# Properties

 Degree 24 Conductor $11^{12}$ Sign $1$ Motivic weight 4 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5·2-s − 6·3-s − 8·4-s − 18·5-s + 30·6-s − 80·7-s + 80·8-s + 140·9-s + 90·10-s − 43·11-s + 48·12-s + 250·13-s + 400·14-s + 108·15-s − 62·16-s − 1.25e3·17-s − 700·18-s − 1.02e3·19-s + 144·20-s + 480·21-s + 215·22-s + 1.68e3·23-s − 480·24-s + 1.19e3·25-s − 1.25e3·26-s − 1.51e3·27-s + 640·28-s + ⋯
 L(s)  = 1 − 5/4·2-s − 2/3·3-s − 1/2·4-s − 0.719·5-s + 5/6·6-s − 1.63·7-s + 5/4·8-s + 1.72·9-s + 9/10·10-s − 0.355·11-s + 1/3·12-s + 1.47·13-s + 2.04·14-s + 0.479·15-s − 0.242·16-s − 4.32·17-s − 2.16·18-s − 2.83·19-s + 9/25·20-s + 1.08·21-s + 0.444·22-s + 3.18·23-s − 5/6·24-s + 1.91·25-s − 1.84·26-s − 2.07·27-s + 0.816·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(11^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$24$$ $$N$$ = $$11^{12}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : induced by $\chi_{11} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(24,\ 11^{12} ,\ ( \ : [2]^{12} ),\ 1 )$ $L(\frac{5}{2})$ $\approx$ $0.00230344$ $L(\frac12)$ $\approx$ $0.00230344$ $L(3)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $$F_p(T)$$ is a polynomial of degree 24. If $p = 11$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad11 $$1 + 43 T + 1721 p T^{2} - 2065 p^{2} T^{3} + 85 p^{3} T^{4} - 3665602 p^{4} T^{5} - 1477606 p^{6} T^{6} - 3665602 p^{8} T^{7} + 85 p^{11} T^{8} - 2065 p^{14} T^{9} + 1721 p^{17} T^{10} + 43 p^{20} T^{11} + p^{24} T^{12}$$
good2 $$1 + 5 T + 33 T^{2} + 125 T^{3} + 551 T^{4} + 905 p T^{5} + 4305 p T^{6} + 12735 p^{2} T^{7} + 58435 p^{2} T^{8} + 141605 p^{3} T^{9} + 544261 p^{3} T^{10} + 1103405 p^{4} T^{11} + 3982909 p^{4} T^{12} + 1103405 p^{8} T^{13} + 544261 p^{11} T^{14} + 141605 p^{15} T^{15} + 58435 p^{18} T^{16} + 12735 p^{22} T^{17} + 4305 p^{25} T^{18} + 905 p^{29} T^{19} + 551 p^{32} T^{20} + 125 p^{36} T^{21} + 33 p^{40} T^{22} + 5 p^{44} T^{23} + p^{48} T^{24}$$
3 $$1 + 2 p T - 104 T^{2} + 46 T^{3} + 10081 T^{4} - 30398 p T^{5} - 163646 p^{2} T^{6} + 204208 p^{3} T^{7} + 233522 p^{5} T^{8} - 262802 p^{7} T^{9} + 16246 p^{8} T^{10} + 6196316 p^{8} T^{11} + 30056416 p^{8} T^{12} + 6196316 p^{12} T^{13} + 16246 p^{16} T^{14} - 262802 p^{19} T^{15} + 233522 p^{21} T^{16} + 204208 p^{23} T^{17} - 163646 p^{26} T^{18} - 30398 p^{29} T^{19} + 10081 p^{32} T^{20} + 46 p^{36} T^{21} - 104 p^{40} T^{22} + 2 p^{45} T^{23} + p^{48} T^{24}$$
