| L(s) = 1 | + 15·2-s − 29·4-s + 192·5-s + 800·7-s − 1.36e3·8-s + 2.88e3·10-s − 5.01e3·11-s − 2.20e3·13-s + 1.20e4·14-s + 2.01e3·16-s − 1.96e4·17-s + 1.12e4·19-s − 5.56e3·20-s − 7.52e4·22-s + 1.54e5·23-s − 5.54e4·25-s − 3.30e4·26-s − 2.32e4·28-s + 1.18e5·29-s + 4.50e5·31-s + 2.40e5·32-s − 2.94e5·34-s + 1.53e5·35-s + 7.11e5·37-s + 1.68e5·38-s − 2.62e5·40-s + 1.02e6·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.226·4-s + 0.686·5-s + 0.881·7-s − 0.942·8-s + 0.910·10-s − 1.13·11-s − 0.277·13-s + 1.16·14-s + 0.122·16-s − 0.968·17-s + 0.375·19-s − 0.155·20-s − 1.50·22-s + 2.64·23-s − 0.710·25-s − 0.368·26-s − 0.199·28-s + 0.903·29-s + 2.71·31-s + 1.29·32-s − 1.28·34-s + 0.605·35-s + 2.31·37-s + 0.498·38-s − 0.647·40-s + 2.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(17.48885231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.48885231\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2 \wr S_4$ | \( 1 - 15 T + 127 p T^{2} - 45 p^{6} T^{3} + 1755 p^{4} T^{4} - 45 p^{13} T^{5} + 127 p^{15} T^{6} - 15 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 192 T + 92354 T^{2} + 723744 p T^{3} + 216356139 p^{2} T^{4} + 723744 p^{8} T^{5} + 92354 p^{14} T^{6} - 192 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 800 T + 183088 p T^{2} + 64459600 T^{3} + 385257219262 T^{4} + 64459600 p^{7} T^{5} + 183088 p^{15} T^{6} - 800 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 456 p T + 25197536 T^{2} + 75921315912 T^{3} + 585464211883566 T^{4} + 75921315912 p^{7} T^{5} + 25197536 p^{14} T^{6} + 456 p^{22} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2200 T + 124193434 T^{2} - 338829581360 T^{3} + 7001267069365579 T^{4} - 338829581360 p^{7} T^{5} + 124193434 p^{14} T^{6} + 2200 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 19620 T + 1301361674 T^{2} + 1236874914000 p T^{3} + 766841815719977235 T^{4} + 1236874914000 p^{8} T^{5} + 1301361674 p^{14} T^{6} + 19620 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 11240 T + 1602943552 T^{2} - 51330484659800 T^{3} + 1208606124510317518 T^{4} - 51330484659800 p^{7} T^{5} + 1602943552 p^{14} T^{6} - 11240 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6720 p T + 19596410096 T^{2} - 1510567007222160 T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - 1510567007222160 p^{7} T^{5} + 19596410096 p^{14} T^{6} - 6720 p^{22} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 118680 T + 53339664170 T^{2} - 5457368696083920 T^{3} + \)\(12\!\cdots\!87\)\( T^{4} - 5457368696083920 p^{7} T^{5} + 53339664170 p^{14} T^{6} - 118680 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 450464 T + 173357954716 T^{2} - 39405543439094048 T^{3} + \)\(79\!\cdots\!86\)\( T^{4} - 39405543439094048 p^{7} T^{5} + 173357954716 p^{14} T^{6} - 450464 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 711800 T + 422855082730 T^{2} - 173395087775794640 T^{3} + \)\(63\!\cdots\!51\)\( T^{4} - 173395087775794640 p^{7} T^{5} + 422855082730 p^{14} T^{6} - 711800 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 1027776 T + 640957024796 T^{2} - 244416969219661632 T^{3} + \)\(10\!\cdots\!26\)\( T^{4} - 244416969219661632 p^{7} T^{5} + 640957024796 p^{14} T^{6} - 1027776 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1259000 T + 1385774668672 T^{2} - 940234041347976200 T^{3} + \)\(58\!\cdots\!94\)\( T^{4} - 940234041347976200 p^{7} T^{5} + 1385774668672 p^{14} T^{6} - 1259000 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 561840 T + 1628123369708 T^{2} - 914023092198697200 T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - 914023092198697200 p^{7} T^{5} + 1628123369708 p^{14} T^{6} - 561840 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 3858840 T + 8944517645420 T^{2} + 14537431046582227080 T^{3} + \)\(18\!