# Properties

 Degree 56 Conductor $2^{56} \cdot 5^{28} \cdot 401^{28}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 28

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·3-s − 28·5-s − 4·7-s − 29·9-s + 2·11-s + 3·13-s − 84·15-s − 10·17-s − 2·19-s − 12·21-s − 23·23-s + 406·25-s − 96·27-s − 37·29-s − 11·31-s + 6·33-s + 112·35-s − 3·37-s + 9·39-s − 30·41-s + 13·43-s + 812·45-s − 15·47-s − 84·49-s − 30·51-s − 35·53-s − 56·55-s + ⋯
 L(s)  = 1 + 1.73·3-s − 12.5·5-s − 1.51·7-s − 9.66·9-s + 0.603·11-s + 0.832·13-s − 21.6·15-s − 2.42·17-s − 0.458·19-s − 2.61·21-s − 4.79·23-s + 81.1·25-s − 18.4·27-s − 6.87·29-s − 1.97·31-s + 1.04·33-s + 18.9·35-s − 0.493·37-s + 1.44·39-s − 4.68·41-s + 1.98·43-s + 121.·45-s − 2.18·47-s − 12·49-s − 4.20·51-s − 4.80·53-s − 7.55·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{28} \cdot 401^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{28} \cdot 401^{28}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$56$$ $$N$$ = $$2^{56} \cdot 5^{28} \cdot 401^{28}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{8020} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 28 Selberg data = $(56,\ 2^{56} \cdot 5^{28} \cdot 401^{28} ,\ ( \ : [1/2]^{28} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;401\}$, $$F_p$$ is a polynomial of degree 56. If $p \in \{2,\;5,\;401\}$, then $F_p$ is a polynomial of degree at most 55.
$p$$F_p$
bad2 $$1$$
5 $$( 1 + T )^{28}$$
401 $$( 1 - T )^{28}$$
good3 $$1 - p T + 38 T^{2} - 35 p T^{3} + 737 T^{4} - 1903 T^{5} + 9710 T^{6} - 23677 T^{7} + 97565 T^{8} - 226495 T^{9} + 265325 p T^{10} - 1770529 T^{11} + 5483035 T^{12} - 11745319 T^{13} + 3638966 p^{2} T^{14} - 67829053 T^{15} + 172922746 T^{16} - 115755263 p T^{17} + 818875408 T^{18} - 1597574602 T^{19} + 3517265206 T^{20} - 6671663828 T^{21} + 4608028253 p T^{22} - 25486156643 T^{23} + 50054373656 T^{24} - 89552659477 T^{25} + 18641510120 p^{2} T^{26} - 290492084273 T^{27} + 522102242732 T^{28} - 290492084273 p T^{29} + 18641510120 p^{4} T^{30} - 89552659477 p^{3} T^{31} + 50054373656 p^{4} T^{32} - 25486156643 p^{5} T^{33} + 4608028253 p^{7} T^{34} - 6671663828 p^{7} T^{35} + 3517265206 p^{8} T^{36} - 1597574602 p^{9} T^{37} + 818875408 p^{10} T^{38} - 115755263 p^{12} T^{39} + 172922746 p^{12} T^{40} - 67829053 p^{13} T^{41} + 3638966 p^{16} T^{42} - 11745319 p^{15} T^{43} + 5483035 p^{16} T^{44} - 1770529 p^{17} T^{45} + 265325 p^{19} T^{46} - 226495 p^{19} T^{47} + 97565 p^{20} T^{48} - 23677 p^{21} T^{49} + 9710 p^{22} T^{50} - 1903 p^{23} T^{51} + 737 p^{24} T^{52} - 35 p^{26} T^{53} + 38 p^{26} T^{54} - p^{28} T^{55} + p^{28} T^{56}$$
7 $$1 + 4 T + 100 T^{2} + 374 T^{3} + 5058 T^{4} + 17676 T^{5} + 172135 T^{6} + 563742 T^{7} + 4430280 T^{8} + 13654399 T^{9} + 91931127 T^{10} + 267782054 T^{11} + 1600892869 T^{12} + 4423817631 T^{13} + 24034635777 T^{14} + 63204479027 T^{15} + 317041570729 T^{16} + 113628156136 p T^{17} + 3725119469593 T^{18} + 8932939801474 T^{19} + 39378620776536 T^{20} + 90385435138888 T^{21} + 53896408986127 p T^{22} + 829577017524448 T^{23} + 3293092867200243 T^{24} + 991341290685365 p T^{25} + 26279981315466325 T^{26} + 53062580543521874 T^{27} + 192132782716477500 T^{28} + 53062580543521874 p T^{29} + 26279981315466325 p^{2} T^{30} + 991341290685365 p^{4} T^{31} + 3293092867200243 p^{4} T^{32} + 829577017524448 p^{5} T^{33} + 53896408986127 p^{7} T^{34} + 90385435138888 p^{7} T^{35} + 39378620776536 p^{8} T^{36} + 8932939801474 p^{9} T^{37} + 3725119469593 p^{10} T^{38} + 113628156136 p^{12} T^{39} + 317041570729 p^{12} T^{40} + 63204479027 p^{13} T^{41} + 24034635777 p^{14} T^{42} + 4423817631 p^{15} T^{43} + 1600892869 p^{16} T^{44} + 267782054 p^{17} T^{45} + 91931127 p^{18} T^{46} + 13654399 p^{19} T^{47} + 4430280 p^{20} T^{48} + 563742 p^{21} T^{49} + 172135 p^{22} T^{50} + 17676 p^{23} T^{51} + 5058 p^{24} T^{52} + 374 p^{25} T^{53} + 100 p^{26} T^{54} + 4 p^{27} T^{55} + p^{28} T^{56}$$