5 $$1 + 18 T - 874 T^{2} - 1798 p T^{3} + 184869 p T^{4} - 2498082 T^{5} - 756868316 T^{6} + 6088380428 T^{7} + 94508025588 p T^{8} - 885497879674 p T^{9} - 231003059512494 T^{10} + 256199910578148 T^{11} + 91821549616068396 T^{12} + 256199910578148 p^{4} T^{13} - 231003059512494 p^{8} T^{14} - 885497879674 p^{13} T^{15} + 94508025588 p^{17} T^{16} + 6088380428 p^{20} T^{17} - 756868316 p^{24} T^{18} - 2498082 p^{28} T^{19} + 184869 p^{33} T^{20} - 1798 p^{37} T^{21} - 874 p^{40} T^{22} + 18 p^{44} T^{23} + p^{48} T^{24}$$
7 $$1 + 80 T + 6388 T^{2} + 701930 T^{3} + 49215681 T^{4} + 3279603800 T^{5} + 239232786210 T^{6} + 14513581978500 T^{7} + 843606939238690 T^{8} + 7237661951983100 p T^{9} + 54529399676115082 p^{2} T^{10} + 404074251544709930 p^{3} T^{11} + 3002897775232982024 p^{4} T^{12} + 404074251544709930 p^{7} T^{13} + 54529399676115082 p^{10} T^{14} + 7237661951983100 p^{13} T^{15} + 843606939238690 p^{16} T^{16} + 14513581978500 p^{20} T^{17} + 239232786210 p^{24} T^{18} + 3279603800 p^{28} T^{19} + 49215681 p^{32} T^{20} + 701930 p^{36} T^{21} + 6388 p^{40} T^{22} + 80 p^{44} T^{23} + p^{48} T^{24}$$
13 $$1 - 250 T + 98548 T^{2} - 25239340 T^{3} + 4920135411 T^{4} - 1056434505070 T^{5} + 134058098615820 T^{6} - 14343324685617900 T^{7} + 158785723907559280 T^{8} +$$$$58\!\cdots\!50$$$$T^{9} -$$$$16\!\cdots\!92$$$$T^{10} +$$$$38\!\cdots\!10$$$$T^{11} -$$$$73\!\cdots\!16$$$$T^{12} +$$$$38\!\cdots\!10$$$$p^{4} T^{13} -$$$$16\!\cdots\!92$$$$p^{8} T^{14} +$$$$58\!\cdots\!50$$$$p^{12} T^{15} + 158785723907559280 p^{16} T^{16} - 14343324685617900 p^{20} T^{17} + 134058098615820 p^{24} T^{18} - 1056434505070 p^{28} T^{19} + 4920135411 p^{32} T^{20} - 25239340 p^{36} T^{21} + 98548 p^{40} T^{22} - 250 p^{44} T^{23} + p^{48} T^{24}$$