\cdots\!38\)\( T^{4} + 14537431046582227080 p^{7} T^{5} + 8944517645420 p^{14} T^{6} + 3858840 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 341472 T + 9437277676316 T^{2} - 2601217461434069088 T^{3} + \)\(34\!\cdots\!70\)\( T^{4} - 2601217461434069088 p^{7} T^{5} + 9437277676316 p^{14} T^{6} - 341472 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 2271896 T + 7230517688266 T^{2} - 12150996167043616592 T^{3} + \)\(28\!\cdots\!11\)\( T^{4} - 12150996167043616592 p^{7} T^{5} + 7230517688266 p^{14} T^{6} - 2271896 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 1649560 T + 10523443023616 T^{2} + 559383501501586600 T^{3} + \)\(51\!\cdots\!54\)\( T^{4} + 559383501501586600 p^{7} T^{5} + 10523443023616 p^{14} T^{6} + 1649560 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 6584688 T + 36563787447152 T^{2} - \)\(13\!\cdots\!12\)\( T^{3} + \)\(48\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!12\)\( p^{7} T^{5} + 36563787447152 p^{14} T^{6} - 6584688 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 358100 T + 19901575268146 T^{2} - 2125649109553166720 T^{3} + \)\(32\!\cdots\!95\)\( T^{4} - 2125649109553166720 p^{7} T^{5} + 19901575268146 p^{14} T^{6} - 358100 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1436336 T + 11027528910736 T^{2} - 16104890696208508640 T^{3} - \)\(27\!\cdots\!70\)\( T^{4} - 16104890696208508640 p^{7} T^{5} + 11027528910736 p^{14} T^{6} - 1436336 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1394160 T + 68352020140700 T^{2} + 30072234493339537680 T^{3} + \)\(21\!\cdots\!58\)\( T^{4} + 30072234493339537680 p^{7} T^{5} + 68352020140700 p^{14} T^{6} - 1394160 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 14935932 T + 126025413039050 T^{2} - \)\(71\!\cdots\!32\)\( T^{3} + \)\(37\!\cdots\!39\)\( T^{4} - \)\(71\!\cdots\!32\)\( p^{7} T^{5} + 126025413039050 p^{14} T^{6} - 14935932 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 4456880 T + 159308978479900 T^{2} + \)\(41\!\cdots\!60\)\( T^{3} + \)\(92\!\cdots\!38\)\( T^{4} + \)\(41\!\cdots\!60\)\( p^{7} T^{5} + 159308978479900 p^{14} T^{6} - 4456880 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−9.387832279203102126464510514074, −8.992397663294926429166411167055, −8.376241607509491206735135391453, −8.126227073166142940743201405276, −7.952988636763126334680029160235, −7.66748687903918330444749452125, −7.42549647556439517406096930476, −6.68212791771222363442499456650, −6.57756422520650887650836700488, −6.11026621859386531979965961885, −5.95372074341530846723628981792, −5.34124364874127904307153739526, −5.23298706429237778414135286052, −4.80793697734854904549245256252, −4.45184784569101331431664158098, −4.34745818632405118719647770808, −4.30688969584781298625789606302, −3.27020186165431743858297017527, −2.89015812168771611511143843380, −2.82683937048845536819983361393, −2.23817621398221591425456544984, −1.91325485600227863559364017806, −0.886358819742471063481970737808, −0.816956881093942160120170048963, −0.67827091219371484691222536080,
0.67827091219371484691222536080, 0.816956881093942160120170048963, 0.886358819742471063481970737808, 1.91325485600227863559364017806, 2.23817621398221591425456544984, 2.82683937048845536819983361393, 2.89015812168771611511143843380, 3.27020186165431743858297017527, 4.30688969584781298625789606302, 4.34745818632405118719647770808, 4.45184784569101331431664158098, 4.80793697734854904549245256252, 5.23298706429237778414135286052, 5.34124364874127904307153739526, 5.95372074341530846723628981792, 6.11026621859386531979965961885, 6.57756422520650887650836700488, 6.68212791771222363442499456650, 7.42549647556439517406096930476, 7.66748687903918330444749452125, 7.952988636763126334680029160235, 8.126227073166142940743201405276, 8.376241607509491206735135391453, 8.992397663294926429166411167055, 9.387832279203102126464510514074