11 $$1 - 2 T + 15 p T^{2} - 314 T^{3} + 13788 T^{4} - 25100 T^{5} + 776069 T^{6} - 1358043 T^{7} + 33024843 T^{8} - 55788688 T^{9} + 1130844351 T^{10} - 1850612209 T^{11} + 32387343265 T^{12} - 51476311299 T^{13} + 796224873681 T^{14} - 1230972326097 T^{15} + 17113731396126 T^{16} - 25746594974831 T^{17} + 325905094038990 T^{18} - 476787942776581 T^{19} + 5553197203200737 T^{20} - 7886034701019231 T^{21} + 85278421922336869 T^{22} - 117222511511158575 T^{23} + 1186401899787746267 T^{24} - 1572621859478728385 T^{25} + 15005141512860999779 T^{26} - 19091944867364739257 T^{27} +$$$$17\!\cdots\!50$$$$T^{28} - 19091944867364739257 p T^{29} + 15005141512860999779 p^{2} T^{30} - 1572621859478728385 p^{3} T^{31} + 1186401899787746267 p^{4} T^{32} - 117222511511158575 p^{5} T^{33} + 85278421922336869 p^{6} T^{34} - 7886034701019231 p^{7} T^{35} + 5553197203200737 p^{8} T^{36} - 476787942776581 p^{9} T^{37} + 325905094038990 p^{10} T^{38} - 25746594974831 p^{11} T^{39} + 17113731396126 p^{12} T^{40} - 1230972326097 p^{13} T^{41} + 796224873681 p^{14} T^{42} - 51476311299 p^{15} T^{43} + 32387343265 p^{16} T^{44} - 1850612209 p^{17} T^{45} + 1130844351 p^{18} T^{46} - 55788688 p^{19} T^{47} + 33024843 p^{20} T^{48} - 1358043 p^{21} T^{49} + 776069 p^{22} T^{50} - 25100 p^{23} T^{51} + 13788 p^{24} T^{52} - 314 p^{25} T^{53} + 15 p^{27} T^{54} - 2 p^{27} T^{55} + p^{28} T^{56}$$
13 $$1 - 3 T + 175 T^{2} - 651 T^{3} + 15371 T^{4} - 65376 T^{5} + 909481 T^{6} - 4153067 T^{7} + 40742297 T^{8} - 190609285 T^{9} + 1464841083 T^{10} - 6801389493 T^{11} + 43686057034 T^{12} - 197404541185 T^{13} + 1103873303451 T^{14} - 4801663153033 T^{15} + 24007316675643 T^{16} - 7693213449330 p T^{17} + 455498064599778 T^{18} - 1816028446543747 T^{19} + 7646215924496207 T^{20} - 29277990688260554 T^{21} + 115587387511849489 T^{22} - 428535393777289949 T^{23} + 1612516862857119994 T^{24} - 5859365166062638050 T^{25} + 21421325174076595635 T^{26} - 77240691694187941173 T^{27} +$$$$27\!\cdots\!98$$$$T^{28} - 77240691694187941173 p T^{29} + 21421325174076595635 p^{2} T^{30} - 5859365166062638050 p^{3} T^{31} + 1612516862857119994 p^{4} T^{32} - 428535393777289949 p^{5} T^{33} + 115587387511849489 p^{6} T^{34} - 29277990688260554 p^{7} T^{35} + 7646215924496207 p^{8} T^{36} - 1816028446543747 p^{9} T^{37} + 455498064599778 p^{10} T^{38} - 7693213449330 p^{12} T^{39} + 24007316675643 p^{12} T^{40} - 4801663153033 p^{13} T^{41} + 1103873303451 p^{14} T^{42} - 197404541185 p^{15} T^{43} + 43686057034 p^{16} T^{44} - 6801389493 p^{17} T^{45} + 1464841083 p^{18} T^{46} - 190609285 p^{19} T^{47} + 40742297 p^{20} T^{48} - 4153067 p^{21} T^{49} + 909481 p^{22} T^{50} - 65376 p^{23} T^{51} + 15371 p^{24} T^{52} - 651 p^{25} T^{53} + 175 p^{26} T^{54} - 3 p^{27} T^{55} + p^{28} T^{56}$$
17 $$1 + 10 T + 329 T^{2} + 3049 T^{3} + 53361 T^{4} + 453448 T^{5} + 5640976 T^{6} + 43775763 T^{7} + 434302530 T^{8} + 3077926180 T^{9} + 25837092138 T^{10} + 167575524368 T^{11} + 1231908185277 T^{12} + 7333168371216 T^{13} + 48255291825739 T^{14} + 264419486700183 T^{15} + 1581411889077322 T^{16} + 8000744775791735 T^{17} + 44015285290264483 T^{18} + 206335921380264445 T^{19} + 1056150217849473591 T^{20} + 4612635345539456884 T^{21} + 22258662871700008629 T^{22} + 91433740355543844189 T^{23} +$$$$42\!\cdots\!26$$$$T^{24} +$$$$16\!\cdots\!85$$$$T^{25} +$$$$75\!\cdots\!34$$$$T^{26} +$$$$28\!\cdots\!69$$$$T^{27} +$$$$12\!\cdots\!16$$$$T^{28} +$$$$28\!\cdots\!69$$$$p T^{29} +$$$$75\!\cdots\!34$$$$p^{2} T^{30} +$$$$16\!\cdots\!85$$$$p^{3} T^{31} +$$$$42\!\cdots\!26$$$$p^{4} T^{32} + 91433740355543844189 p^{5} T^{33} + 22258662871700008629 p^{6} T^{34} + 4612635345539456884 p^{7} T^{35} + 1056150217849473591 p^{8} T^{36} + 206335921380264445 p^{9} T^{37} + 44015285290264483 p^{10} T^{38} + 8000744775791735 p^{11} T^{39} + 1581411889077322 p^{12} T^{40} + 264419486700183 p^{13} T^{41} + 48255291825739 p^{14} T^{42} + 7333168371216 p^{15} T^{43} + 1231908185277 p^{16} T^{44} + 167575524368 p^{17} T^{45} + 25837092138 p^{18} T^{46} + 3077926180 p^{19} T^{47} + 434302530 p^{20} T^{48} + 43775763 p^{21} T^{49} + 5640976 p^{22} T^{50} + 453448 p^{23} T^{51} + 53361 p^{24} T^{52} + 3049 p^{25} T^{53} + 329 p^{26} T^{54} + 10 p^{27} T^{55} + p^{28} T^{56}$$
19 $$1 + 2 T + 319 T^{2} + 696 T^{3} + 50410 T^{4} + 120221 T^{5} + 5258442 T^{6} + 719396 p T^{7} + 407225302 T^{8} + 60339360 p T^{9} + 1314395516 p T^{10} + 3972308202 p T^{11} + 1263671743892 T^{12} + 4055478124164 T^{13} + 54289876501995 T^{14} + 182754482348572 T^{15} + 2022470618868320 T^{16} + 7046562918159143 T^{17} + 66386710262653507 T^{18} + 236098171003727549 T^{19} + 1943819870950422725 T^{20} + 6958857184563344228 T^{21} + 51245388589551809331 T^{22} +$$$$18\!\cdots\!05$$$$T^{23} +$$$$12\!\cdots\!63$$$$T^{24} +$$$$42\!\cdots\!25$$$$T^{25} +$$$$26\!\cdots\!34$$$$T^{26} +$$$$89\!\cdots\!23$$$$T^{27} +$$$$52\!\cdots\!38$$$$T^{28} +$$$$89\!\cdots\!23$$$$p T^{29} +$$$$26\!\cdots\!34$$$$p^{2} T^{30} +$$$$42\!\cdots\!25$$$$p^{3} T^{31} +$$$$12\!\cdots\!63$$$$p^{4} T^{32} +$$$$18\!\cdots\!05$$$$p^{5} T^{33} + 51245388589551809331 p^{6} T^{34} + 6958857184563344228 p^{7} T^{35} + 1943819870950422725 p^{8} T^{36} + 236098171003727549 p^{9} T^{37} + 66386710262653507 p^{10} T^{38} + 7046562918159143 p^{11} T^{39} + 2022470618868320 p^{12} T^{40} + 182754482348572 p^{13} T^{41} + 54289876501995 p^{14} T^{42} + 4055478124164 p^{15} T^{43} + 1263671743892 p^{16} T^{44} + 3972308202 p^{18} T^{45} + 1314395516 p^{19} T^{46} + 60339360 p^{20} T^{47} + 407225302 p^{20} T^{48} + 719396 p^{22} T^{49} + 5258442 p^{22} T^{50} + 120221 p^{23} T^{51} + 50410 p^{24} T^{52} + 696 p^{25} T^{53} + 319 p^{26} T^{54} + 2 p^{27} T^{55} + p^{28} T^{56}$$
