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

Origins of factors

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Normalization:  

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 \( 1 - 2 T + 1502 T^{2} - 493 T^{3} + 1126698 T^{4} + 1289864 T^{5} + 566171738 T^{6} + 1365579071 T^{7} + 215083749502 T^{8} + 744924666546 T^{9} + 65920683687675 T^{10} + 283378112885765 T^{11} + 16952045377449017 T^{12} + 83569186839515136 T^{13} + 3751170624567366257 T^{14} + 20169680755011925205 T^{15} + \)\(72\!\cdots\!90\)\( T^{16} + \)\(41\!\cdots\!94\)\( T^{17} + \)\(12\!\cdots\!32\)\( T^{18} + \)\(72\!\cdots\!00\)\( T^{19} + \)\(19\!\cdots\!82\)\( T^{20} + \)\(11\!\cdots\!36\)\( T^{21} + \)\(26\!\cdots\!95\)\( T^{22} + \)\(15\!\cdots\!71\)\( T^{23} + \)\(32\!\cdots\!44\)\( T^{24} + \)\(18\!\cdots\!94\)\( T^{25} + \)\(36\!\cdots\!25\)\( T^{26} + \)\(20\!\cdots\!33\)\( T^{27} + \)\(37\!\cdots\!48\)\( T^{28} + \)\(20\!\cdots\!33\)\( p T^{29} + \)\(36\!\cdots\!25\)\( p^{2} T^{30} + \)\(18\!\cdots\!94\)\( p^{3} T^{31} + \)\(32\!\cdots\!44\)\( p^{4} T^{32} + \)\(15\!\cdots\!71\)\( p^{5} T^{33} + \)\(26\!\cdots\!95\)\( p^{6} T^{34} + \)\(11\!\cdots\!36\)\( p^{7} T^{35} + \)\(19\!\cdots\!82\)\( p^{8} T^{36} + \)\(72\!\cdots\!00\)\( p^{9} T^{37} + \)\(12\!\cdots\!32\)\( p^{10} T^{38} + \)\(41\!\cdots\!94\)\( p^{11} T^{39} + \)\(72\!\cdots\!90\)\( p^{12} T^{40} + 20169680755011925205 p^{13} T^{41} + 3751170624567366257 p^{14} T^{42} + 83569186839515136 p^{15} T^{43} + 16952045377449017 p^{16} T^{44} + 283378112885765 p^{17} T^{45} + 65920683687675 p^{18} T^{46} + 744924666546 p^{19} T^{47} + 215083749502 p^{20} T^{48} + 1365579071 p^{21} T^{49} + 566171738 p^{22} T^{50} + 1289864 p^{23} T^{51} + 1126698 p^{24} T^{52} - 493 p^{25} T^{53} + 1502 p^{26} T^{54} - 2 p^{27} T^{55} + p^{28} T^{56} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.56503940487372944419387662519, −1.54219253529821082629997779570, −1.50251394353462875842998383170, −1.41402069269485335909508544716, −1.38635712212457435354054933021, −1.38523354952156025633250165150, −1.38444681704797425284282462107, −1.38332726423575101759535163903, −1.34547215080914100432521626491, −1.31455535381201338664025949617, −1.31341892314447239851968841596, −1.16051571896793194894995080154, −1.13948588617764530255932568053, −1.13115990512992049218609362628, −1.09497315953622543086315223578, −1.09247427952807017573011427261, −1.06822696942609025334577132040, −1.06075026593322353374370857803, −1.05237591476129300001690471069, −0.939350364232261011605637152371, −0.937635391250608470685619627633, −0.932539278313875518478159747716, −0.857228154443025368527627935911, −0.814548852009464511332244847293, −0.74183695435515405195452736787, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.74183695435515405195452736787, 0.814548852009464511332244847293, 0.857228154443025368527627935911, 0.932539278313875518478159747716, 0.937635391250608470685619627633, 0.939350364232261011605637152371, 1.05237591476129300001690471069, 1.06075026593322353374370857803, 1.06822696942609025334577132040, 1.09247427952807017573011427261, 1.09497315953622543086315223578, 1.13115990512992049218609362628, 1.13948588617764530255932568053, 1.16051571896793194894995080154, 1.31341892314447239851968841596, 1.31455535381201338664025949617, 1.34547215080914100432521626491, 1.38332726423575101759535163903, 1.38444681704797425284282462107, 1.38523354952156025633250165150, 1.38635712212457435354054933021, 1.41402069269485335909508544716, 1.50251394353462875842998383170, 1.54219253529821082629997779570, 1.56503940487372944419387662519

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.