Properties

Degree 52
Conductor $ 2^{52} \cdot 3^{26} \cdot 503^{26} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 26·3-s + 6·5-s + 5·7-s + 351·9-s − 11·11-s + 13·13-s − 156·15-s + 12·17-s − 19-s − 130·21-s − 22·23-s − 23·25-s − 3.27e3·27-s + 6·29-s + 19·31-s + 286·33-s + 30·35-s + 20·37-s − 338·39-s + 25·41-s + 4·43-s + 2.10e3·45-s + 8·47-s − 45·49-s − 312·51-s − 5·53-s − 66·55-s + ⋯
L(s)  = 1  − 15.0·3-s + 2.68·5-s + 1.88·7-s + 117·9-s − 3.31·11-s + 3.60·13-s − 40.2·15-s + 2.91·17-s − 0.229·19-s − 28.3·21-s − 4.58·23-s − 4.59·25-s − 630.·27-s + 1.11·29-s + 3.41·31-s + 49.7·33-s + 5.07·35-s + 3.28·37-s − 54.1·39-s + 3.90·41-s + 0.609·43-s + 313.·45-s + 1.16·47-s − 6.42·49-s − 43.6·51-s − 0.686·53-s − 8.89·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{52} \cdot 3^{26} \cdot 503^{26}\right)^{s/2} \, \Gamma_{\C}(s)^{26} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{52} \cdot 3^{26} \cdot 503^{26}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{26} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(52\)
\( N \)  =  \(2^{52} \cdot 3^{26} \cdot 503^{26}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6036} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(52,\ 2^{52} \cdot 3^{26} \cdot 503^{26} ,\ ( \ : [1/2]^{26} ),\ 1 )$
$L(1)$  $\approx$  $3.445190807$
$L(\frac12)$  $\approx$  $3.445190807$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \(F_p\) is a polynomial of degree 52. If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 51.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 + T )^{26} \)
503 \( ( 1 + T )^{26} \)
good5 \( 1 - 6 T + 59 T^{2} - 284 T^{3} + 337 p T^{4} - 6914 T^{5} + 31587 T^{6} - 114166 T^{7} + 438647 T^{8} - 1424896 T^{9} + 4809194 T^{10} - 14234326 T^{11} + 43297179 T^{12} - 117957183 T^{13} + 329040486 T^{14} - 832070156 T^{15} + 2158559669 T^{16} - 5108681244 T^{17} + 2498940498 p T^{18} - 1118176209 p^{2} T^{19} + 65494917014 T^{20} - 140489207488 T^{21} + 321724426541 T^{22} - 675114522734 T^{23} + 1546416819888 T^{24} - 650280490648 p T^{25} + 7584497702934 T^{26} - 650280490648 p^{2} T^{27} + 1546416819888 p^{2} T^{28} - 675114522734 p^{3} T^{29} + 321724426541 p^{4} T^{30} - 140489207488 p^{5} T^{31} + 65494917014 p^{6} T^{32} - 1118176209 p^{9} T^{33} + 2498940498 p^{9} T^{34} - 5108681244 p^{9} T^{35} + 2158559669 p^{10} T^{36} - 832070156 p^{11} T^{37} + 329040486 p^{12} T^{38} - 117957183 p^{13} T^{39} + 43297179 p^{14} T^{40} - 14234326 p^{15} T^{41} + 4809194 p^{16} T^{42} - 1424896 p^{17} T^{43} + 438647 p^{18} T^{44} - 114166 p^{19} T^{45} + 31587 p^{20} T^{46} - 6914 p^{21} T^{47} + 337 p^{23} T^{48} - 284 p^{23} T^{49} + 59 p^{24} T^{50} - 6 p^{25} T^{51} + p^{26} T^{52} \)
7 \( 1 - 5 T + 10 p T^{2} - 331 T^{3} + 2610 T^{4} - 1629 p T^{5} + 9606 p T^{6} - 271695 T^{7} + 1333672 T^{8} - 5018266 T^{9} + 21647776 T^{10} - 76471230 T^{11} + 299218120 T^{12} - 999944769 T^{13} + 3621575431 T^{14} - 11525166943 T^{15} + 5596392296 p T^{16} - 119356743701 T^{17} + 384517351668 T^{18} - 1126295043491 T^{19} + 3463713297125 T^{20} - 9783134484400 T^{21} + 28870795941151 T^{22} - 78780931643086 T^{23} + 223925546129968 T^{24} - 590825483722872 T^{25} + 1621418905260524 T^{26} - 590825483722872 p T^{27} + 223925546129968 p^{2} T^{28} - 78780931643086 p^{3} T^{29} + 28870795941151 p^{4} T^{30} - 9783134484400 p^{5} T^{31} + 3463713297125 p^{6} T^{32} - 1126295043491 p^{7} T^{33} + 384517351668 p^{8} T^{34} - 119356743701 p^{9} T^{35} + 5596392296 p^{11} T^{36} - 11525166943 p^{11} T^{37} + 3621575431 p^{12} T^{38} - 999944769 p^{13} T^{39} + 299218120 p^{14} T^{40} - 76471230 p^{15} T^{41} + 21647776 p^{16} T^{42} - 5018266 p^{17} T^{43} + 1333672 p^{18} T^{44} - 271695 p^{19} T^{45} + 9606 p^{21} T^{46} - 1629 p^{22} T^{47} + 2610 p^{22} T^{48} - 331 p^{23} T^{49} + 10 p^{25} T^{50} - 5 p^{25} T^{51} + p^{26} T^{52} \)