23 $$1 + p T + 623 T^{2} + 10165 T^{3} + 166623 T^{4} + 2151787 T^{5} + 26927612 T^{6} + 291466846 T^{7} + 3030758778 T^{8} + 28472273569 T^{9} + 256833999140 T^{10} + 93213363215 p T^{11} + 17222269421189 T^{12} + 129946408243192 T^{13} + 946560789535518 T^{14} + 6542688426508581 T^{15} + 43812268857414451 T^{16} + 280523146754735336 T^{17} + 1746374331962341007 T^{18} + 10458700819648257380 T^{19} + 61116448648481092050 T^{20} +$$$$34\!\cdots\!74$$$$T^{21} +$$$$19\!\cdots\!50$$$$T^{22} +$$$$10\!\cdots\!51$$$$T^{23} +$$$$54\!\cdots\!19$$$$T^{24} +$$$$27\!\cdots\!59$$$$T^{25} +$$$$14\!\cdots\!14$$$$T^{26} +$$$$69\!\cdots\!48$$$$T^{27} +$$$$33\!\cdots\!06$$$$T^{28} +$$$$69\!\cdots\!48$$$$p T^{29} +$$$$14\!\cdots\!14$$$$p^{2} T^{30} +$$$$27\!\cdots\!59$$$$p^{3} T^{31} +$$$$54\!\cdots\!19$$$$p^{4} T^{32} +$$$$10\!\cdots\!51$$$$p^{5} T^{33} +$$$$19\!\cdots\!50$$$$p^{6} T^{34} +$$$$34\!\cdots\!74$$$$p^{7} T^{35} + 61116448648481092050 p^{8} T^{36} + 10458700819648257380 p^{9} T^{37} + 1746374331962341007 p^{10} T^{38} + 280523146754735336 p^{11} T^{39} + 43812268857414451 p^{12} T^{40} + 6542688426508581 p^{13} T^{41} + 946560789535518 p^{14} T^{42} + 129946408243192 p^{15} T^{43} + 17222269421189 p^{16} T^{44} + 93213363215 p^{18} T^{45} + 256833999140 p^{18} T^{46} + 28472273569 p^{19} T^{47} + 3030758778 p^{20} T^{48} + 291466846 p^{21} T^{49} + 26927612 p^{22} T^{50} + 2151787 p^{23} T^{51} + 166623 p^{24} T^{52} + 10165 p^{25} T^{53} + 623 p^{26} T^{54} + p^{28} T^{55} + p^{28} T^{56}$$
29 $$1 + 37 T + 1104 T^{2} + 23349 T^{3} + 430246 T^{4} + 6694325 T^{5} + 3251767 p T^{6} + 1189567334 T^{7} + 13894583204 T^{8} + 149567731203 T^{9} + 1512869414779 T^{10} + 14344837861508 T^{11} + 129143757218966 T^{12} + 1102322319034896 T^{13} + 9003251652643796 T^{14} + 70290169572495558 T^{15} + 528298122210964745 T^{16} + 3819208786256296018 T^{17} + 26711515324097790131 T^{18} +$$$$18\!\cdots\!40$$$$T^{19} +$$$$11\!\cdots\!84$$$$T^{20} +$$$$75\!\cdots\!91$$$$T^{21} +$$$$47\!\cdots\!08$$$$T^{22} +$$$$28\!\cdots\!94$$$$T^{23} +$$$$16\!\cdots\!61$$$$T^{24} +$$$$97\!\cdots\!65$$$$T^{25} +$$$$55\!\cdots\!85$$$$T^{26} +$$$$30\!\cdots\!62$$$$T^{27} +$$$$16\!\cdots\!02$$$$T^{28} +$$$$30\!\cdots\!62$$$$p T^{29} +$$$$55\!\cdots\!85$$$$p^{2} T^{30} +$$$$97\!\cdots\!65$$$$p^{3} T^{31} +$$$$16\!\cdots\!61$$$$p^{4} T^{32} +$$$$28\!\cdots\!94$$$$p^{5} T^{33} +$$$$47\!\cdots\!08$$$$p^{6} T^{34} +$$$$75\!\cdots\!91$$$$p^{7} T^{35} +$$$$11\!\cdots\!84$$$$p^{8} T^{36} +$$$$18\!\cdots\!40$$$$p^{9} T^{37} + 26711515324097790131 p^{10} T^{38} + 3819208786256296018 p^{11} T^{39} + 528298122210964745 p^{12} T^{40} + 70290169572495558 p^{13} T^{41} + 9003251652643796 p^{14} T^{42} + 1102322319034896 p^{15} T^{43} + 129143757218966 p^{16} T^{44} + 14344837861508 p^{17} T^{45} + 1512869414779 p^{18} T^{46} + 149567731203 p^{19} T^{47} + 13894583204 p^{20} T^{48} + 1189567334 p^{21} T^{49} + 3251767 p^{23} T^{50} + 6694325 p^{23} T^{51} + 430246 p^{24} T^{52} + 23349 p^{25} T^{53} + 1104 p^{26} T^{54} + 37 p^{27} T^{55} + p^{28} T^{56}$$
31 $$1 + 11 T + 544 T^{2} + 4959 T^{3} + 138494 T^{4} + 1082203 T^{5} + 723159 p T^{6} + 154371357 T^{7} + 2633676626 T^{8} + 16403579555 T^{9} + 242246420024 T^{10} + 1401331456981 T^{11} + 18321585351527 T^{12} + 101122237115267 T^{13} + 1178216083581342 T^{14} + 205354929609304 p T^{15} + 65968741037199475 T^{16} + 356656589350173246 T^{17} + 3272866412331013458 T^{18} + 17983423747913650860 T^{19} +$$$$14\!\cdots\!79$$$$T^{20} +$$$$82\!\cdots\!92$$$$T^{21} +$$$$59\!\cdots\!42$$$$T^{22} +$$$$33\!\cdots\!29$$$$T^{23} +$$$$22\!\cdots\!18$$$$T^{24} +$$$$12\!\cdots\!91$$$$T^{25} +$$$$75\!\cdots\!61$$$$T^{26} +$$$$43\!\cdots\!87$$$$T^{27} +$$$$24\!\cdots\!88$$$$T^{28} +$$$$43\!\cdots\!87$$$$p T^{29} +$$$$75\!\cdots\!61$$$$p^{2} T^{30} +$$$$12\!\cdots\!91$$$$p^{3} T^{31} +$$$$22\!\cdots\!18$$$$p^{4} T^{32} +$$$$33\!\cdots\!29$$$$p^{5} T^{33} +$$$$59\!\cdots\!42$$$$p^{6} T^{34} +$$$$82\!\cdots\!92$$$$p^{7} T^{35} +$$$$14\!\cdots\!79$$$$p^{8} T^{36} + 17983423747913650860 p^{9} T^{37} + 3272866412331013458 p^{10} T^{38} + 356656589350173246 p^{11} T^{39} + 65968741037199475 p^{12} T^{40} + 205354929609304 p^{14} T^{41} + 1178216083581342 p^{14} T^{42} + 101122237115267 p^{15} T^{43} + 18321585351527 p^{16} T^{44} + 1401331456981 p^{17} T^{45} + 242246420024 p^{18} T^{46} + 16403579555 p^{19} T^{47} + 2633676626 p^{20} T^{48} + 154371357 p^{21} T^{49} + 723159 p^{23} T^{50} + 1082203 p^{23} T^{51} + 138494 p^{24} T^{52} + 4959 p^{25} T^{53} + 544 p^{26} T^{54} + 11 p^{27} T^{55} + p^{28} T^{56}$$
