794162277620, −4.89972819466503816971833937316, −3.89012226026541001686739810425, −3.81635072447011408723329986672, −3.03215615826968310429192992718, −1.69489235694202723256593919769, −0.39109703914146516006631361426,
0.39109703914146516006631361426, 1.69489235694202723256593919769, 3.03215615826968310429192992718, 3.81635072447011408723329986672, 3.89012226026541001686739810425, 4.89972819466503816971833937316, 5.27010059882829304794162277620, 6.00885093534105990113428260288, 6.65206773995398280644786319028, 6.88194435028054545177384372176, 7.985977100662061605231820694448, 8.452912891128604124224958869563, 8.672333404717320328734594196579, 9.361802549948283547341716553575, 9.400396772325168850754347239669, 10.25380916613010875090088803739, 10.63022711122821205794223649941, 10.90225577865438231117238712801, 11.72105439464693920321220333112, 12.44206299126176279306719509631