17 $$1 + 1250 T + 992428 T^{2} + 582264010 T^{3} + 283914863531 T^{4} + 119803876725030 T^{5} + 46223383217047820 T^{6} + 16729167329461506640 T^{7} +$$$$58\!\cdots\!20$$$$T^{8} +$$$$19\!\cdots\!90$$$$T^{9} +$$$$62\!\cdots\!28$$$$T^{10} +$$$$19\!\cdots\!80$$$$T^{11} +$$$$56\!\cdots\!04$$$$T^{12} +$$$$19\!\cdots\!80$$$$p^{4} T^{13} +$$$$62\!\cdots\!28$$$$p^{8} T^{14} +$$$$19\!\cdots\!90$$$$p^{12} T^{15} +$$$$58\!\cdots\!20$$$$p^{16} T^{16} + 16729167329461506640 p^{20} T^{17} + 46223383217047820 p^{24} T^{18} + 119803876725030 p^{28} T^{19} + 283914863531 p^{32} T^{20} + 582264010 p^{36} T^{21} + 992428 p^{40} T^{22} + 1250 p^{44} T^{23} + p^{48} T^{24}$$
19 $$1 + 1025 T + 619378 T^{2} + 353530160 T^{3} + 166899801006 T^{4} + 66114359518655 T^{5} + 21592456954609665 T^{6} + 5902640524678934610 T^{7} +$$$$15\!\cdots\!25$$$$T^{8} +$$$$24\!\cdots\!45$$$$T^{9} -$$$$10\!\cdots\!67$$$$T^{10} -$$$$11\!\cdots\!95$$$$T^{11} -$$$$61\!\cdots\!56$$$$T^{12} -$$$$11\!\cdots\!95$$$$p^{4} T^{13} -$$$$10\!\cdots\!67$$$$p^{8} T^{14} +$$$$24\!\cdots\!45$$$$p^{12} T^{15} +$$$$15\!\cdots\!25$$$$p^{16} T^{16} + 5902640524678934610 p^{20} T^{17} + 21592456954609665 p^{24} T^{18} + 66114359518655 p^{28} T^{19} + 166899801006 p^{32} T^{20} + 353530160 p^{36} T^{21} + 619378 p^{40} T^{22} + 1025 p^{44} T^{23} + p^{48} T^{24}$$
23 $$( 1 - 842 T + 1237966 T^{2} - 916714810 T^{3} + 731144219855 T^{4} - 446744415652252 T^{5} + 258141193233856964 T^{6} - 446744415652252 p^{4} T^{7} + 731144219855 p^{8} T^{8} - 916714810 p^{12} T^{9} + 1237966 p^{16} T^{10} - 842 p^{20} T^{11} + p^{24} T^{12} )^{2}$$
29 $$1 + 2690 T + 5931748 T^{2} + 10598261380 T^{3} + 16033149600451 T^{4} + 21947778968733670 T^{5} + 27405046849774036980 T^{6} +$$$$31\!\cdots\!20$$$$T^{7} +$$$$34\!\cdots\!60$$$$T^{8} +$$$$34\!\cdots\!50$$$$T^{9} +$$$$33\!\cdots\!68$$$$T^{10} +$$$$30\!\cdots\!90$$$$T^{11} +$$$$26\!\cdots\!84$$$$T^{12} +$$$$30\!\cdots\!90$$$$p^{4} T^{13} +$$$$33\!\cdots\!68$$$$p^{8} T^{14} +$$$$34\!\cdots\!50$$$$p^{12} T^{15} +$$$$34\!\cdots\!60$$$$p^{16} T^{16} +$$$$31\!\cdots\!20$$$$p^{20} T^{17} + 27405046849774036980 p^{24} T^{18} + 21947778968733670 p^{28} T^{19} + 16033149600451 p^{32} T^{20} + 10598261380 p^{36} T^{21} + 5931748 p^{40} T^{22} + 2690 p^{44} T^{23} + p^{48} T^{24}$$