37 $$1 + 3 T + 552 T^{2} + 1461 T^{3} + 151121 T^{4} + 355318 T^{5} + 27301402 T^{6} + 57607330 T^{7} + 3654822442 T^{8} + 7031835250 T^{9} + 386104245658 T^{10} + 693521521213 T^{11} + 33486988615817 T^{12} + 57973198457574 T^{13} + 2450636395850445 T^{14} + 4248977833932948 T^{15} + 154491492845220139 T^{16} + 279249763661748835 T^{17} + 8536523535794689833 T^{18} + 16660803250731554100 T^{19} +$$$$42\!\cdots\!05$$$$T^{20} +$$$$90\!\cdots\!41$$$$T^{21} +$$$$18\!\cdots\!04$$$$T^{22} +$$$$44\!\cdots\!08$$$$T^{23} +$$$$77\!\cdots\!90$$$$T^{24} +$$$$20\!\cdots\!11$$$$T^{25} +$$$$30\!\cdots\!14$$$$T^{26} +$$$$82\!\cdots\!04$$$$T^{27} +$$$$11\!\cdots\!98$$$$T^{28} +$$$$82\!\cdots\!04$$$$p T^{29} +$$$$30\!\cdots\!14$$$$p^{2} T^{30} +$$$$20\!\cdots\!11$$$$p^{3} T^{31} +$$$$77\!\cdots\!90$$$$p^{4} T^{32} +$$$$44\!\cdots\!08$$$$p^{5} T^{33} +$$$$18\!\cdots\!04$$$$p^{6} T^{34} +$$$$90\!\cdots\!41$$$$p^{7} T^{35} +$$$$42\!\cdots\!05$$$$p^{8} T^{36} + 16660803250731554100 p^{9} T^{37} + 8536523535794689833 p^{10} T^{38} + 279249763661748835 p^{11} T^{39} + 154491492845220139 p^{12} T^{40} + 4248977833932948 p^{13} T^{41} + 2450636395850445 p^{14} T^{42} + 57973198457574 p^{15} T^{43} + 33486988615817 p^{16} T^{44} + 693521521213 p^{17} T^{45} + 386104245658 p^{18} T^{46} + 7031835250 p^{19} T^{47} + 3654822442 p^{20} T^{48} + 57607330 p^{21} T^{49} + 27301402 p^{22} T^{50} + 355318 p^{23} T^{51} + 151121 p^{24} T^{52} + 1461 p^{25} T^{53} + 552 p^{26} T^{54} + 3 p^{27} T^{55} + p^{28} T^{56}$$
41 $$1 + 30 T + 1021 T^{2} + 20991 T^{3} + 433066 T^{4} + 6975284 T^{5} + 2682152 p T^{6} + 1483542378 T^{7} + 19500776410 T^{8} + 229448669789 T^{9} + 2633637428015 T^{10} + 677467365860 p T^{11} + 286579933315658 T^{12} + 2762922794543536 T^{13} + 26136504865850762 T^{14} + 233695672186813090 T^{15} + 2055693463910342342 T^{16} + 17230012901170637489 T^{17} +$$$$14\!\cdots\!68$$$$T^{18} +$$$$11\!\cdots\!13$$$$T^{19} +$$$$88\!\cdots\!61$$$$T^{20} +$$$$66\!\cdots\!04$$$$T^{21} +$$$$49\!\cdots\!25$$$$T^{22} +$$$$35\!\cdots\!96$$$$T^{23} +$$$$25\!\cdots\!41$$$$T^{24} +$$$$17\!\cdots\!89$$$$T^{25} +$$$$11\!\cdots\!95$$$$T^{26} +$$$$77\!\cdots\!07$$$$T^{27} +$$$$50\!\cdots\!42$$$$T^{28} +$$$$77\!\cdots\!07$$$$p T^{29} +$$$$11\!\cdots\!95$$$$p^{2} T^{30} +$$$$17\!\cdots\!89$$$$p^{3} T^{31} +$$$$25\!\cdots\!41$$$$p^{4} T^{32} +$$$$35\!\cdots\!96$$$$p^{5} T^{33} +$$$$49\!\cdots\!25$$$$p^{6} T^{34} +$$$$66\!\cdots\!04$$$$p^{7} T^{35} +$$$$88\!\cdots\!61$$$$p^{8} T^{36} +$$$$11\!\cdots\!13$$$$p^{9} T^{37} +$$$$14\!\cdots\!68$$$$p^{10} T^{38} + 17230012901170637489 p^{11} T^{39} + 2055693463910342342 p^{12} T^{40} + 233695672186813090 p^{13} T^{41} + 26136504865850762 p^{14} T^{42} + 2762922794543536 p^{15} T^{43} + 286579933315658 p^{16} T^{44} + 677467365860 p^{18} T^{45} + 2633637428015 p^{18} T^{46} + 229448669789 p^{19} T^{47} + 19500776410 p^{20} T^{48} + 1483542378 p^{21} T^{49} + 2682152 p^{23} T^{50} + 6975284 p^{23} T^{51} + 433066 p^{24} T^{52} + 20991 p^{25} T^{53} + 1021 p^{26} T^{54} + 30 p^{27} T^{55} + p^{28} T^{56}$$
43 $$1 - 13 T + 650 T^{2} - 7176 T^{3} + 206788 T^{4} - 2002698 T^{5} + 43471085 T^{6} - 377307620 T^{7} + 6843025379 T^{8} - 54032287529 T^{9} + 864015605410 T^{10} - 6275244754705 T^{11} + 91346656401286 T^{12} - 615455692369315 T^{13} + 8323297632712368 T^{14} - 52377411283002350 T^{15} + 666931081905648491 T^{16} - 3941872190540293102 T^{17} + 47674153357244953051 T^{18} -$$$$26\!\cdots\!06$$$$T^{19} +$$$$30\!\cdots\!62$$$$T^{20} -$$$$16\!\cdots\!74$$$$T^{21} +$$$$17\!\cdots\!04$$$$T^{22} -$$$$90\!\cdots\!93$$$$T^{23} +$$$$95\!\cdots\!79$$$$T^{24} -$$$$45\!\cdots\!08$$$$T^{25} +$$$$46\!\cdots\!50$$$$T^{26} -$$$$21\!\cdots\!17$$$$T^{27} +$$$$21\!\cdots\!40$$$$T^{28} -$$$$21\!\cdots\!17$$$$p T^{29} +$$$$46\!\cdots\!50$$$$p^{2} T^{30} -$$$$45\!\cdots\!08$$$$p^{3} T^{31} +$$$$95\!