11 \( 1 + p T + 173 T^{2} + 1439 T^{3} + 13766 T^{4} + 95589 T^{5} + 709460 T^{6} + 4321507 T^{7} + 27200271 T^{8} + 149527755 T^{9} + 834160134 T^{10} + 4213664381 T^{11} + 21373596823 T^{12} + 100427436367 T^{13} + 470773304860 T^{14} + 188670583355 p T^{15} + 9089769393768 T^{16} + 37831392235599 T^{17} + 1289268325590 p^{2} T^{18} + 615741024036961 T^{19} + 218504787215266 p T^{20} + 9025565833007326 T^{21} + 33478686085858235 T^{22} + 119853818921837342 T^{23} + 423545272346396905 T^{24} + 1447259156071921652 T^{25} + 4879913191003027360 T^{26} + 1447259156071921652 p T^{27} + 423545272346396905 p^{2} T^{28} + 119853818921837342 p^{3} T^{29} + 33478686085858235 p^{4} T^{30} + 9025565833007326 p^{5} T^{31} + 218504787215266 p^{7} T^{32} + 615741024036961 p^{7} T^{33} + 1289268325590 p^{10} T^{34} + 37831392235599 p^{9} T^{35} + 9089769393768 p^{10} T^{36} + 188670583355 p^{12} T^{37} + 470773304860 p^{12} T^{38} + 100427436367 p^{13} T^{39} + 21373596823 p^{14} T^{40} + 4213664381 p^{15} T^{41} + 834160134 p^{16} T^{42} + 149527755 p^{17} T^{43} + 27200271 p^{18} T^{44} + 4321507 p^{19} T^{45} + 709460 p^{20} T^{46} + 95589 p^{21} T^{47} + 13766 p^{22} T^{48} + 1439 p^{23} T^{49} + 173 p^{24} T^{50} + p^{26} T^{51} + p^{26} T^{52} \)
13 \( 1 - p T + 191 T^{2} - 1591 T^{3} + 14051 T^{4} - 88605 T^{5} + 590349 T^{6} - 2997364 T^{7} + 1259256 p T^{8} - 68503165 T^{9} + 322659882 T^{10} - 86180319 p T^{11} + 4855329372 T^{12} - 14302850112 T^{13} + 65147905374 T^{14} - 183266274272 T^{15} + 1005453118377 T^{16} - 3119035359890 T^{17} + 18195625011788 T^{18} - 57629355503368 T^{19} + 303784416808895 T^{20} - 867320809570544 T^{21} + 4105980314234883 T^{22} - 9965059389122832 T^{23} + 46692669230194787 T^{24} - 101193666551834413 T^{25} + 548809484688968484 T^{26} - 101193666551834413 p T^{27} + 46692669230194787 p^{2} T^{28} - 9965059389122832 p^{3} T^{29} + 4105980314234883 p^{4} T^{30} - 867320809570544 p^{5} T^{31} + 303784416808895 p^{6} T^{32} - 57629355503368 p^{7} T^{33} + 18195625011788 p^{8} T^{34} - 3119035359890 p^{9} T^{35} + 1005453118377 p^{10} T^{36} - 183266274272 p^{11} T^{37} + 65147905374 p^{12} T^{38} - 14302850112 p^{13} T^{39} + 4855329372 p^{14} T^{40} - 86180319 p^{16} T^{41} + 322659882 p^{16} T^{42} - 68503165 p^{17} T^{43} + 1259256 p^{19} T^{44} - 2997364 p^{19} T^{45} + 590349 p^{20} T^{46} - 88605 p^{21} T^{47} + 14051 p^{22} T^{48} - 1591 p^{23} T^{49} + 191 p^{24} T^{50} - p^{26} T^{51} + p^{26} T^{52} \)
17 \( 1 - 12 T + 181 T^{2} - 1611 T^{3} + 15233 T^{4} - 113698 T^{5} + 860703 T^{6} - 5699408 T^{7} + 37579759 T^{8} - 226834321 T^{9} + 1352159341 T^{10} - 7561178747 T^{11} + 41633288515 T^{12} - 218371321186 T^{13} + 1127335672000 T^{14} - 5599476925440 T^{15} + 27398796451663 T^{16} - 129865541303929 T^{17} + 607579855880737 T^{18} - 2767352394398108 T^{19} + 12473489968238791 T^{20} - 54941816364304962 T^{21} + 240099260368925457 T^{22} - 1028120204203741786 T^{23} + 4377325263745299014 T^{24} - 18292496615342116472 T^{25} + 76106280276350641498 T^{26} - 18292496615342116472 p T^{27} + 4377325263745299014 p^{2} T^{28} - 1028120204203741786 p^{3} T^{29} + 240099260368925457 p^{4} T^{30} - 54941816364304962 p^{5} T^{31} + 12473489968238791 p^{6} T^{32} - 2767352394398108 p^{7} T^{33} + 607579855880737 p^{8} T^{34} - 129865541303929 p^{9} T^{35} + 27398796451663 p^{10} T^{36} - 5599476925440 p^{11} T^{37} + 1127335672000 p^{12} T^{38} - 218371321186 p^{13} T^{39} + 41633288515 p^{14} T^{40} - 7561178747 p^{15} T^{41} + 1352159341 p^{16} T^{42} - 226834321 p^{17} T^{43} + 37579759 p^{18} T^{44} - 5699408 p^{19} T^{45} + 860703 p^{20} T^{46} - 113698 p^{21} T^{47} + 15233 p^{22} T^{48} - 1611 p^{23} T^{49} + 181 p^{24} T^{50} - 12 p^{25} T^{51} + p^{26} T^{52} \)