31 $$1 + 1136 T - 230384 T^{2} - 239729514 T^{3} + 1267435488321 T^{4} + 2677537350048516 T^{5} + 1095876078080394266 T^{6} -$$$$12\!\cdots\!44$$$$T^{7} +$$$$51\!\cdots\!26$$$$T^{8} +$$$$24\!\cdots\!96$$$$T^{9} +$$$$25\!\cdots\!46$$$$T^{10} +$$$$85\!\cdots\!26$$$$T^{11} -$$$$77\!\cdots\!44$$$$T^{12} +$$$$85\!\cdots\!26$$$$p^{4} T^{13} +$$$$25\!\cdots\!46$$$$p^{8} T^{14} +$$$$24\!\cdots\!96$$$$p^{12} T^{15} +$$$$51\!\cdots\!26$$$$p^{16} T^{16} -$$$$12\!\cdots\!44$$$$p^{20} T^{17} + 1095876078080394266 p^{24} T^{18} + 2677537350048516 p^{28} T^{19} + 1267435488321 p^{32} T^{20} - 239729514 p^{36} T^{21} - 230384 p^{40} T^{22} + 1136 p^{44} T^{23} + p^{48} T^{24}$$
37 $$1 + 336 T + 72946 T^{2} - 4475893024 T^{3} + 1626992896581 T^{4} - 4706417114238944 T^{5} + 10320605504003112056 T^{6} -$$$$11\!\cdots\!84$$$$T^{7} +$$$$22\!\cdots\!56$$$$T^{8} -$$$$44\!\cdots\!44$$$$T^{9} +$$$$30\!\cdots\!46$$$$T^{10} -$$$$47\!\cdots\!04$$$$T^{11} +$$$$15\!\cdots\!76$$$$T^{12} -$$$$47\!\cdots\!04$$$$p^{4} T^{13} +$$$$30\!\cdots\!46$$$$p^{8} T^{14} -$$$$44\!\cdots\!44$$$$p^{12} T^{15} +$$$$22\!\cdots\!56$$$$p^{16} T^{16} -$$$$11\!\cdots\!84$$$$p^{20} T^{17} + 10320605504003112056 p^{24} T^{18} - 4706417114238944 p^{28} T^{19} + 1626992896581 p^{32} T^{20} - 4475893024 p^{36} T^{21} + 72946 p^{40} T^{22} + 336 p^{44} T^{23} + p^{48} T^{24}$$
41 $$1 + 4550 T + 14401928 T^{2} + 23922427770 T^{3} + 36931794819271 T^{4} + 40163345738311410 T^{5} +$$$$12\!\cdots\!20$$$$T^{6} +$$$$30\!\cdots\!20$$$$T^{7} +$$$$86\!\cdots\!60$$$$T^{8} +$$$$14\!\cdots\!70$$$$T^{9} +$$$$24\!\cdots\!68$$$$T^{10} +$$$$25\!\cdots\!80$$$$T^{11} +$$$$43\!\cdots\!24$$$$T^{12} +$$$$25\!\cdots\!80$$$$p^{4} T^{13} +$$$$24\!\cdots\!68$$$$p^{8} T^{14} +$$$$14\!\cdots\!70$$$$p^{12} T^{15} +$$$$86\!\cdots\!60$$$$p^{16} T^{16} +$$$$30\!\cdots\!20$$$$p^{20} T^{17} +$$$$12\!\cdots\!20$$$$p^{24} T^{18} + 40163345738311410 p^{28} T^{19} + 36931794819271 p^{32} T^{20} + 23922427770 p^{36} T^{21} + 14401928 p^{40} T^{22} + 4550 p^{44} T^{23} + p^{48} T^{24}$$
43 $$1 - 23356307 T^{2} + 286833712371781 T^{4} -$$$$23\!\cdots\!15$$$$T^{6} +$$$$14\!\cdots\!15$$$$T^{8} -$$$$69\!\cdots\!82$$$$T^{10} +$$$$26\!\cdots\!14$$$$T^{12} -$$$$69\!\cdots\!82$$$$p^{8} T^{14} +$$$$14\!\cdots\!15$$$$p^{16} T^{16} -$$$$23\!\cdots\!15$$$$p^{24} T^{18} + 286833712371781 p^{32} T^{20} - 23356307 p^{40} T^{22} + p^{48} T^{24}$$