\cdots\!79$$$$p^{4} T^{32} -$$$$90\!\cdots\!93$$$$p^{5} T^{33} +$$$$17\!\cdots\!04$$$$p^{6} T^{34} -$$$$16\!\cdots\!74$$$$p^{7} T^{35} +$$$$30\!\cdots\!62$$$$p^{8} T^{36} -$$$$26\!\cdots\!06$$$$p^{9} T^{37} + 47674153357244953051 p^{10} T^{38} - 3941872190540293102 p^{11} T^{39} + 666931081905648491 p^{12} T^{40} - 52377411283002350 p^{13} T^{41} + 8323297632712368 p^{14} T^{42} - 615455692369315 p^{15} T^{43} + 91346656401286 p^{16} T^{44} - 6275244754705 p^{17} T^{45} + 864015605410 p^{18} T^{46} - 54032287529 p^{19} T^{47} + 6843025379 p^{20} T^{48} - 377307620 p^{21} T^{49} + 43471085 p^{22} T^{50} - 2002698 p^{23} T^{51} + 206788 p^{24} T^{52} - 7176 p^{25} T^{53} + 650 p^{26} T^{54} - 13 p^{27} T^{55} + p^{28} T^{56}$$
47 $$1 + 15 T + 860 T^{2} + 11322 T^{3} + 353962 T^{4} + 4178966 T^{5} + 93690990 T^{6} + 1008348233 T^{7} + 18053568726 T^{8} + 179455852285 T^{9} + 2715991872525 T^{10} + 25205678533962 T^{11} + 333949175084461 T^{12} + 2919804711465234 T^{13} + 34676492540891762 T^{14} + 287776407381370833 T^{15} + 3116532095712630771 T^{16} + 24694198463591536943 T^{17} +$$$$24\!\cdots\!19$$$$T^{18} +$$$$18\!\cdots\!67$$$$T^{19} +$$$$17\!\cdots\!05$$$$T^{20} +$$$$12\!\cdots\!09$$$$T^{21} +$$$$11\!\cdots\!72$$$$T^{22} +$$$$78\!\cdots\!30$$$$T^{23} +$$$$65\!\cdots\!22$$$$T^{24} +$$$$44\!\cdots\!48$$$$T^{25} +$$$$35\!\cdots\!34$$$$T^{26} +$$$$22\!\cdots\!65$$$$T^{27} +$$$$17\!\cdots\!32$$$$T^{28} +$$$$22\!\cdots\!65$$$$p T^{29} +$$$$35\!\cdots\!34$$$$p^{2} T^{30} +$$$$44\!\cdots\!48$$$$p^{3} T^{31} +$$$$65\!\cdots\!22$$$$p^{4} T^{32} +$$$$78\!\cdots\!30$$$$p^{5} T^{33} +$$$$11\!\cdots\!72$$$$p^{6} T^{34} +$$$$12\!\cdots\!09$$$$p^{7} T^{35} +$$$$17\!\cdots\!05$$$$p^{8} T^{36} +$$$$18\!\cdots\!67$$$$p^{9} T^{37} +$$$$24\!\cdots\!19$$$$p^{10} T^{38} + 24694198463591536943 p^{11} T^{39} + 3116532095712630771 p^{12} T^{40} + 287776407381370833 p^{13} T^{41} + 34676492540891762 p^{14} T^{42} + 2919804711465234 p^{15} T^{43} + 333949175084461 p^{16} T^{44} + 25205678533962 p^{17} T^{45} + 2715991872525 p^{18} T^{46} + 179455852285 p^{19} T^{47} + 18053568726 p^{20} T^{48} + 1008348233 p^{21} T^{49} + 93690990 p^{22} T^{50} + 4178966 p^{23} T^{51} + 353962 p^{24} T^{52} + 11322 p^{25} T^{53} + 860 p^{26} T^{54} + 15 p^{27} T^{55} + p^{28} T^{56}$$
53 $$1 + 35 T + 1362 T^{2} + 628 p T^{3} + 795399 T^{4} + 15331623 T^{5} + 282441258 T^{6} + 4566620095 T^{7} + 70407419638 T^{8} + 990262175693 T^{9} + 13317813281211 T^{10} + 166855397127278 T^{11} + 2006566391782088 T^{12} + 22770700939466102 T^{13} + 248980606618551144 T^{14} + 2591419720813656079 T^{15} + 26083203996112553221 T^{16} +$$$$25\!\cdots\!52$$$$T^{17} +$$$$23\!\cdots\!99$$$$T^{18} +$$$$21\!\cdots\!26$$$$T^{19} +$$$$18\!\cdots\!53$$$$T^{20} +$$$$15\!\cdots\!02$$$$T^{21} +$$$$13\!\cdots\!83$$$$T^{22} +$$$$10\!\cdots\!88$$$$T^{23} +$$$$84\!\cdots\!34$$$$T^{24} +$$$$65\!\cdots\!83$$$$T^{25} +$$$$49\!\cdots\!35$$$$T^{26} +$$$$36\!\cdots\!16$$$$T^{27} +$$$$27\!\cdots\!04$$$$T^{28} +$$$$36\!\cdots\!16$$$$p T^{29} +$$$$49\!\cdots\!35$$$$p^{2} T^{30} +$$$$65\!\cdots\!83$$$$p^{3} T^{31} +$$$$84\!\cdots\!34$$$$p^{4} T^{32} +$$$$10\!\cdots\!88$$$$p^{5} T^{33} +$$$$13\!\cdots\!83$$$$p^{6} T^{34} +$$$$15\!\cdots\!02$$$$p^{7} T^{35} +$$$$18\!\cdots\!53$$$$p^{8} T^{36} +$$$$21\!\cdots\!26$$$$p^{9} T^{37} +$$$$23\!\cdots\!99$$$$p^{10} T^{38} +$$$$25\!\cdots\!52$$$$p^{11} T^{39} + 26083203996112553221 p^{12} T^{40} + 2591419720813656079 p^{13} T^{41} + 248980606618551144 p^{14} T^{42} + 22770700939466102 p^{15} T^{43} + 2006566391782088 p^{16} T^{44} + 166855397127278 p^{17} T^{45} + 13317813281211 p^{18} T^{46} + 990262175693 p^{19} T^{47} + 70407419638 p^{20} T^{48} + 4566620095 p^{21} T^{49} + 282441258 p^{22} T^{50} + 15331623 p^{23} T^{51} + 795399 p^{24} T^{52} + 628 p^{26} T^{53} + 1362 p^{26} T^{54} + 35 p^{27} T^{55} + p^{28} T^{56}$$
