| L(s) = 1 | + 8·5-s − 16·23-s + 24·25-s − 8·29-s − 16·43-s + 16·47-s + 4·49-s + 8·53-s − 48·67-s − 16·71-s − 16·73-s + 32·97-s − 8·101-s − 128·115-s + 28·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.33·23-s + 24/5·25-s − 1.48·29-s − 2.43·43-s + 2.33·47-s + 4/7·49-s + 1.09·53-s − 5.86·67-s − 1.89·71-s − 1.87·73-s + 3.24·97-s − 0.796·101-s − 11.9·115-s + 2.54·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224379880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224379880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7330 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 746 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 20566 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 20450 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.21334430589839571823284236589, −6.02824321065049013239422844552, −5.91608311452727762746857227518, −5.82469466625548586968915220488, −5.74321423432464136038343047358, −5.60954251541041735328909275286, −5.32822658451853229592401122351, −4.95590953664081951571103748715, −4.73052820495714928541401268023, −4.61679690207275194865865715882, −4.19161724723050316145218186929, −4.10046254065773747266944469205, −3.83748474723901262805046475903, −3.68172223563678435722019297893, −3.09378886361479788636458209660, −3.05166164318882793353193493911, −2.78637633829103657988950457794, −2.32251120395997807349617784552, −2.31309741102351129722389550244, −1.88315417137459952637792387924, −1.70625626132795416293482729059, −1.61123607492311535459949866335, −1.60036584838745865650294246436, −0.826585340299214730326717437238, −0.14521879525570596710742023155,
0.14521879525570596710742023155, 0.826585340299214730326717437238, 1.60036584838745865650294246436, 1.61123607492311535459949866335, 1.70625626132795416293482729059, 1.88315417137459952637792387924, 2.31309741102351129722389550244, 2.32251120395997807349617784552, 2.78637633829103657988950457794, 3.05166164318882793353193493911, 3.09378886361479788636458209660, 3.68172223563678435722019297893, 3.83748474723901262805046475903, 4.10046254065773747266944469205, 4.19161724723050316145218186929, 4.61679690207275194865865715882, 4.73052820495714928541401268023, 4.95590953664081951571103748715, 5.32822658451853229592401122351, 5.60954251541041735328909275286, 5.74321423432464136038343047358, 5.82469466625548586968915220488, 5.91608311452727762746857227518, 6.02824321065049013239422844552, 6.21334430589839571823284236589
