| L(s) = 1 | + 32·2-s + 640·4-s + 288·5-s + 1.37e3·7-s + 1.02e4·8-s + 9.21e3·10-s + 2.30e3·11-s + 1.07e4·13-s + 4.39e4·14-s + 1.43e5·16-s + 2.40e4·17-s − 3.55e3·19-s + 1.84e5·20-s + 7.37e4·22-s + 6.88e4·23-s − 7.13e4·25-s + 3.42e5·26-s + 8.78e5·28-s + 5.00e4·29-s + 4.55e4·31-s + 1.83e6·32-s + 7.68e5·34-s + 3.95e5·35-s + 7.13e4·37-s − 1.13e5·38-s + 2.94e6·40-s + 6.84e5·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.03·5-s + 1.51·7-s + 7.07·8-s + 2.91·10-s + 0.521·11-s + 1.35·13-s + 4.27·14-s + 35/4·16-s + 1.18·17-s − 0.118·19-s + 5.15·20-s + 1.47·22-s + 1.17·23-s − 0.913·25-s + 3.82·26-s + 7.55·28-s + 0.380·29-s + 0.274·31-s + 9.89·32-s + 3.35·34-s + 1.55·35-s + 0.231·37-s − 0.336·38-s + 7.28·40-s + 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(316.8571286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(316.8571286\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 288 T + 154334 T^{2} - 6314112 p T^{3} + 609686763 p^{2} T^{4} - 6314112 p^{8} T^{5} + 154334 p^{14} T^{6} - 288 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2304 T + 29116646 T^{2} - 99420229440 T^{3} + 375994430811195 T^{4} - 99420229440 p^{7} T^{5} + 29116646 p^{14} T^{6} - 2304 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 824 p T + 281193772 T^{2} - 2038483209224 T^{3} + 27407016399407638 T^{4} - 2038483209224 p^{7} T^{5} + 281193772 p^{14} T^{6} - 824 p^{22} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 24000 T + 897588068 T^{2} - 20683506159936 T^{3} + 500004331312646022 T^{4} - 20683506159936 p^{7} T^{5} + 897588068 p^{14} T^{6} - 24000 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 3556 T + 1032935698 T^{2} + 7941066463816 T^{3} + 1713969400888843387 T^{4} + 7941066463816 p^{7} T^{5} + 1032935698 p^{14} T^{6} + 3556 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 68832 T + 8253375734 T^{2} - 212475597723072 T^{3} + 23984237082170738427 T^{4} - 212475597723072 p^{7} T^{5} + 8253375734 p^{14} T^{6} - 68832 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 50016 T + 50536858364 T^{2} - 3225619221033888 T^{3} + \)\(11\!\cdots\!22\)\( T^{4} - 3225619221033888 p^{7} T^{5} + 50536858364 p^{14} T^{6} - 50016 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 45596 T + 56337162466 T^{2} - 1427464116950024 T^{3} + \)\(19\!\cdots\!71\)\( T^{4} - 1427464116950024 p^{7} T^{5} + 56337162466 p^{14} T^{6} - 45596 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 71396 T + 229175914906 T^{2} - 6021563698237664 T^{3} + \)\(25\!\cdots\!59\)\( T^{4} - 6021563698237664 p^{7} T^{5} + 229175914906 p^{14} T^{6} - 71396 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 684672 T + 294775930334 T^{2} - 166872811015127232 T^{3} + \)\(98\!\cdots\!03\)\( T^{4} - 166872811015127232 p^{7} T^{5} + 294775930334 p^{14} T^{6} - 684672 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 258656 T + 44443515724 T^{2} - 77665308600296288 T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - 77665308600296288 p^{7} T^{5} + 44443515724 p^{14} T^{6} - 258656 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 869280 T + 388780058492 T^{2} + 111625250434227936 T^{3} - \)\(38\!\cdots\!38\)\( T^{4} + 111625250434227936 p^{7} T^{5} + 388780058492 p^{14} T^{6} - 869280 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 862080 T + 2997358103156 T^{2} - 2244394535328996480 T^{3} + \)\(42\!\cdots\!58\)\( T^{4} - 2244394535328996480 p^{7} T^{5} + 2997358103156 p^{14} T^{6} - 862080 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 240672 T + 5573843409548 T^{2} + 3048813682041350880 T^{3} + \)\(13\!