59 $$1 + T + 1009 T^{2} + 1156 T^{3} + 502318 T^{4} + 616532 T^{5} + 164536866 T^{6} + 205836808 T^{7} + 39890887853 T^{8} + 48830729935 T^{9} + 7634674457668 T^{10} + 8819202398463 T^{11} + 1201514243764571 T^{12} + 1265433374429716 T^{13} + 159953792943397582 T^{14} + 148404764012199856 T^{15} + 18396261751222078764 T^{16} + 14515958149766504114 T^{17} +$$$$18\!\cdots\!54$$$$T^{18} +$$$$12\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!07$$$$T^{20} +$$$$86\!\cdots\!38$$$$T^{21} +$$$$13\!\cdots\!06$$$$T^{22} +$$$$54\!\cdots\!91$$$$T^{23} +$$$$16\!\cdots\!06$$$$p T^{24} +$$$$31\!\cdots\!24$$$$T^{25} +$$$$66\!\cdots\!27$$$$T^{26} +$$$$18\!\cdots\!94$$$$T^{27} +$$$$40\!\cdots\!24$$$$T^{28} +$$$$18\!\cdots\!94$$$$p T^{29} +$$$$66\!\cdots\!27$$$$p^{2} T^{30} +$$$$31\!\cdots\!24$$$$p^{3} T^{31} +$$$$16\!\cdots\!06$$$$p^{5} T^{32} +$$$$54\!\cdots\!91$$$$p^{5} T^{33} +$$$$13\!\cdots\!06$$$$p^{6} T^{34} +$$$$86\!\cdots\!38$$$$p^{7} T^{35} +$$$$16\!\cdots\!07$$$$p^{8} T^{36} +$$$$12\!\cdots\!60$$$$p^{9} T^{37} +$$$$18\!\cdots\!54$$$$p^{10} T^{38} + 14515958149766504114 p^{11} T^{39} + 18396261751222078764 p^{12} T^{40} + 148404764012199856 p^{13} T^{41} + 159953792943397582 p^{14} T^{42} + 1265433374429716 p^{15} T^{43} + 1201514243764571 p^{16} T^{44} + 8819202398463 p^{17} T^{45} + 7634674457668 p^{18} T^{46} + 48830729935 p^{19} T^{47} + 39890887853 p^{20} T^{48} + 205836808 p^{21} T^{49} + 164536866 p^{22} T^{50} + 616532 p^{23} T^{51} + 502318 p^{24} T^{52} + 1156 p^{25} T^{53} + 1009 p^{26} T^{54} + p^{27} T^{55} + p^{28} T^{56}$$
61 $$1 + 33 T + 1480 T^{2} + 36357 T^{3} + 980982 T^{4} + 19674479 T^{5} + 406107727 T^{6} + 6986158876 T^{7} + 120507095646 T^{8} + 1832652714542 T^{9} + 27620570610148 T^{10} + 378877441525436 T^{11} + 5122282557480817 T^{12} + 64280251299061197 T^{13} + 793048508128018286 T^{14} + 9199606659991280381 T^{15} +$$$$10\!\cdots\!14$$$$T^{16} +$$$$11\!\cdots\!95$$$$T^{17} +$$$$12\!\cdots\!85$$$$T^{18} +$$$$12\!\cdots\!48$$$$T^{19} +$$$$19\!\cdots\!38$$$$p T^{20} +$$$$11\!\cdots\!96$$$$T^{21} +$$$$10\!\cdots\!96$$$$T^{22} +$$$$97\!\cdots\!66$$$$T^{23} +$$$$86\!\cdots\!85$$$$T^{24} +$$$$73\!\cdots\!88$$$$T^{25} +$$$$61\!\cdots\!60$$$$T^{26} +$$$$49\!\cdots\!58$$$$T^{27} +$$$$39\!\cdots\!02$$$$T^{28} +$$$$49\!\cdots\!58$$$$p T^{29} +$$$$61\!\cdots\!60$$$$p^{2} T^{30} +$$$$73\!\cdots\!88$$$$p^{3} T^{31} +$$$$86\!\cdots\!85$$$$p^{4} T^{32} +$$$$97\!\cdots\!66$$$$p^{5} T^{33} +$$$$10\!\cdots\!96$$$$p^{6} T^{34} +$$$$11\!\cdots\!96$$$$p^{7} T^{35} +$$$$19\!\cdots\!38$$$$p^{9} T^{36} +$$$$12\!\cdots\!48$$$$p^{9} T^{37} +$$$$12\!\cdots\!85$$$$p^{10} T^{38} +$$$$11\!\cdots\!95$$$$p^{11} T^{39} +$$$$10\!\cdots\!14$$$$p^{12} T^{40} + 9199606659991280381 p^{13} T^{41} + 793048508128018286 p^{14} T^{42} + 64280251299061197 p^{15} T^{43} + 5122282557480817 p^{16} T^{44} + 378877441525436 p^{17} T^{45} + 27620570610148 p^{18} T^{46} + 1832652714542 p^{19} T^{47} + 120507095646 p^{20} T^{48} + 6986158876 p^{21} T^{49} + 406107727 p^{22} T^{50} + 19674479 p^{23} T^{51} + 980982 p^{24} T^{52} + 36357 p^{25} T^{53} + 1480 p^{26} T^{54} + 33 p^{27} T^{55} + p^{28} T^{56}$$
67 $$1 - 19 T + 1208 T^{2} - 19657 T^{3} + 10392 p T^{4} - 9991941 T^{5} + 258876093 T^{6} - 3348285421 T^{7} + 70590831981 T^{8} - 836560343032 T^{9} + 15174996234826 T^{10} - 166852739397707 T^{11} + 2692740916653214 T^{12} - 27728135379287506 T^{13} + 406842361420690859 T^{14} - 3950579812818239948 T^{15} + 53485288659486443710 T^{16} -$$$$49\!\cdots\!40$$$$T^{17} +$$$$62\!\cdots\!17$$$$T^{18} -$$$$54\!\cdots\!04$$$$T^{19} +$$$$64\!\cdots\!77$$$$T^{20} -$$$$53\!\cdots\!81$$$$T^{21} +$$$$60\!\cdots\!97$$$$T^{22} -$$$$48\!\cdots\!19$$$$T^{23} +$$$$51\!\cdots\!86$$$$T^{24} -$$$$38\!\cdots\!06$$$$T^{25} +$$$$39\!\cdots\!92$$$$T^{26} -$$$$28\!\cdots\!97$$$$T^{27} +$$$$27\!\cdots\!26$$$$T^{28} -$$$$28\!\cdots\!97$$$$p T^{29} +$$$$39\!\cdots\!92$$$$p^{2} T^{30} -$$$$38\!\cdots\!06$$$$p^{3} T^{31} +$$$$51\!\cdots\!86$$$$p^{4} T^{32} -$$$$48\!\cdots\!19$$$$p^{5} T^{33} +$$$$60\!\cdots\!97$$$$p^{6} T^{34} -$$$$53\!\cdots\!81$$$$p^{7} T^{35} +$$$$64\!\cdots\!77$$$$p^{8} T^{36} -$$$$54\!\cdots\!04$$$$p^{9} T^{37} +$$$$62\!\cdots\!17$$$$p^{10} T^{38} -$$$$49\!\cdots\!40$$$$p^{11} T^{39} + 53485288659486443710 p^{12} T^{40} - 3950579812818239948 p^{13} T^{41} + 406842361420690859 p^{14} T^{42} - 27728135379287506 p^{15} T^{43} + 2692740916653214 p^{16} T^{44} - 166852739397707 p^{17} T^{45} + 15174996234826 p^{18} T^{46} - 836560343032 p^{19} T^{47} + 70590831981 p^{20} T^{48} - 3348285421 p^{21} T^{49} + 258876093 p^{22} T^{50} - 9991941 p^{23} T^{51} + 10392 p^{25} T^{52} - 19657 p^{25} T^{53} + 1208 p^{26} T^{54} - 19 p^{27} T^{55} + p^{28} T^{56}$$