47 $$1 - 24 T - 10068774 T^{2} - 8140867214 T^{3} + 53779050778611 T^{4} + 233185473449483776 T^{5} - 81521520124710716194 T^{6} -$$$$19\!\cdots\!44$$$$T^{7} -$$$$12\!\cdots\!74$$$$T^{8} +$$$$86\!\cdots\!36$$$$T^{9} +$$$$22\!\cdots\!96$$$$T^{10} -$$$$14\!\cdots\!34$$$$T^{11} -$$$$15\!\cdots\!64$$$$T^{12} -$$$$14\!\cdots\!34$$$$p^{4} T^{13} +$$$$22\!\cdots\!96$$$$p^{8} T^{14} +$$$$86\!\cdots\!36$$$$p^{12} T^{15} -$$$$12\!\cdots\!74$$$$p^{16} T^{16} -$$$$19\!\cdots\!44$$$$p^{20} T^{17} - 81521520124710716194 p^{24} T^{18} + 233185473449483776 p^{28} T^{19} + 53779050778611 p^{32} T^{20} - 8140867214 p^{36} T^{21} - 10068774 p^{40} T^{22} - 24 p^{44} T^{23} + p^{48} T^{24}$$
53 $$1 - 414 T - 13725134 T^{2} - 7014732824 T^{3} + 165958055930181 T^{4} + 208417442032136246 T^{5} -$$$$14\!\cdots\!44$$$$T^{6} -$$$$22\!\cdots\!24$$$$T^{7} +$$$$12\!\cdots\!16$$$$T^{8} +$$$$14\!\cdots\!66$$$$T^{9} -$$$$14\!\cdots\!58$$$$p T^{10} -$$$$33\!\cdots\!74$$$$T^{11} +$$$$54\!\cdots\!16$$$$T^{12} -$$$$33\!\cdots\!74$$$$p^{4} T^{13} -$$$$14\!\cdots\!58$$$$p^{9} T^{14} +$$$$14\!\cdots\!66$$$$p^{12} T^{15} +$$$$12\!\cdots\!16$$$$p^{16} T^{16} -$$$$22\!\cdots\!24$$$$p^{20} T^{17} -$$$$14\!\cdots\!44$$$$p^{24} T^{18} + 208417442032136246 p^{28} T^{19} + 165958055930181 p^{32} T^{20} - 7014732824 p^{36} T^{21} - 13725134 p^{40} T^{22} - 414 p^{44} T^{23} + p^{48} T^{24}$$
59 $$1 + 10011 T + 24606746 T^{2} - 48335046044 T^{3} - 85055058906234 T^{4} + 1848432652355091201 T^{5} +$$$$68\!\cdots\!21$$$$T^{6} -$$$$10\!\cdots\!34$$$$T^{7} -$$$$82\!\cdots\!39$$$$T^{8} +$$$$15\!\cdots\!31$$$$T^{9} +$$$$13\!\cdots\!41$$$$T^{10} -$$$$16\!\cdots\!29$$$$T^{11} -$$$$23\!\cdots\!24$$$$T^{12} -$$$$16\!\cdots\!29$$$$p^{4} T^{13} +$$$$13\!\cdots\!41$$$$p^{8} T^{14} +$$$$15\!\cdots\!31$$$$p^{12} T^{15} -$$$$82\!\cdots\!39$$$$p^{16} T^{16} -$$$$10\!\cdots\!34$$$$p^{20} T^{17} +$$$$68\!\cdots\!21$$$$p^{24} T^{18} + 1848432652355091201 p^{28} T^{19} - 85055058906234 p^{32} T^{20} - 48335046044 p^{36} T^{21} + 24606746 p^{40} T^{22} + 10011 p^{44} T^{23} + p^{48} T^{24}$$
61 $$1 - 9460 T + 57744648 T^{2} - 309062215210 T^{3} + 1619001804085751 T^{4} - 7693654169306140700 T^{5} +$$$$35\!\cdots\!40$$$$T^{6} -$$$$16\!\cdots\!00$$$$T^{7} +$$$$77\!\cdots\!60$$$$T^{8} -$$$$32\!\cdots\!20$$$$T^{9} +$$$$12\!\cdots\!28$$$$T^{10} -$$$$48\!\cdots\!10$$$$T^{11} +$$$$18\!\cdots\!84$$$$T^{12} -$$$$48\!\cdots\!10$$$$p^{4} T^{13} +$$$$12\!\cdots\!28$$$$p^{8} T^{14} -$$$$32\!\cdots\!20$$$$p^{12} T^{15} +$$$$77\!\cdots\!60$$$$p^{16} T^{16} -$$$$16\!\cdots\!00$$$$p^{20} T^{17} +$$$$35\!\cdots\!40$$$$p^{24} T^{18} - 7693654169306140700 p^{28} T^{19} + 1619001804085751 p^{32} T^{20} - 309062215210 p^{36} T^{21} + 57744648 p^{40} T^{22} - 9460 p^{44} T^{23} + p^{48} T^{24}$$