19 \( 1 + T + 168 T^{2} + 83 T^{3} + 14505 T^{4} + 1953 T^{5} + 871508 T^{6} - 40733 T^{7} + 41105889 T^{8} - 2499289 T^{9} + 1611663375 T^{10} + 105288208 T^{11} + 54207271189 T^{12} + 17072752398 T^{13} + 84108402590 p T^{14} + 1088135021175 T^{15} + 42001972705954 T^{16} + 48402038917147 T^{17} + 999443669116658 T^{18} + 1703231482605274 T^{19} + 21890035521474048 T^{20} + 49961736849532097 T^{21} + 449877751959212857 T^{22} + 1254142466101232044 T^{23} + 8856167227615727912 T^{24} + 27307406239785133992 T^{25} + \)\(16\!\cdots\!40\)\( T^{26} + 27307406239785133992 p T^{27} + 8856167227615727912 p^{2} T^{28} + 1254142466101232044 p^{3} T^{29} + 449877751959212857 p^{4} T^{30} + 49961736849532097 p^{5} T^{31} + 21890035521474048 p^{6} T^{32} + 1703231482605274 p^{7} T^{33} + 999443669116658 p^{8} T^{34} + 48402038917147 p^{9} T^{35} + 42001972705954 p^{10} T^{36} + 1088135021175 p^{11} T^{37} + 84108402590 p^{13} T^{38} + 17072752398 p^{13} T^{39} + 54207271189 p^{14} T^{40} + 105288208 p^{15} T^{41} + 1611663375 p^{16} T^{42} - 2499289 p^{17} T^{43} + 41105889 p^{18} T^{44} - 40733 p^{19} T^{45} + 871508 p^{20} T^{46} + 1953 p^{21} T^{47} + 14505 p^{22} T^{48} + 83 p^{23} T^{49} + 168 p^{24} T^{50} + p^{25} T^{51} + p^{26} T^{52} \)
23 \( 1 + 22 T + 450 T^{2} + 6381 T^{3} + 83310 T^{4} + 918304 T^{5} + 9457921 T^{6} + 87809673 T^{7} + 771676330 T^{8} + 6305405664 T^{9} + 49251947027 T^{10} + 364030116410 T^{11} + 112656272119 p T^{12} + 17645114299194 T^{13} + 116355212282471 T^{14} + 739516336367702 T^{15} + 4569956585272404 T^{16} + 27358811998112340 T^{17} + 159737619179730687 T^{18} + 906765945452689140 T^{19} + 5031176656321627704 T^{20} + 27208096226621421724 T^{21} + \)\(14\!\cdots\!01\)\( T^{22} + \)\(74\!\cdots\!89\)\( T^{23} + \)\(37\!\cdots\!30\)\( T^{24} + \)\(18\!\cdots\!23\)\( T^{25} + \)\(90\!\cdots\!58\)\( T^{26} + \)\(18\!\cdots\!23\)\( p T^{27} + \)\(37\!\cdots\!30\)\( p^{2} T^{28} + \)\(74\!\cdots\!89\)\( p^{3} T^{29} + \)\(14\!\cdots\!01\)\( p^{4} T^{30} + 27208096226621421724 p^{5} T^{31} + 5031176656321627704 p^{6} T^{32} + 906765945452689140 p^{7} T^{33} + 159737619179730687 p^{8} T^{34} + 27358811998112340 p^{9} T^{35} + 4569956585272404 p^{10} T^{36} + 739516336367702 p^{11} T^{37} + 116355212282471 p^{12} T^{38} + 17645114299194 p^{13} T^{39} + 112656272119 p^{15} T^{40} + 364030116410 p^{15} T^{41} + 49251947027 p^{16} T^{42} + 6305405664 p^{17} T^{43} + 771676330 p^{18} T^{44} + 87809673 p^{19} T^{45} + 9457921 p^{20} T^{46} + 918304 p^{21} T^{47} + 83310 p^{22} T^{48} + 6381 p^{23} T^{49} + 450 p^{24} T^{50} + 22 p^{25} T^{51} + p^{26} T^{52} \)
29 \( 1 - 6 T + 392 T^{2} - 2401 T^{3} + 78302 T^{4} - 481078 T^{5} + 10592749 T^{6} - 64374673 T^{7} + 1087988711 T^{8} - 6473357493 T^{9} + 90184489573 T^{10} - 521644234122 T^{11} + 6263797361272 T^{12} - 35060001600096 T^{13} + 373815973370170 T^{14} - 2018584670996708 T^{15} + 19510994817012723 T^{16} - 101425963671856290 T^{17} + 902086273464094292 T^{18} - 4506346926131165452 T^{19} + 37287932481330335177 T^{20} - \)\(17\!\cdots\!56\)\( T^{21} + \)\(13\!\cdots\!79\)\( T^{22} - \)\(63\!\cdots\!72\)\( T^{23} + \)\(46\!\cdots\!83\)\( T^{24} - \)\(20\!\cdots\!67\)\( T^{25} + \)\(14\!\cdots\!98\)\( T^{26} - \)\(20\!\cdots\!67\)\( p T^{27} + \)\(46\!\cdots\!83\)\( p^{2} T^{28} - \)\(63\!\cdots\!72\)\( p^{3} T^{29} + \)\(13\!\cdots\!79\)\( p^{4} T^{30} - \)\(17\!\cdots\!56\)\( p^{5} T^{31} + 37287932481330335177 p^{6} T^{32} - 4506346926131165452 p^{7} T^{33} + 902086273464094292 p^{8} T^{34} - 101425963671856290 p^{9} T^{35} + 19510994817012723 p^{10} T^{36} - 2018584670996708 p^{11} T^{37} + 373815973370170 p^{12} T^{38} - 35060001600096 p^{13} T^{39} + 6263797361272 p^{14} T^{40} - 521644234122 p^{15} T^{41} + 90184489573 p^{16} T^{42} - 6473357493 p^{17} T^{43} + 1087988711 p^{18} T^{44} - 64374673 p^{19} T^{45} + 10592749 p^{20} T^{46} - 481078 p^{21} T^{47} + 78302 p^{22} T^{48} - 2401 p^{23} T^{49} + 392 p^{24} T^{50} - 6 p^{25} T^{51} + p^{26} T^{52} \)
31 \( 1 - 19 T + 573 T^{2} - 8439 T^{3} + 148270 T^{4} - 1813596 T^{5} + 23857861 T^{6} - 252391569 T^{7} + 2733328288 T^{8} - 25689001293 T^{9} + 240712402689 T^{10} - 2049414464128 T^{11} + 17145990324252 T^{12} - 134277702716468 T^{13} + 33093842647546 p T^{14} - 7484239556678636 T^{15} + 53137958966472274 T^{16} - 364954069244859148 T^{17} + 2441048532718743894 T^{18} - 15917689415028030486 T^{19} + \)\(10\!\cdots\!51\)\( T^{20} - \)\(63\!\cdots\!65\)\( T^{21} + \)\(38\!\cdots\!08\)\( T^{22} - \)\(22\!\cdots\!82\)\( T^{23} + \)\(13\!\cdots\!54\)\( T^{24} - \)\(77\!\cdots\!55\)\( T^{25} + \)\(43\!\cdots\!42\)\( T^{26} - \)\(77\!\cdots\!55\)\( p T^{27} + \)\(13\!\cdots\!54\)\( p^{2} T^{28} - \)\(22\!