| L(s) = 1 | + 2-s + 3-s − 3·4-s − 3·5-s + 6-s − 7-s − 5·8-s − 3·10-s − 3·12-s − 7·13-s − 14-s − 3·15-s − 12·17-s + 10·19-s + 9·20-s − 21-s − 8·23-s − 5·24-s + 10·25-s − 7·26-s + 3·28-s − 6·29-s − 3·30-s − 12·31-s + 9·32-s − 12·34-s + 3·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 3/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s − 1.76·8-s − 0.948·10-s − 0.866·12-s − 1.94·13-s − 0.267·14-s − 0.774·15-s − 2.91·17-s + 2.29·19-s + 2.01·20-s − 0.218·21-s − 1.66·23-s − 1.02·24-s + 2·25-s − 1.37·26-s + 0.566·28-s − 1.11·29-s − 0.547·30-s − 2.15·31-s + 1.59·32-s − 2.05·34-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09469498683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09469498683\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 13 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 49 p T^{5} + 6 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 63 T^{2} + 14 p T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + 63 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 307 p T^{5} - 6 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 3 T - 22 T^{2} - 171 T^{3} + 2215 T^{4} - 171 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 17 T + 67 T^{2} + 335 T^{3} - 4344 T^{4} + 335 p T^{5} + 67 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 160 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 222 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 21 T + 245 T^{2} - 2559 T^{3} + 23404 T^{4} - 2559 p T^{5} + 245 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 1455 p T^{5} + 119 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 850 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 407 p T^{5} + 42 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 421 p T^{5} - 14 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 765 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.360363740201522801495811347616, −8.123787834656179143775809837971, −7.77810010278450428346438900742, −7.35754831211684885040875314103, −7.32634544624290186172783555577, −7.14461928997515118705016356647, −7.05466099334094111026054064516, −6.58504836279860561466500977430, −6.10556596136151687791095745468, −5.88576372364379481721535902749, −5.62647799432999924947370077821, −5.33000089846511745604554363672, −4.97342856024598073191749865083, −4.83245424417012716936761698911, −4.49412252142244487831620565722, −4.15234130499552035711375873530, −4.06415307846894545888638803819, −3.84990407601444485678462766639, −3.45868565095073252218064502822, −3.13171424632067351604230909286, −2.45379306537636276647045538437, −2.42973866780133081041971408440, −2.21446844940188821761293792449, −0.987766414688239641000841586789, −0.13007060242265709801906984920,
0.13007060242265709801906984920, 0.987766414688239641000841586789, 2.21446844940188821761293792449, 2.42973866780133081041971408440, 2.45379306537636276647045538437, 3.13171424632067351604230909286, 3.45868565095073252218064502822, 3.84990407601444485678462766639, 4.06415307846894545888638803819, 4.15234130499552035711375873530, 4.49412252142244487831620565722, 4.83245424417012716936761698911, 4.97342856024598073191749865083, 5.33000089846511745604554363672, 5.62647799432999924947370077821, 5.88576372364379481721535902749, 6.10556596136151687791095745468, 6.58504836279860561466500977430, 7.05466099334094111026054064516, 7.14461928997515118705016356647, 7.32634544624290186172783555577, 7.35754831211684885040875314103, 7.77810010278450428346438900742, 8.123787834656179143775809837971, 8.360363740201522801495811347616