\cdots\!02\)\( T^{4} + 3048813682041350880 p^{7} T^{5} + 5573843409548 p^{14} T^{6} - 240672 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1887872 T + 8655208401364 T^{2} - 6806217635983413632 T^{3} + \)\(28\!\cdots\!42\)\( T^{4} - 6806217635983413632 p^{7} T^{5} + 8655208401364 p^{14} T^{6} - 1887872 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 928568 T + 22343672178436 T^{2} - 15989629213195179032 T^{3} + \)\(19\!\cdots\!50\)\( T^{4} - 15989629213195179032 p^{7} T^{5} + 22343672178436 p^{14} T^{6} - 928568 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 842976 T + 32348866315046 T^{2} - 16827464750902276032 T^{3} + \)\(42\!\cdots\!35\)\( T^{4} - 16827464750902276032 p^{7} T^{5} + 32348866315046 p^{14} T^{6} - 842976 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 4220864 T + 15897248547364 T^{2} - 73711921078744060736 T^{3} + \)\(35\!\cdots\!42\)\( T^{4} - 73711921078744060736 p^{7} T^{5} + 15897248547364 p^{14} T^{6} - 4220864 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 7265840 T + 58697684033884 T^{2} - \)\(31\!\cdots\!92\)\( T^{3} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(31\!\cdots\!92\)\( p^{7} T^{5} + 58697684033884 p^{14} T^{6} - 7265840 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1015392 T + 71423124327188 T^{2} - 93626490508519360608 T^{3} + \)\(24\!\cdots\!66\)\( T^{4} - 93626490508519360608 p^{7} T^{5} + 71423124327188 p^{14} T^{6} - 1015392 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 7378176 T + 94916537445470 T^{2} - \)\(57\!\cdots\!36\)\( T^{3} + \)\(62\!\cdots\!23\)\( T^{4} - \)\(57\!\cdots\!36\)\( p^{7} T^{5} + 94916537445470 p^{14} T^{6} - 7378176 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 16390544 T + 202665772348900 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + \)\(27\!\cdots\!82\)\( T^{4} - \)\(25\!\cdots\!68\)\( p^{7} T^{5} + 202665772348900 p^{14} T^{6} - 16390544 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.91009372230412560996744754730, −6.40347128076628512539761816307, −6.38784903069235081721831851504, −6.15999567877678467403172630444, −5.87157366115488502729598216455, −5.60779286012031726372755317188, −5.35118989102945665951393345777, −5.33570837184290790885968551615, −5.15052321600729272450261076297, −4.48560043433882076123681435773, −4.37401351182238978866460226205, −4.24034230952602631081277351758, −4.16583800488065202730301425878, −3.49636002308594514976812962146, −3.34187277453971136840707645500, −3.21875467913064049320510988184, −2.96117314548182177559925173161, −2.17121160865021398415345443848, −2.10924038049036684071342496016, −2.07742295733976024148231361293, −1.92101610788102910670070650643, −1.13012509360748887471812147970, −1.01147070701418000460063496064, −0.838969045167554884685771236295, −0.67356149042877946197752078825,
0.67356149042877946197752078825, 0.838969045167554884685771236295, 1.01147070701418000460063496064, 1.13012509360748887471812147970, 1.92101610788102910670070650643, 2.07742295733976024148231361293, 2.10924038049036684071342496016, 2.17121160865021398415345443848, 2.96117314548182177559925173161, 3.21875467913064049320510988184, 3.34187277453971136840707645500, 3.49636002308594514976812962146, 4.16583800488065202730301425878, 4.24034230952602631081277351758, 4.37401351182238978866460226205, 4.48560043433882076123681435773, 5.15052321600729272450261076297, 5.33570837184290790885968551615, 5.35118989102945665951393345777, 5.60779286012031726372755317188, 5.87157366115488502729598216455, 6.15999567877678467403172630444, 6.38784903069235081721831851504, 6.40347128076628512539761816307, 6.91009372230412560996744754730