\cdots\!82\)\( p^{3} T^{29} + \)\(38\!\cdots\!08\)\( p^{4} T^{30} - \)\(63\!\cdots\!65\)\( p^{5} T^{31} + \)\(10\!\cdots\!51\)\( p^{6} T^{32} - 15917689415028030486 p^{7} T^{33} + 2441048532718743894 p^{8} T^{34} - 364954069244859148 p^{9} T^{35} + 53137958966472274 p^{10} T^{36} - 7484239556678636 p^{11} T^{37} + 33093842647546 p^{13} T^{38} - 134277702716468 p^{13} T^{39} + 17145990324252 p^{14} T^{40} - 2049414464128 p^{15} T^{41} + 240712402689 p^{16} T^{42} - 25689001293 p^{17} T^{43} + 2733328288 p^{18} T^{44} - 252391569 p^{19} T^{45} + 23857861 p^{20} T^{46} - 1813596 p^{21} T^{47} + 148270 p^{22} T^{48} - 8439 p^{23} T^{49} + 573 p^{24} T^{50} - 19 p^{25} T^{51} + p^{26} T^{52} \)
37 \( 1 - 20 T + 546 T^{2} - 8180 T^{3} + 134629 T^{4} - 1661619 T^{5} + 21052314 T^{6} - 225308154 T^{7} + 2410627536 T^{8} - 23113208559 T^{9} + 219259537312 T^{10} - 1927161721488 T^{11} + 16703212776743 T^{12} - 136864496504467 T^{13} + 1105386990473156 T^{14} - 8545953100790970 T^{15} + 65149188029237400 T^{16} - 479082137356815555 T^{17} + 3475194901053707010 T^{18} - 24430672508803117285 T^{19} + \)\(16\!\cdots\!14\)\( T^{20} - \)\(11\!\cdots\!07\)\( T^{21} + \)\(76\!\cdots\!26\)\( T^{22} - \)\(49\!\cdots\!67\)\( T^{23} + \)\(31\!\cdots\!53\)\( T^{24} - \)\(19\!\cdots\!89\)\( T^{25} + \)\(12\!\cdots\!80\)\( T^{26} - \)\(19\!\cdots\!89\)\( p T^{27} + \)\(31\!\cdots\!53\)\( p^{2} T^{28} - \)\(49\!\cdots\!67\)\( p^{3} T^{29} + \)\(76\!\cdots\!26\)\( p^{4} T^{30} - \)\(11\!\cdots\!07\)\( p^{5} T^{31} + \)\(16\!\cdots\!14\)\( p^{6} T^{32} - 24430672508803117285 p^{7} T^{33} + 3475194901053707010 p^{8} T^{34} - 479082137356815555 p^{9} T^{35} + 65149188029237400 p^{10} T^{36} - 8545953100790970 p^{11} T^{37} + 1105386990473156 p^{12} T^{38} - 136864496504467 p^{13} T^{39} + 16703212776743 p^{14} T^{40} - 1927161721488 p^{15} T^{41} + 219259537312 p^{16} T^{42} - 23113208559 p^{17} T^{43} + 2410627536 p^{18} T^{44} - 225308154 p^{19} T^{45} + 21052314 p^{20} T^{46} - 1661619 p^{21} T^{47} + 134629 p^{22} T^{48} - 8180 p^{23} T^{49} + 546 p^{24} T^{50} - 20 p^{25} T^{51} + p^{26} T^{52} \)
41 \( 1 - 25 T + 976 T^{2} - 18310 T^{3} + 415805 T^{4} - 6340536 T^{5} + 107379214 T^{6} - 1394313216 T^{7} + 19352559730 T^{8} - 220737812010 T^{9} + 2639384895607 T^{10} - 27052217228944 T^{11} + 287518546263103 T^{12} - 2694547144522855 T^{13} + 26008976955844387 T^{14} - 225869868818602460 T^{15} + 2010453124804563423 T^{16} - 16338368463024837306 T^{17} + \)\(13\!\cdots\!77\)\( T^{18} - \)\(10\!\cdots\!71\)\( T^{19} + \)\(80\!\cdots\!44\)\( T^{20} - \)\(58\!\cdots\!25\)\( T^{21} + \)\(43\!\cdots\!87\)\( T^{22} - \)\(29\!\cdots\!34\)\( T^{23} + \)\(20\!\cdots\!09\)\( T^{24} - \)\(13\!\cdots\!88\)\( T^{25} + \)\(21\!\cdots\!36\)\( p T^{26} - \)\(13\!\cdots\!88\)\( p T^{27} + \)\(20\!\cdots\!09\)\( p^{2} T^{28} - \)\(29\!\cdots\!34\)\( p^{3} T^{29} + \)\(43\!\cdots\!87\)\( p^{4} T^{30} - \)\(58\!\cdots\!25\)\( p^{5} T^{31} + \)\(80\!\cdots\!44\)\( p^{6} T^{32} - \)\(10\!\cdots\!71\)\( p^{7} T^{33} + \)\(13\!\cdots\!77\)\( p^{8} T^{34} - 16338368463024837306 p^{9} T^{35} + 2010453124804563423 p^{10} T^{36} - 225869868818602460 p^{11} T^{37} + 26008976955844387 p^{12} T^{38} - 2694547144522855 p^{13} T^{39} + 287518546263103 p^{14} T^{40} - 27052217228944 p^{15} T^{41} + 2639384895607 p^{16} T^{42} - 220737812010 p^{17} T^{43} + 19352559730 p^{18} T^{44} - 1394313216 p^{19} T^{45} + 107379214 p^{20} T^{46} - 6340536 p^{21} T^{47} + 415805 p^{22} T^{48} - 18310 p^{23} T^{49} + 976 p^{24} T^{50} - 25 p^{25} T^{51} + p^{26} T^{52} \)
