| L(s) = 1 | − 8·4-s + 28·7-s − 3·8-s − 21·11-s + 5·13-s + 18·16-s − 99·17-s + 72·19-s − 102·23-s − 224·28-s + 240·29-s + 351·31-s + 24·32-s + 399·37-s − 381·41-s + 460·43-s + 168·44-s − 60·47-s + 490·49-s − 40·52-s − 873·53-s − 84·56-s + 855·59-s + 687·61-s − 279·64-s − 503·67-s + 792·68-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s − 0.132·8-s − 0.575·11-s + 0.106·13-s + 9/32·16-s − 1.41·17-s + 0.869·19-s − 0.924·23-s − 1.51·28-s + 1.53·29-s + 2.03·31-s + 0.132·32-s + 1.77·37-s − 1.45·41-s + 1.63·43-s + 0.575·44-s − 0.186·47-s + 10/7·49-s − 0.106·52-s − 2.26·53-s − 0.200·56-s + 1.88·59-s + 1.44·61-s − 0.544·64-s − 0.917·67-s + 1.41·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.411915773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.411915773\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p^{3} T^{2} + 3 T^{3} + 23 p T^{4} + 3 p^{3} T^{5} + p^{9} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 21 T + 2681 T^{2} + 22236 T^{3} + 3289690 T^{4} + 22236 p^{3} T^{5} + 2681 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 1848 T^{2} + 69653 T^{3} - 988138 T^{4} + 69653 p^{3} T^{5} + 1848 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 99 T + 13298 T^{2} + 508917 T^{3} + 56706418 T^{4} + 508917 p^{3} T^{5} + 13298 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 72 T + 25164 T^{2} - 1424592 T^{3} + 252125750 T^{4} - 1424592 p^{3} T^{5} + 25164 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 102 T + 38528 T^{2} + 3477336 T^{3} + 641077657 T^{4} + 3477336 p^{3} T^{5} + 38528 p^{6} T^{6} + 102 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 240 T + 86430 T^{2} - 14441664 T^{3} + 3156579983 T^{4} - 14441664 p^{3} T^{5} + 86430 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 351 T + 102010 T^{2} - 21726711 T^{3} + 4440563682 T^{4} - 21726711 p^{3} T^{5} + 102010 p^{6} T^{6} - 351 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 399 T + 123385 T^{2} - 25924734 T^{3} + 6891314910 T^{4} - 25924734 p^{3} T^{5} + 123385 p^{6} T^{6} - 399 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 381 T + 273692 T^{2} + 71526783 T^{3} + 28134293782 T^{4} + 71526783 p^{3} T^{5} + 273692 p^{6} T^{6} + 381 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 460 T + 241338 T^{2} - 78643232 T^{3} + 29430041627 T^{4} - 78643232 p^{3} T^{5} + 241338 p^{6} T^{6} - 460 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 60 T + 130548 T^{2} + 11848116 T^{3} + 16872903254 T^{4} + 11848116 p^{3} T^{5} + 130548 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 873 T + 701906 T^{2} + 368908875 T^{3} + 161980949146 T^{4} + 368908875 p^{3} T^{5} + 701906 p^{6} T^{6} + 873 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 855 T + 767616 T^{2} - 344685519 T^{3} + 195821792246 T^{4} - 344685519 p^{3} T^{5} + 767616 p^{6} T^{6} - 855 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 687 T + 715504 T^{2} - 385624593 T^{3} + 235308406830 T^{4} - 385624593 p^{3} T^{5} + 715504 p^{6} T^{6} - 687 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 503 T + 6753 p T^{2} + 193160692 T^{3} + 211333484900 T^{4} + 193160692 p^{3} T^{5} + 6753 p^{7} T^{6} + 503 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 681 T + 1310019 T^{2} - 668258544 T^{3} + 678493576820 T^{4} - 668258544 p^{3} T^{5} + 1310019 p^{6} T^{6} - 681 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1228 T + 618972 T^{2} + 228294532 T^{3} - 319091414218 T^{4} + 228294532 p^{3} T^{5} + 618972 p^{6} T^{6} - 1228 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 345 T + 1235121 T^{2} - 721148448 T^{3} + 716751093026 T^{4} - 721148448 p^{3} T^{5} + 1235121 p^{6} T^{6} - 345 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 1509 T + 2187990 T^{2} + 1588008513 T^{3} + 1474125264218 T^{4} + 1588008513 p^{3} T^{5} + 2187990 p^{6} T^{6} + 1509 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 198 T + 1184884 T^{2} + 492558966 T^{3} + 650980125750 T^{4} + 492558966 p^{3} T^{5} + 1184884 p^{6} T^{6} - 198 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 372 T + 1910724 T^{2} + 1429297356 T^{3} + 2012229372278 T^{4} + 1429297356 p^{3} T^{5} + 1910724 p^{6} T^{6} + 372 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.43426336740754356236685980547, −6.08585292040275716448233716152, −5.76550558178173522107636173244, −5.70068427247550463944700885442, −5.49126069476146095339390712982, −5.06857525044980487288968828340, −4.95646536809536067163428523803, −4.80832114782018388224413184361, −4.62179352516964987083629579962, −4.35330143877642546574617851620, −4.23862940236413818815328166314, −3.98506986205537041382411820736, −3.83117405511955459300967708044, −3.26899224861962938088793056437, −3.11579181911685273235131350913, −2.93402174721032865563932579535, −2.48448647528250359621338834028, −2.26969771098484712920137327649, −2.25273825681696634583412902810, −1.67090541981498614793239259294, −1.52414060992694602984362130928, −1.07461484927796161767912244916, −0.74725289111070947176102852662, −0.52495703997550388500132063913, −0.40142611412861849310104114298,
0.40142611412861849310104114298, 0.52495703997550388500132063913, 0.74725289111070947176102852662, 1.07461484927796161767912244916, 1.52414060992694602984362130928, 1.67090541981498614793239259294, 2.25273825681696634583412902810, 2.26969771098484712920137327649, 2.48448647528250359621338834028, 2.93402174721032865563932579535, 3.11579181911685273235131350913, 3.26899224861962938088793056437, 3.83117405511955459300967708044, 3.98506986205537041382411820736, 4.23862940236413818815328166314, 4.35330143877642546574617851620, 4.62179352516964987083629579962, 4.80832114782018388224413184361, 4.95646536809536067163428523803, 5.06857525044980487288968828340, 5.49126069476146095339390712982, 5.70068427247550463944700885442, 5.76550558178173522107636173244, 6.08585292040275716448233716152, 6.43426336740754356236685980547