71 $$1 + 31 T + 1461 T^{2} + 33690 T^{3} + 932065 T^{4} + 17409846 T^{5} + 360982586 T^{6} + 5723624276 T^{7} + 97393742250 T^{8} + 1349708956652 T^{9} + 19739488007383 T^{10} + 243929339023814 T^{11} + 3152577837450824 T^{12} + 35272200098590966 T^{13} + 410587737983340817 T^{14} + 4215840826506786586 T^{15} + 44872064280763796175 T^{16} +$$$$42\!\cdots\!51$$$$T^{17} +$$$$42\!\cdots\!54$$$$T^{18} +$$$$38\!\cdots\!01$$$$T^{19} +$$$$35\!\cdots\!78$$$$T^{20} +$$$$31\!\cdots\!59$$$$T^{21} +$$$$27\!\cdots\!24$$$$T^{22} +$$$$24\!\cdots\!03$$$$T^{23} +$$$$20\!\cdots\!93$$$$T^{24} +$$$$18\!\cdots\!74$$$$T^{25} +$$$$15\!\cdots\!95$$$$T^{26} +$$$$13\!\cdots\!41$$$$T^{27} +$$$$10\!\cdots\!60$$$$T^{28} +$$$$13\!\cdots\!41$$$$p T^{29} +$$$$15\!\cdots\!95$$$$p^{2} T^{30} +$$$$18\!\cdots\!74$$$$p^{3} T^{31} +$$$$20\!\cdots\!93$$$$p^{4} T^{32} +$$$$24\!\cdots\!03$$$$p^{5} T^{33} +$$$$27\!\cdots\!24$$$$p^{6} T^{34} +$$$$31\!\cdots\!59$$$$p^{7} T^{35} +$$$$35\!\cdots\!78$$$$p^{8} T^{36} +$$$$38\!\cdots\!01$$$$p^{9} T^{37} +$$$$42\!\cdots\!54$$$$p^{10} T^{38} +$$$$42\!\cdots\!51$$$$p^{11} T^{39} + 44872064280763796175 p^{12} T^{40} + 4215840826506786586 p^{13} T^{41} + 410587737983340817 p^{14} T^{42} + 35272200098590966 p^{15} T^{43} + 3152577837450824 p^{16} T^{44} + 243929339023814 p^{17} T^{45} + 19739488007383 p^{18} T^{46} + 1349708956652 p^{19} T^{47} + 97393742250 p^{20} T^{48} + 5723624276 p^{21} T^{49} + 360982586 p^{22} T^{50} + 17409846 p^{23} T^{51} + 932065 p^{24} T^{52} + 33690 p^{25} T^{53} + 1461 p^{26} T^{54} + 31 p^{27} T^{55} + p^{28} T^{56}$$
73 $$1 - 31 T + 1601 T^{2} - 41004 T^{3} + 1240555 T^{4} - 27101520 T^{5} + 622444858 T^{6} - 11892009187 T^{7} + 228019383569 T^{8} - 3884167912556 T^{9} + 65132850471393 T^{10} - 1004092203606562 T^{11} + 15118338527272646 T^{12} - 213358340500576932 T^{13} + 2932360049577945495 T^{14} - 38219871483509841779 T^{15} +$$$$48\!\cdots\!46$$$$T^{16} -$$$$58\!\cdots\!44$$$$T^{17} +$$$$69\!\cdots\!89$$$$T^{18} -$$$$78\!\cdots\!79$$$$T^{19} +$$$$86\!\cdots\!40$$$$T^{20} -$$$$92\!\cdots\!03$$$$T^{21} +$$$$95\!\cdots\!04$$$$T^{22} -$$$$95\!\cdots\!16$$$$T^{23} +$$$$92\!\cdots\!47$$$$T^{24} -$$$$87\!\cdots\!58$$$$T^{25} +$$$$80\!\cdots\!34$$$$T^{26} -$$$$71\!\cdots\!21$$$$T^{27} +$$$$62\!\cdots\!40$$$$T^{28} -$$$$71\!\cdots\!21$$$$p T^{29} +$$$$80\!\cdots\!34$$$$p^{2} T^{30} -$$$$87\!\cdots\!58$$$$p^{3} T^{31} +$$$$92\!\cdots\!47$$$$p^{4} T^{32} -$$$$95\!\cdots\!16$$$$p^{5} T^{33} +$$$$95\!\cdots\!04$$$$p^{6} T^{34} -$$$$92\!\cdots\!03$$$$p^{7} T^{35} +$$$$86\!\cdots\!40$$$$p^{8} T^{36} -$$$$78\!\cdots\!79$$$$p^{9} T^{37} +$$$$69\!\cdots\!89$$$$p^{10} T^{38} -$$$$58\!\cdots\!44$$$$p^{11} T^{39} +$$$$48\!\cdots\!46$$$$p^{12} T^{40} - 38219871483509841779 p^{13} T^{41} + 2932360049577945495 p^{14} T^{42} - 213358340500576932 p^{15} T^{43} + 15118338527272646 p^{16} T^{44} - 1004092203606562 p^{17} T^{45} + 65132850471393 p^{18} T^{46} - 3884167912556 p^{19} T^{47} + 228019383569 p^{20} T^{48} - 11892009187 p^{21} T^{49} + 622444858 p^{22} T^{50} - 27101520 p^{23} T^{51} + 1240555 p^{24} T^{52} - 41004 p^{25} T^{53} + 1601 p^{26} T^{54} - 31 p^{27} T^{55} + p^{28} T^{56}$$
79 $$1 + 29 T + 1388 T^{2} + 30218 T^{3} + 851286 T^{4} + 15084172 T^{5} + 319813972 T^{6} + 4818722069 T^{7} + 84400015203 T^{8} + 1111857520988 T^{9} + 16920341139212 T^{10} + 198974572592968 T^{11} + 2723449315633923 T^{12} + 29123913842233903 T^{13} + 368755478348295844 T^{14} + 3654942417631209729 T^{15} + 43851253320901778213 T^{16} +$$$$41\!\cdots\!28$$$$T^{17} +$$$$47\!\cdots\!39$$$$T^{18} +$$$$42\!\cdots\!17$$$$T^{19} +$$$$48\!\cdots\!80$$$$T^{20} +$$$$41\!\cdots\!11$$$$T^{21} +$$$$46\!\cdots\!71$$$$T^{22} +$$$$39\!\cdots\!76$$$$T^{23} +$$$$41\!\cdots\!52$$$$T^{24} +$$$$34\!\cdots\!64$$$$T^{25} +$$$$35\!\cdots\!90$$$$T^{26} +$$$$28\!\cdots\!02$$$$T^{27} +$$$$28\!\cdots\!68$$$$T^{28} +$$$$28\!\cdots\!02$$$$p T^{29} +$$$$35\!\cdots\!90$$$$p^{2} T^{30} +$$$$34\!\cdots\!64$$$$p^{3} T^{31} +$$$$41\!\cdots\!52$$$$p^{4} T^{32} +$$$$39\!\cdots\!76$$$$p^{5} T^{33} +$$$$46\!\cdots\!71$$$$p^{6} T^{34} +$$$$41\!\cdots\!11$$$$p^{7} T^{35} +$$$$48\!\cdots\!80$$$$p^{8} T^{36} +$$$$42\!\cdots\!17$$$$p^{9} T^{37} +$$$$47\!\cdots\!39$$$$p^{10} T^{38} +$$$$41\!\cdots\!28$$$$p^{11} T^{39} + 43851253320901778213 p^{12} T^{40} + 3654942417631209729 p^{13} T^{41} + 368755478348295844 p^{14} T^{42} + 29123913842233903 p^{15} T^{43} + 2723449315633923 p^{16} T^{44} + 198974572592968 p^{17} T^{45} + 16920341139212 p^{18} T^{46} + 1111857520988 p^{19} T^{47} + 84400015203 p^{20} T^{48} + 4818722069 p^{21} T^{49} + 319813972 p^{22} T^{50} + 15084172 p^{23} T^{51} + 851286 p^{24} T^{52} + 30218 p^{25} T^{53} + 1388 p^{26} T^{54} + 29 p^{27} T^{55} + p^{28} T^{56}$$