43 \( 1 - 4 T + 514 T^{2} - 2042 T^{3} + 134214 T^{4} - 549355 T^{5} + 23740418 T^{6} - 102687879 T^{7} + 3196780873 T^{8} - 14819811548 T^{9} + 349105647383 T^{10} - 1742679965362 T^{11} + 32176579649520 T^{12} - 172481643990799 T^{13} + 2572856719149156 T^{14} - 14685685240985035 T^{15} + 182082817085574561 T^{16} - 1092725702722853684 T^{17} + 11575098119617967398 T^{18} - 71900529881159510646 T^{19} + \)\(66\!\cdots\!16\)\( T^{20} - \)\(42\!\cdots\!64\)\( T^{21} + \)\(35\!\cdots\!58\)\( T^{22} - \)\(22\!\cdots\!08\)\( T^{23} + \)\(17\!\cdots\!79\)\( T^{24} - \)\(10\!\cdots\!22\)\( T^{25} + \)\(76\!\cdots\!22\)\( T^{26} - \)\(10\!\cdots\!22\)\( p T^{27} + \)\(17\!\cdots\!79\)\( p^{2} T^{28} - \)\(22\!\cdots\!08\)\( p^{3} T^{29} + \)\(35\!\cdots\!58\)\( p^{4} T^{30} - \)\(42\!\cdots\!64\)\( p^{5} T^{31} + \)\(66\!\cdots\!16\)\( p^{6} T^{32} - 71900529881159510646 p^{7} T^{33} + 11575098119617967398 p^{8} T^{34} - 1092725702722853684 p^{9} T^{35} + 182082817085574561 p^{10} T^{36} - 14685685240985035 p^{11} T^{37} + 2572856719149156 p^{12} T^{38} - 172481643990799 p^{13} T^{39} + 32176579649520 p^{14} T^{40} - 1742679965362 p^{15} T^{41} + 349105647383 p^{16} T^{42} - 14819811548 p^{17} T^{43} + 3196780873 p^{18} T^{44} - 102687879 p^{19} T^{45} + 23740418 p^{20} T^{46} - 549355 p^{21} T^{47} + 134214 p^{22} T^{48} - 2042 p^{23} T^{49} + 514 p^{24} T^{50} - 4 p^{25} T^{51} + p^{26} T^{52} \)
47 \( 1 - 8 T + 591 T^{2} - 4673 T^{3} + 178438 T^{4} - 1397655 T^{5} + 36700610 T^{6} - 284811346 T^{7} + 5774410000 T^{8} - 44343461651 T^{9} + 739220023664 T^{10} - 5605360207845 T^{11} + 79935112212715 T^{12} - 596861114910326 T^{13} + 159212530388108 p T^{14} - 54841797518249558 T^{15} + 616785649601921666 T^{16} - 4420711143971075360 T^{17} + 45300767864763049244 T^{18} - \)\(31\!\cdots\!22\)\( T^{19} + \)\(29\!\cdots\!86\)\( T^{20} - \)\(43\!\cdots\!71\)\( p T^{21} + \)\(17\!\cdots\!19\)\( T^{22} - \)\(11\!\cdots\!46\)\( T^{23} + \)\(96\!\cdots\!46\)\( T^{24} - \)\(60\!\cdots\!17\)\( T^{25} + \)\(47\!\cdots\!56\)\( T^{26} - \)\(60\!\cdots\!17\)\( p T^{27} + \)\(96\!\cdots\!46\)\( p^{2} T^{28} - \)\(11\!\cdots\!46\)\( p^{3} T^{29} + \)\(17\!\cdots\!19\)\( p^{4} T^{30} - \)\(43\!\cdots\!71\)\( p^{6} T^{31} + \)\(29\!\cdots\!86\)\( p^{6} T^{32} - \)\(31\!\cdots\!22\)\( p^{7} T^{33} + 45300767864763049244 p^{8} T^{34} - 4420711143971075360 p^{9} T^{35} + 616785649601921666 p^{10} T^{36} - 54841797518249558 p^{11} T^{37} + 159212530388108 p^{13} T^{38} - 596861114910326 p^{13} T^{39} + 79935112212715 p^{14} T^{40} - 5605360207845 p^{15} T^{41} + 739220023664 p^{16} T^{42} - 44343461651 p^{17} T^{43} + 5774410000 p^{18} T^{44} - 284811346 p^{19} T^{45} + 36700610 p^{20} T^{46} - 1397655 p^{21} T^{47} + 178438 p^{22} T^{48} - 4673 p^{23} T^{49} + 591 p^{24} T^{50} - 8 p^{25} T^{51} + p^{26} T^{52} \)
53 \( 1 + 5 T + 565 T^{2} + 3501 T^{3} + 172239 T^{4} + 1216005 T^{5} + 37091297 T^{6} + 283352674 T^{7} + 6270419843 T^{8} + 50005307886 T^{9} + 878340204995 T^{10} + 7129257044900 T^{11} + 105235080812033 T^{12} + 853893137082898 T^{13} + 11005907807193822 T^{14} + 88143342647789876 T^{15} + 1018756446635655827 T^{16} + 7979463315192386490 T^{17} + 84281900155423572478 T^{18} + \)\(64\!\cdots\!32\)\( T^{19} + \)\(62\!\cdots\!20\)\( T^{20} + \)\(46\!\cdots\!47\)\( T^{21} + \)\(42\!\cdots\!26\)\( T^{22} + \)\(29\!\cdots\!07\)\( T^{23} + \)\(25\!\cdots\!02\)\( T^{24} + \)\(17\!\cdots\!73\)\( T^{25} + \)\(14\!\cdots\!42\)\( T^{26} + \)\(17\!\cdots\!73\)\( p T^{27} + \)\(25\!\cdots\!02\)\( p^{2} T^{28} + \)\(29\!\cdots\!07\)\( p^{3} T^{29} + \)\(42\!\cdots\!26\)\( p^{4} T^{30} + \)\(46\!\cdots\!47\)\( p^{5} T^{31} + \)\(62\!\cdots\!20\)\( p^{6} T^{32} + \)\(64\!\cdots\!32\)\( p^{7} T^{33} + 84281900155423572478 p^{8} T^{34} + 7979463315192386490 p^{9} T^{35} + 1018756446635655827 p^{10} T^{36} + 88143342647789876 p^{11} T^{37} + 11005907807193822 p^{12} T^{38} + 853893137082898 p^{13} T^{39} + 105235080812033 p^{14} T^{40} + 7129257044900 p^{15} T^{41} + 878340204995 p^{16} T^{42} + 50005307886 p^{17} T^{43} + 6270419843 p^{18} T^{44} + 283352674 p^{19} T^{45} + 37091297 p^{20} T^{46} + 1216005 p^{21} T^{47} + 172239 p^{22} T^{48} + 3501 p^{23} T^{49} + 565 p^{24} T^{50} + 5 p^{25} T^{51} + p^{26} T^{52} \)
