| L(s) = 1 | + 2-s − 3·4-s − 14·5-s + 20·7-s + 19·8-s − 14·10-s + 32·13-s + 20·14-s + 49·16-s − 92·17-s − 34·19-s + 42·20-s + 26·23-s + 15·25-s + 32·26-s − 60·28-s − 174·29-s + 422·31-s + 95·32-s − 92·34-s − 280·35-s + 518·37-s − 34·38-s − 266·40-s − 428·41-s − 550·43-s + 26·46-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 3/8·4-s − 1.25·5-s + 1.07·7-s + 0.839·8-s − 0.442·10-s + 0.682·13-s + 0.381·14-s + 0.765·16-s − 1.31·17-s − 0.410·19-s + 0.469·20-s + 0.235·23-s + 3/25·25-s + 0.241·26-s − 0.404·28-s − 1.11·29-s + 2.44·31-s + 0.524·32-s − 0.464·34-s − 1.35·35-s + 2.30·37-s − 0.145·38-s − 1.05·40-s − 1.63·41-s − 1.95·43-s + 0.0833·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1825826152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1825826152\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + p^{2} T^{2} - 13 p T^{3} + p^{3} T^{4} - 13 p^{4} T^{5} + p^{8} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 14 T + 181 T^{2} + 34 T^{3} - 3712 T^{4} + 34 p^{3} T^{5} + 181 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 20 T + 745 T^{2} - 250 p T^{3} + 151460 T^{4} - 250 p^{4} T^{5} + 745 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 32 T + 6283 T^{2} - 172324 T^{3} + 19678376 T^{4} - 172324 p^{3} T^{5} + 6283 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 92 T + 10015 T^{2} + 1708 p^{2} T^{3} + 48261836 T^{4} + 1708 p^{5} T^{5} + 10015 p^{6} T^{6} + 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 34 T + 7789 T^{2} + 491934 T^{3} + 100192812 T^{4} + 491934 p^{3} T^{5} + 7789 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 26 T + 5800 T^{2} - 628762 T^{3} + 272602574 T^{4} - 628762 p^{3} T^{5} + 5800 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 6 p T + 89405 T^{2} + 10079334 T^{3} + 3059530596 T^{4} + 10079334 p^{3} T^{5} + 89405 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 422 T + 121621 T^{2} - 813586 p T^{3} + 4715438036 T^{4} - 813586 p^{4} T^{5} + 121621 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 14 p T + 222196 T^{2} - 69969108 T^{3} + 17042241381 T^{4} - 69969108 p^{3} T^{5} + 222196 p^{6} T^{6} - 14 p^{10} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 428 T + 281623 T^{2} + 72996448 T^{3} + 27843065288 T^{4} + 72996448 p^{3} T^{5} + 281623 p^{6} T^{6} + 428 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 550 T + 356860 T^{2} + 122234838 T^{3} + 43628494230 T^{4} + 122234838 p^{3} T^{5} + 356860 p^{6} T^{6} + 550 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 556 T + 464992 T^{2} + 159161852 T^{3} + 73095550526 T^{4} + 159161852 p^{3} T^{5} + 464992 p^{6} T^{6} + 556 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 882 T + 753641 T^{2} + 356933250 T^{3} + 170508810204 T^{4} + 356933250 p^{3} T^{5} + 753641 p^{6} T^{6} + 882 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 158 T + 455704 T^{2} + 10644746 T^{3} + 99834013694 T^{4} + 10644746 p^{3} T^{5} + 455704 p^{6} T^{6} - 158 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 290 T + 780649 T^{2} - 160512114 T^{3} + 249052940940 T^{4} - 160512114 p^{3} T^{5} + 780649 p^{6} T^{6} - 290 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 992 T + 1325281 T^{2} - 870459746 T^{3} + 613021090972 T^{4} - 870459746 p^{3} T^{5} + 1325281 p^{6} T^{6} - 992 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 42 T + 203372 T^{2} + 102129390 T^{3} - 39343991994 T^{4} + 102129390 p^{3} T^{5} + 203372 p^{6} T^{6} - 42 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1274 T + 1771225 T^{2} - 1360918918 T^{3} + 1080945673244 T^{4} - 1360918918 p^{3} T^{5} + 1771225 p^{6} T^{6} - 1274 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 362 T + 1421725 T^{2} - 271468310 T^{3} + 902647752772 T^{4} - 271468310 p^{3} T^{5} + 1421725 p^{6} T^{6} - 362 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 1500 T + 2217920 T^{2} + 1968382332 T^{3} + 1823815616910 T^{4} + 1968382332 p^{3} T^{5} + 2217920 p^{6} T^{6} + 1500 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1428 T + 2335535 T^{2} + 2459628540 T^{3} + 2471453298972 T^{4} + 2459628540 p^{3} T^{5} + 2335535 p^{6} T^{6} + 1428 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1052 T + 2038366 T^{2} - 1884713816 T^{3} + 2895539158591 T^{4} - 1884713816 p^{3} T^{5} + 2038366 p^{6} T^{6} - 1052 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.64078833074032819780713417675, −6.60503773418916631990735926463, −6.30023963786866843391387264330, −6.12753901282023116049784914478, −5.73038090531625026645834424949, −5.20160514428955354444797021340, −5.14596739563894989522745833992, −5.03466935541653114772000941192, −4.96129556224600008063533601035, −4.53070409372661797450175192247, −4.23829008575134041331940215244, −4.13980192724655415578655991850, −4.06736486715988690884826241000, −3.68015745001260408022595079854, −3.48974632589735637103206752358, −2.97674748959884954339611357102, −2.94913844120577588414260112786, −2.39444829151434016195014962437, −2.38561710040078021671152448616, −1.65640257487525828946262342991, −1.56774132742732966099618592176, −1.37946314011773625874413579541, −1.00924828277065645429831560027, −0.49921104366563106115214353221, −0.05681487010249071672656973119,
0.05681487010249071672656973119, 0.49921104366563106115214353221, 1.00924828277065645429831560027, 1.37946314011773625874413579541, 1.56774132742732966099618592176, 1.65640257487525828946262342991, 2.38561710040078021671152448616, 2.39444829151434016195014962437, 2.94913844120577588414260112786, 2.97674748959884954339611357102, 3.48974632589735637103206752358, 3.68015745001260408022595079854, 4.06736486715988690884826241000, 4.13980192724655415578655991850, 4.23829008575134041331940215244, 4.53070409372661797450175192247, 4.96129556224600008063533601035, 5.03466935541653114772000941192, 5.14596739563894989522745833992, 5.20160514428955354444797021340, 5.73038090531625026645834424949, 6.12753901282023116049784914478, 6.30023963786866843391387264330, 6.60503773418916631990735926463, 6.64078833074032819780713417675