67 $$( 1 - 6077 T + 35718991 T^{2} - 18219600105 T^{3} + 122534918608155 T^{4} + 2701855134548306278 T^{5} -$$$$61\!\cdots\!46$$$$T^{6} + 2701855134548306278 p^{4} T^{7} + 122534918608155 p^{8} T^{8} - 18219600105 p^{12} T^{9} + 35718991 p^{16} T^{10} - 6077 p^{20} T^{11} + p^{24} T^{12} )^{2}$$
71 $$1 - 17574 T + 104286776 T^{2} - 424728504824 T^{3} + 5029513991381961 T^{4} - 44618257451712179514 T^{5} +$$$$20\!\cdots\!06$$$$T^{6} -$$$$10\!\cdots\!44$$$$T^{7} +$$$$80\!\cdots\!46$$$$T^{8} -$$$$47\!\cdots\!54$$$$T^{9} +$$$$20\!\cdots\!26$$$$T^{10} -$$$$11\!\cdots\!94$$$$T^{11} +$$$$68\!\cdots\!36$$$$T^{12} -$$$$11\!\cdots\!94$$$$p^{4} T^{13} +$$$$20\!\cdots\!26$$$$p^{8} T^{14} -$$$$47\!\cdots\!54$$$$p^{12} T^{15} +$$$$80\!\cdots\!46$$$$p^{16} T^{16} -$$$$10\!\cdots\!44$$$$p^{20} T^{17} +$$$$20\!\cdots\!06$$$$p^{24} T^{18} - 44618257451712179514 p^{28} T^{19} + 5029513991381961 p^{32} T^{20} - 424728504824 p^{36} T^{21} + 104286776 p^{40} T^{22} - 17574 p^{44} T^{23} + p^{48} T^{24}$$
73 $$1 - 27950 T + 454822888 T^{2} - 5337429236470 T^{3} + 50136942858006191 T^{4} -$$$$39\!\cdots\!10$$$$T^{5} +$$$$27\!\cdots\!20$$$$T^{6} -$$$$17\!\cdots\!80$$$$T^{7} +$$$$99\!\cdots\!60$$$$T^{8} -$$$$54\!\cdots\!10$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} -$$$$15\!\cdots\!80$$$$T^{11} +$$$$84\!\cdots\!04$$$$T^{12} -$$$$15\!\cdots\!80$$$$p^{4} T^{13} +$$$$29\!\cdots\!08$$$$p^{8} T^{14} -$$$$54\!\cdots\!10$$$$p^{12} T^{15} +$$$$99\!\cdots\!60$$$$p^{16} T^{16} -$$$$17\!\cdots\!80$$$$p^{20} T^{17} +$$$$27\!\cdots\!20$$$$p^{24} T^{18} -$$$$39\!\cdots\!10$$$$p^{28} T^{19} + 50136942858006191 p^{32} T^{20} - 5337429236470 p^{36} T^{21} + 454822888 p^{40} T^{22} - 27950 p^{44} T^{23} + p^{48} T^{24}$$