59 \( 1 + 18 T + 818 T^{2} + 13594 T^{3} + 345115 T^{4} + 5210789 T^{5} + 97949598 T^{6} + 1342515357 T^{7} + 20780145149 T^{8} + 259852771843 T^{9} + 3490019368782 T^{10} + 40084472458319 T^{11} + 481325786913095 T^{12} + 5112374446741000 T^{13} + 55948085332114207 T^{14} + 553215654171291038 T^{15} + 5594511466634020530 T^{16} + 51844977677000497986 T^{17} + \)\(48\!\cdots\!75\)\( T^{18} + \)\(42\!\cdots\!06\)\( T^{19} + \)\(38\!\cdots\!56\)\( T^{20} + \)\(31\!\cdots\!37\)\( T^{21} + \)\(27\!\cdots\!04\)\( T^{22} + \)\(21\!\cdots\!95\)\( T^{23} + \)\(17\!\cdots\!66\)\( T^{24} + \)\(13\!\cdots\!52\)\( T^{25} + \)\(10\!\cdots\!04\)\( T^{26} + \)\(13\!\cdots\!52\)\( p T^{27} + \)\(17\!\cdots\!66\)\( p^{2} T^{28} + \)\(21\!\cdots\!95\)\( p^{3} T^{29} + \)\(27\!\cdots\!04\)\( p^{4} T^{30} + \)\(31\!\cdots\!37\)\( p^{5} T^{31} + \)\(38\!\cdots\!56\)\( p^{6} T^{32} + \)\(42\!\cdots\!06\)\( p^{7} T^{33} + \)\(48\!\cdots\!75\)\( p^{8} T^{34} + 51844977677000497986 p^{9} T^{35} + 5594511466634020530 p^{10} T^{36} + 553215654171291038 p^{11} T^{37} + 55948085332114207 p^{12} T^{38} + 5112374446741000 p^{13} T^{39} + 481325786913095 p^{14} T^{40} + 40084472458319 p^{15} T^{41} + 3490019368782 p^{16} T^{42} + 259852771843 p^{17} T^{43} + 20780145149 p^{18} T^{44} + 1342515357 p^{19} T^{45} + 97949598 p^{20} T^{46} + 5210789 p^{21} T^{47} + 345115 p^{22} T^{48} + 13594 p^{23} T^{49} + 818 p^{24} T^{50} + 18 p^{25} T^{51} + p^{26} T^{52} \)
61 \( 1 - 43 T + 1848 T^{2} - 51104 T^{3} + 1345290 T^{4} - 28524725 T^{5} + 575601079 T^{6} - 10117320213 T^{7} + 170186390160 T^{8} - 42460185301 p T^{9} + 37938238610354 T^{10} - 513841705027729 T^{11} + 6730410199332763 T^{12} - 82648151857460878 T^{13} + 985254450721438258 T^{14} - 11113947362041712667 T^{15} + \)\(12\!\cdots\!77\)\( T^{16} - \)\(12\!\cdots\!99\)\( T^{17} + \)\(13\!\cdots\!54\)\( T^{18} - \)\(12\!\cdots\!28\)\( T^{19} + \)\(12\!\cdots\!36\)\( T^{20} - \)\(11\!\cdots\!76\)\( T^{21} + \)\(99\!\cdots\!54\)\( T^{22} - \)\(85\!\cdots\!33\)\( T^{23} + \)\(72\!\cdots\!42\)\( T^{24} - \)\(58\!\cdots\!92\)\( T^{25} + \)\(46\!\cdots\!08\)\( T^{26} - \)\(58\!\cdots\!92\)\( p T^{27} + \)\(72\!\cdots\!42\)\( p^{2} T^{28} - \)\(85\!\cdots\!33\)\( p^{3} T^{29} + \)\(99\!\cdots\!54\)\( p^{4} T^{30} - \)\(11\!\cdots\!76\)\( p^{5} T^{31} + \)\(12\!\cdots\!36\)\( p^{6} T^{32} - \)\(12\!\cdots\!28\)\( p^{7} T^{33} + \)\(13\!\cdots\!54\)\( p^{8} T^{34} - \)\(12\!\cdots\!99\)\( p^{9} T^{35} + \)\(12\!\cdots\!77\)\( p^{10} T^{36} - 11113947362041712667 p^{11} T^{37} + 985254450721438258 p^{12} T^{38} - 82648151857460878 p^{13} T^{39} + 6730410199332763 p^{14} T^{40} - 513841705027729 p^{15} T^{41} + 37938238610354 p^{16} T^{42} - 42460185301 p^{18} T^{43} + 170186390160 p^{18} T^{44} - 10117320213 p^{19} T^{45} + 575601079 p^{20} T^{46} - 28524725 p^{21} T^{47} + 1345290 p^{22} T^{48} - 51104 p^{23} T^{49} + 1848 p^{24} T^{50} - 43 p^{25} T^{51} + p^{26} T^{52} \)
67 \( 1 - 5 T + 814 T^{2} - 4478 T^{3} + 341665 T^{4} - 1997385 T^{5} + 97975177 T^{6} - 595584161 T^{7} + 21493258810 T^{8} - 133822277673 T^{9} + 3833722517484 T^{10} - 24171905691343 T^{11} + 577402398534893 T^{12} - 3653370147874373 T^{13} + 75318515417736652 T^{14} - 474593680464763413 T^{15} + 8662788468296758752 T^{16} - 53992357312291735524 T^{17} + \)\(88\!\cdots\!80\)\( T^{18} - \)\(54\!\cdots\!07\)\( T^{19} + \)\(82\!\cdots\!42\)\( T^{20} - \)\(49\!\cdots\!56\)\( T^{21} + \)\(69\!\cdots\!31\)\( T^{22} - \)\(40\!\cdots\!96\)\( T^{23} + \)\(53\!\cdots\!97\)\( T^{24} - \)\(29\!\cdots\!66\)\( T^{25} + \)\(37\!\cdots\!60\)\( T^{26} - \)\(29\!\cdots\!66\)\( p T^{27} + \)\(53\!\cdots\!97\)\( p^{2} T^{28} - \)\(40\!\cdots\!96\)\( p^{3} T^{29} + \)\(69\!\cdots\!31\)\( p^{4} T^{30} - \)\(49\!\cdots\!56\)\( p^{5} T^{31} + \)\(82\!\cdots\!42\)\( p^{6} T^{32} - \)\(54\!\cdots\!07\)\( p^{7} T^{33} + \)\(88\!\cdots\!80\)\( p^{8} T^{34} - 53992357312291735524 p^{9} T^{35} + 8662788468296758752 p^{10} T^{36} - 474593680464763413 p^{11} T^{37} + 75318515417736652 p^{12} T^{38} - 3653370147874373 p^{13} T^{39} + 577402398534893 p^{14} T^{40} - 24171905691343 p^{15} T^{41} + 3833722517484 p^{16} T^{42} - 133822277673 p^{17} T^{43} + 21493258810 p^{18} T^{44} - 595584161 p^{19} T^{45} + 97975177 p^{20} T^{46} - 1997385 p^{21} T^{47} + 341665 p^{22} T^{48} - 4478 p^{23} T^{49} + 814 p^{24} T^{50} - 5 p^{25} T^{51} + p^{26} T^{52} \)