83 $$1 - 14 T + 1287 T^{2} - 17741 T^{3} + 837324 T^{4} - 11262975 T^{5} + 366315979 T^{6} - 4776470256 T^{7} + 120936832633 T^{8} - 1521665084563 T^{9} + 32061065447944 T^{10} - 388109757569647 T^{11} + 7092645234580713 T^{12} - 82451987034091908 T^{13} + 1343560750254561547 T^{14} - 14982647626570325870 T^{15} +$$$$22\!\cdots\!77$$$$T^{16} -$$$$23\!\cdots\!35$$$$T^{17} +$$$$32\!\cdots\!94$$$$T^{18} -$$$$33\!\cdots\!11$$$$T^{19} +$$$$42\!\cdots\!79$$$$T^{20} -$$$$41\!\cdots\!17$$$$T^{21} +$$$$49\!\cdots\!67$$$$T^{22} -$$$$46\!\cdots\!00$$$$T^{23} +$$$$52\!\cdots\!54$$$$T^{24} -$$$$46\!\cdots\!51$$$$T^{25} +$$$$50\!\cdots\!00$$$$T^{26} -$$$$42\!\cdots\!58$$$$T^{27} +$$$$43\!\cdots\!38$$$$T^{28} -$$$$42\!\cdots\!58$$$$p T^{29} +$$$$50\!\cdots\!00$$$$p^{2} T^{30} -$$$$46\!\cdots\!51$$$$p^{3} T^{31} +$$$$52\!\cdots\!54$$$$p^{4} T^{32} -$$$$46\!\cdots\!00$$$$p^{5} T^{33} +$$$$49\!\cdots\!67$$$$p^{6} T^{34} -$$$$41\!\cdots\!17$$$$p^{7} T^{35} +$$$$42\!\cdots\!79$$$$p^{8} T^{36} -$$$$33\!\cdots\!11$$$$p^{9} T^{37} +$$$$32\!\cdots\!94$$$$p^{10} T^{38} -$$$$23\!\cdots\!35$$$$p^{11} T^{39} +$$$$22\!\cdots\!77$$$$p^{12} T^{40} - 14982647626570325870 p^{13} T^{41} + 1343560750254561547 p^{14} T^{42} - 82451987034091908 p^{15} T^{43} + 7092645234580713 p^{16} T^{44} - 388109757569647 p^{17} T^{45} + 32061065447944 p^{18} T^{46} - 1521665084563 p^{19} T^{47} + 120936832633 p^{20} T^{48} - 4776470256 p^{21} T^{49} + 366315979 p^{22} T^{50} - 11262975 p^{23} T^{51} + 837324 p^{24} T^{52} - 17741 p^{25} T^{53} + 1287 p^{26} T^{54} - 14 p^{27} T^{55} + p^{28} T^{56}$$
89 $$1 + 32 T + 1374 T^{2} + 33875 T^{3} + 898411 T^{4} + 18504023 T^{5} + 384639308 T^{6} + 6927597241 T^{7} + 123069031866 T^{8} + 1994020022523 T^{9} + 31599989230627 T^{10} + 469376059371199 T^{11} + 6801609277209727 T^{12} + 93850521761180129 T^{13} + 1263164442042240211 T^{14} + 16346854041044973984 T^{15} +$$$$20\!\cdots\!43$$$$T^{16} +$$$$25\!\cdots\!64$$$$T^{17} +$$$$30\!\cdots\!82$$$$T^{18} +$$$$35\!\cdots\!02$$$$T^{19} +$$$$39\!\cdots\!41$$$$T^{20} +$$$$43\!\cdots\!66$$$$T^{21} +$$$$47\!\cdots\!11$$$$T^{22} +$$$$50\!\cdots\!51$$$$T^{23} +$$$$52\!\cdots\!42$$$$T^{24} +$$$$53\!\cdots\!92$$$$T^{25} +$$$$52\!\cdots\!29$$$$T^{26} +$$$$51\!\cdots\!83$$$$T^{27} +$$$$48\!\cdots\!70$$$$T^{28} +$$$$51\!\cdots\!83$$$$p T^{29} +$$$$52\!\cdots\!29$$$$p^{2} T^{30} +$$$$53\!\cdots\!92$$$$p^{3} T^{31} +$$$$52\!\cdots\!42$$$$p^{4} T^{32} +$$$$50\!\cdots\!51$$$$p^{5} T^{33} +$$$$47\!\cdots\!11$$$$p^{6} T^{34} +$$$$43\!\cdots\!66$$$$p^{7} T^{35} +$$$$39\!\cdots\!41$$$$p^{8} T^{36} +$$$$35\!\cdots\!02$$$$p^{9} T^{37} +$$$$30\!\cdots\!82$$$$p^{10} T^{38} +$$$$25\!\cdots\!64$$$$p^{11} T^{39} +$$$$20\!\cdots\!43$$$$p^{12} T^{40} + 16346854041044973984 p^{13} T^{41} + 1263164442042240211 p^{14} T^{42} + 93850521761180129 p^{15} T^{43} + 6801609277209727 p^{16} T^{44} + 469376059371199 p^{17} T^{45} + 31599989230627 p^{18} T^{46} + 1994020022523 p^{19} T^{47} + 123069031866 p^{20} T^{48} + 6927597241 p^{21} T^{49} + 384639308 p^{22} T^{50} + 18504023 p^{23} T^{51} + 898411 p^{24} T^{52} + 33875 p^{25} T^{53} + 1374 p^{26} T^{54} + 32 p^{27} T^{55} + p^{28} T^{56}$$
97 $$1 - 2 T + 1502 T^{2} - 493 T^{3} + 1126698 T^{4} + 1289864 T^{5} + 566171738 T^{6} + 1365579071 T^{7} + 215083749502 T^{8} + 744924666546 T^{9} + 65920683687675 T^{10} + 283378112885765 T^{11} + 16952045377449017 T^{12} + 83569186839515136 T^{13} + 3751170624567366257 T^{14} + 20169680755011925205 T^{15} +$$$$72\!\cdots\!90$$$$T^{16} +$$$$41\!\cdots\!94$$$$T^{17} +$$$$12\!\cdots\!32$$$$T^{18} +$$$$72\!\cdots\!00$$$$T^{19} +$$$$19\!\cdots\!82$$$$T^{20} +$$$$11\!\cdots\!36$$$$T^{21} +$$$$26\!\cdots\!95$$$$T^{22} +$$$$15\!\cdots\!71$$$$T^{23} +$$$$32\!\cdots\!44$$$$T^{24} +$$$$18\!\cdots\!94$$$$T^{25} +$$$$36\!\cdots\!25$$$$T^{26} +$$$$20\!\cdots\!33$$$$T^{27} +$$$$37\!\cdots\!48$$$$T^{28} +$$$$20\!\cdots\!33$$$$p T^{29} +$$$$36\!\cdots\!25$$$$p^{2} T^{30} +$$$$18\!\cdots\!94$$$$p^{3} T^{31} +$$$$32\!\cdots\!44$$$$p^{4} T^{32} +$$$$15\!\cdots\!71$$$$p^{5} T^{33} +$$$$26\!\cdots\!95$$$$p^{6} T^{34} +$$$$11\!\cdots\!36$$$$p^{7} T^{35} +$$$$19\!\cdots\!82$$$$p^{8} T^{36} +$$$$72\!\cdots\!00$$$$p^{9} T^{37} +$$$$12\!\cdots\!32$$$$p^{10} T^{38} +$$$$41\!\cdots\!94$$$$p^{11} T^{39} +$$$$72\!\cdots\!90$$$$p^{12} T^{40} + 20169680755011925205 p^{13} T^{41} + 3751170624567366257 p^{14} T^{42} + 83569186839515136 p^{15} T^{43} + 16952045377449017 p^{16} T^{44} + 283378112885765 p^{17} T^{45} + 65920683687675 p^{18} T^{46} + 744924666546 p^{19} T^{47} + 215083749502 p^{20} T^{48} + 1365579071 p^{21} T^{49} + 566171738 p^{22} T^{50} + 1289864 p^{23} T^{51} + 1126698 p^{24} T^{52} - 493 p^{25} T^{53} + 1502 p^{26} T^{54} - 2 p^{27} T^{55} + p^{28} T^{56}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}