79 $$1 + 41540 T + 884578158 T^{2} + 13528269571760 T^{3} + 171031570391119991 T^{4} +$$$$18\!\cdots\!60$$$$T^{5} +$$$$18\!\cdots\!10$$$$T^{6} +$$$$17\!\cdots\!80$$$$T^{7} +$$$$14\!\cdots\!70$$$$T^{8} +$$$$11\!\cdots\!40$$$$T^{9} +$$$$82\!\cdots\!08$$$$T^{10} +$$$$56\!\cdots\!20$$$$T^{11} +$$$$36\!\cdots\!24$$$$T^{12} +$$$$56\!\cdots\!20$$$$p^{4} T^{13} +$$$$82\!\cdots\!08$$$$p^{8} T^{14} +$$$$11\!\cdots\!40$$$$p^{12} T^{15} +$$$$14\!\cdots\!70$$$$p^{16} T^{16} +$$$$17\!\cdots\!80$$$$p^{20} T^{17} +$$$$18\!\cdots\!10$$$$p^{24} T^{18} +$$$$18\!\cdots\!60$$$$p^{28} T^{19} + 171031570391119991 p^{32} T^{20} + 13528269571760 p^{36} T^{21} + 884578158 p^{40} T^{22} + 41540 p^{44} T^{23} + p^{48} T^{24}$$
83 $$1 + 18665 T + 224643628 T^{2} + 792700664890 T^{3} - 7907731388500564 T^{4} -$$$$17\!\cdots\!75$$$$T^{5} -$$$$12\!\cdots\!65$$$$T^{6} -$$$$28\!\cdots\!70$$$$T^{7} +$$$$49\!\cdots\!05$$$$T^{8} +$$$$50\!\cdots\!25$$$$T^{9} +$$$$19\!\cdots\!53$$$$T^{10} -$$$$10\!\cdots\!35$$$$T^{11} -$$$$13\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!35$$$$p^{4} T^{13} +$$$$19\!\cdots\!53$$$$p^{8} T^{14} +$$$$50\!\cdots\!25$$$$p^{12} T^{15} +$$$$49\!\cdots\!05$$$$p^{16} T^{16} -$$$$28\!\cdots\!70$$$$p^{20} T^{17} -$$$$12\!\cdots\!65$$$$p^{24} T^{18} -$$$$17\!\cdots\!75$$$$p^{28} T^{19} - 7907731388500564 p^{32} T^{20} + 792700664890 p^{36} T^{21} + 224643628 p^{40} T^{22} + 18665 p^{44} T^{23} + p^{48} T^{24}$$
89 $$( 1 - 2777 T + 289401541 T^{2} - 984806913445 T^{3} + 37686355645869335 T^{4} -$$$$12\!\cdots\!62$$$$T^{5} +$$$$29\!\cdots\!14$$$$T^{6} -$$$$12\!\cdots\!62$$$$p^{4} T^{7} + 37686355645869335 p^{8} T^{8} - 984806913445 p^{12} T^{9} + 289401541 p^{16} T^{10} - 2777 p^{20} T^{11} + p^{24} T^{12} )^{2}$$
97 $$1 - 20769 T + 424679856 T^{2} - 8132949547004 T^{3} + 119819968442578026 T^{4} -$$$$16\!\cdots\!09$$$$T^{5} +$$$$21\!\cdots\!91$$$$T^{6} -$$$$25\!\cdots\!44$$$$T^{7} +$$$$28\!\cdots\!01$$$$T^{8} -$$$$31\!\cdots\!79$$$$T^{9} +$$$$31\!\cdots\!81$$$$T^{10} -$$$$31\!\cdots\!59$$$$T^{11} +$$$$30\!\cdots\!76$$$$T^{12} -$$$$31\!\cdots\!59$$$$p^{4} T^{13} +$$$$31\!\cdots\!81$$$$p^{8} T^{14} -$$$$31\!\cdots\!79$$$$p^{12} T^{15} +$$$$28\!\cdots\!01$$$$p^{16} T^{16} -$$$$25\!\cdots\!44$$$$p^{20} T^{17} +$$$$21\!\cdots\!91$$$$p^{24} T^{18} -$$$$16\!\cdots\!09$$$$p^{28} T^{19} + 119819968442578026 p^{32} T^{20} - 8132949547004 p^{36} T^{21} + 424679856 p^{40} T^{22} - 20769 p^{44} T^{23} + p^{48} T^{24}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}