71 \( 1 + T + 1126 T^{2} + 1308 T^{3} + 623407 T^{4} + 726184 T^{5} + 226294141 T^{6} + 235199413 T^{7} + 60595635070 T^{8} + 49520145221 T^{9} + 12775848184756 T^{10} + 6796462365306 T^{11} + 2212482429483205 T^{12} + 482598862588590 T^{13} + 324406227210301927 T^{14} - 29193222490805065 T^{15} + 41213188752921021202 T^{16} - 14891190474840065463 T^{17} + \)\(46\!\cdots\!34\)\( T^{18} - \)\(26\!\cdots\!50\)\( T^{19} + \)\(46\!\cdots\!86\)\( T^{20} - \)\(33\!\cdots\!09\)\( T^{21} + \)\(41\!\cdots\!68\)\( T^{22} - \)\(32\!\cdots\!55\)\( T^{23} + \)\(34\!\cdots\!89\)\( T^{24} - \)\(27\!\cdots\!35\)\( T^{25} + \)\(25\!\cdots\!12\)\( T^{26} - \)\(27\!\cdots\!35\)\( p T^{27} + \)\(34\!\cdots\!89\)\( p^{2} T^{28} - \)\(32\!\cdots\!55\)\( p^{3} T^{29} + \)\(41\!\cdots\!68\)\( p^{4} T^{30} - \)\(33\!\cdots\!09\)\( p^{5} T^{31} + \)\(46\!\cdots\!86\)\( p^{6} T^{32} - \)\(26\!\cdots\!50\)\( p^{7} T^{33} + \)\(46\!\cdots\!34\)\( p^{8} T^{34} - 14891190474840065463 p^{9} T^{35} + 41213188752921021202 p^{10} T^{36} - 29193222490805065 p^{11} T^{37} + 324406227210301927 p^{12} T^{38} + 482598862588590 p^{13} T^{39} + 2212482429483205 p^{14} T^{40} + 6796462365306 p^{15} T^{41} + 12775848184756 p^{16} T^{42} + 49520145221 p^{17} T^{43} + 60595635070 p^{18} T^{44} + 235199413 p^{19} T^{45} + 226294141 p^{20} T^{46} + 726184 p^{21} T^{47} + 623407 p^{22} T^{48} + 1308 p^{23} T^{49} + 1126 p^{24} T^{50} + p^{25} T^{51} + p^{26} T^{52} \)
73 \( 1 - 22 T + 1004 T^{2} - 17117 T^{3} + 466510 T^{4} - 6730245 T^{5} + 140513376 T^{6} - 1794074895 T^{7} + 31436814124 T^{8} - 364740776377 T^{9} + 5615721712084 T^{10} - 60248091493856 T^{11} + 837495236401165 T^{12} - 115220917284466 p T^{13} + 107488302270725945 T^{14} - 1019911725504883527 T^{15} + 12143795681252944561 T^{16} - \)\(10\!\cdots\!43\)\( T^{17} + \)\(12\!\cdots\!21\)\( T^{18} - \)\(10\!\cdots\!02\)\( T^{19} + \)\(11\!\cdots\!06\)\( T^{20} - \)\(94\!\cdots\!54\)\( T^{21} + \)\(96\!\cdots\!36\)\( T^{22} - \)\(77\!\cdots\!82\)\( T^{23} + \)\(77\!\cdots\!16\)\( T^{24} - \)\(60\!\cdots\!70\)\( T^{25} + \)\(57\!\cdots\!74\)\( T^{26} - \)\(60\!\cdots\!70\)\( p T^{27} + \)\(77\!\cdots\!16\)\( p^{2} T^{28} - \)\(77\!\cdots\!82\)\( p^{3} T^{29} + \)\(96\!\cdots\!36\)\( p^{4} T^{30} - \)\(94\!\cdots\!54\)\( p^{5} T^{31} + \)\(11\!\cdots\!06\)\( p^{6} T^{32} - \)\(10\!\cdots\!02\)\( p^{7} T^{33} + \)\(12\!\cdots\!21\)\( p^{8} T^{34} - \)\(10\!\cdots\!43\)\( p^{9} T^{35} + 12143795681252944561 p^{10} T^{36} - 1019911725504883527 p^{11} T^{37} + 107488302270725945 p^{12} T^{38} - 115220917284466 p^{14} T^{39} + 837495236401165 p^{14} T^{40} - 60248091493856 p^{15} T^{41} + 5615721712084 p^{16} T^{42} - 364740776377 p^{17} T^{43} + 31436814124 p^{18} T^{44} - 1794074895 p^{19} T^{45} + 140513376 p^{20} T^{46} - 6730245 p^{21} T^{47} + 466510 p^{22} T^{48} - 17117 p^{23} T^{49} + 1004 p^{24} T^{50} - 22 p^{25} T^{51} + p^{26} T^{52} \)
79 \( 1 - 16 T + 1366 T^{2} - 245 p T^{3} + 895023 T^{4} - 11415652 T^{5} + 377925052 T^{6} - 4397043686 T^{7} + 116488484887 T^{8} - 1250054831851 T^{9} + 28125662041939 T^{10} - 280959212923055 T^{11} + 5568355942761631 T^{12} - 52169863460778754 T^{13} + 933067158567629149 T^{14} - 8246959629052579002 T^{15} + \)\(13\!\cdots\!29\)\( T^{16} - \)\(11\!\cdots\!73\)\( T^{17} + \)\(17\!\cdots\!24\)\( T^{18} - \)\(13\!\cdots\!74\)\( T^{19} + \)\(19\!\cdots\!47\)\( T^{20} - \)\(14\!\cdots\!54\)\( T^{21} + \)\(19\!\cdots\!03\)\( T^{22} - \)\(14\!\cdots\!48\)\( T^{23} + \)\(18\!\cdots\!22\)\( T^{24} - \)\(12\!\cdots\!16\)\( T^{25} + \)\(15\!\cdots\!42\)\( T^{26} - \)\(12\!\cdots\!16\)\( p T^{27} + \)\(18\!\cdots\!22\)\( p^{2} T^{28} - \)\(14\!\cdots\!48\)\( p^{3} T^{29} + \)\(19\!\cdots\!03\)\( p^{4} T^{30} - \)\(14\!\cdots\!54\)\( p^{5} T^{31} + \)\(19\!\cdots\!47\)\( p^{6} T^{32} - \)\(13\!\cdots\!74\)\( p^{7} T^{33} + \)\(17\!\cdots\!24\)\( p^{8} T^{34} - \)\(11\!\cdots\!73\)\( p^{9} T^{35} + \)\(13\!\cdots\!29\)\( p^{10} T^{36} - 8246959629052579002 p^{11} T^{37} + 933067158567629149 p^{12} T^{38} - 52169863460778754 p^{13} T^{39} + 5568355942761631 p^{14} T^{40} - 280959212923055 p^{15} T^{41} + 28125662041939 p^{16} T^{42} - 1250054831851 p^{17} T^{43} + 116488484887 p^{18} T^{44} - 4397043686 p^{19} T^{45} + 377925052 p^{20} T^{46} - 11415652 p^{21} T^{47} + 895023 p^{22} T^{48} - 245 p^{24} T^{49} + 1366 p^{24} T^{50} - 16 p^{25} T^{51} + p^{26} T^{52} \)
83 \( 1 + 19 T + 1252 T^{2} + 21084 T^{3} + 762292 T^{4} + 11500199 T^{5} + 301549041 T^{6} + 4115234134 T^{7} + 87419395275 T^{8} + 1088467660538 T^{9} + 19871342862012 T^{10} + 227433761163785 T^{11} + 3702016739363280 T^{12} + 39206742525470963 T^{13} + 583654969144642058 T^{14} + 5754578729412660310 T^{15} + 79841300514619305232 T^{16} + \)\(73\!\cdots\!52\)\( T^{17} + \)\(96\!\cdots\!16\)\( T^{18} + \)\(84\!\cdots\!77\)\( T^{19} + \)\(10\!\cdots\!93\)\( T^{20} + \)\(87\!\cdots\!58\)\( T^{21} + \)\(10\!\cdots\!54\)\( T^{22} + \)\(83\!\cdots\!88\)\( T^{23} + \)\(97\!\cdots\!71\)\( T^{24} + \)\(73\!\cdots\!01\)\( T^{25} + \)\(83\!\cdots\!82\)\( T^{26} + \)\(73\!\cdots\!01\)\( p T^{27} + \)\(97\!\cdots\!71\)\( p^{2} T^{28} + \)\(83\!\cdots\!88\)\( p^{3} T^{29} + \)\(10\!\cdots\!54\)\( p^{4} T^{30} + \)\(87\!\cdots\!58\)\( p^{5} T^{31} + \)\(10\!\cdots\!93\)\( p^{6} T^{32} + \)\(84\!\cdots\!77\)\( p^{7} T^{33} + \)\(96\!\cdots\!16\)\( p^{8} T^{34} + \)\(73\!\cdots\!52\)\( p^{9} T^{35} + 79841300514619305232 p^{10} T^{36} + 5754578729412660310 p^{11} T^{37} + 583654969144642058 p^{12} T^{38} + 39206742525470963 p^{13} T^{39} + 3702016739363280 p^{14} T^{40} + 227433761163785 p^{15} T^{41} + 19871342862012 p^{16} T^{42} + 1088467660538 p^{17} T^{43} + 87419395275 p^{18} T^{44} + 4115234134 p^{19} T^{45} + 301549041 p^{20} T^{46} + 11500199 p^{21} T^{47} + 762292 p^{22} T^{48} + 21084 p^{23} T^{49} + 1252 p^{24} T^{50} + 19 p^{25} T^{51} + p^{26} T^{52} \)
89 \( 1 - 49 T + 2339 T^{2} - 74400 T^{3} + 2210989 T^{4} - 54325536 T^{5} + 1252807650 T^{6} - 25688345270 T^{7} + 498690321223 T^{8} - 8907689373082 T^{9} + 151802616903456 T^{10} - 2426390032948819 T^{11} + 37227686380378050 T^{12} - 542130751706767129 T^{13} + 7613650177128579896 T^{14} - \)\(10\!\cdots\!89\)\( T^{15} + \)\(13\!\cdots\!80\)\( T^{16} - \)\(16\!\cdots\!25\)\( T^{17} + \)\(20\!\cdots\!49\)\( T^{18} - \)\(23\!\cdots\!90\)\( T^{19} + \)\(26\!\cdots\!98\)\( T^{20} - \)\(29\!\cdots\!11\)\( T^{21} + \)\(31\!\cdots\!10\)\( T^{22} - \)\(33\!\cdots\!77\)\( T^{23} + \)\(33\!\cdots\!23\)\( T^{24} - \)\(32\!\cdots\!19\)\( T^{25} + \)\(31\!\cdots\!00\)\( T^{26} - \)\(32\!\cdots\!19\)\( p T^{27} + \)\(33\!\cdots\!23\)\( p^{2} T^{28} - \)\(33\!\cdots\!77\)\( p^{3} T^{29} + \)\(31\!\cdots\!10\)\( p^{4} T^{30} - \)\(29\!\cdots\!11\)\( p^{5} T^{31} + \)\(26\!\cdots\!98\)\( p^{6} T^{32} - \)\(23\!\cdots\!90\)\( p^{7} T^{33} + \)\(20\!\cdots\!49\)\( p^{8} T^{34} - \)\(16\!\cdots\!25\)\( p^{9} T^{35} + \)\(13\!\cdots\!80\)\( p^{10} T^{36} - \)\(10\!\cdots\!89\)\( p^{11} T^{37} + 7613650177128579896 p^{12} T^{38} - 542130751706767129 p^{13} T^{39} + 37227686380378050 p^{14} T^{40} - 2426390032948819 p^{15} T^{41} + 151802616903456 p^{16} T^{42} - 8907689373082 p^{17} T^{43} + 498690321223 p^{18} T^{44} - 25688345270 p^{19} T^{45} + 1252807650 p^{20} T^{46} - 54325536 p^{21} T^{47} + 2210989 p^{22} T^{48} - 74400 p^{23} T^{49} + 2339 p^{24} T^{50} - 49 p^{25} T^{51} + p^{26} T^{52} \)
97 \( 1 - 25 T + 1239 T^{2} - 27619 T^{3} + 803977 T^{4} - 15974866 T^{5} + 356168790 T^{6} - 6384561206 T^{7} + 120043498963 T^{8} - 1965995891697 T^{9} + 32671533011495 T^{10} - 494128651625603 T^{11} + 7454524451434571 T^{12} - 105023542465769154 T^{13} + 1462674238614125863 T^{14} - 19329398787216040573 T^{15} + \)\(25\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!93\)\( T^{17} + \)\(38\!\cdots\!79\)\( T^{18} - \)\(45\!\cdots\!46\)\( T^{19} + \)\(52\!\cdots\!84\)\( T^{20} - \)\(58\!\cdots\!56\)\( T^{21} + \)\(64\!\cdots\!34\)\( T^{22} - \)\(68\!\cdots\!72\)\( T^{23} + \)\(72\!\cdots\!74\)\( T^{24} - \)\(73\!\cdots\!66\)\( T^{25} + \)\(73\!\cdots\!30\)\( T^{26} - \)\(73\!\cdots\!66\)\( p T^{27} + \)\(72\!\cdots\!74\)\( p^{2} T^{28} - \)\(68\!\cdots\!72\)\( p^{3} T^{29} + \)\(64\!\cdots\!34\)\( p^{4} T^{30} - \)\(58\!\cdots\!56\)\( p^{5} T^{31} + \)\(52\!\cdots\!84\)\( p^{6} T^{32} - \)\(45\!\cdots\!46\)\( p^{7} T^{33} + \)\(38\!\cdots\!79\)\( p^{8} T^{34} - \)\(31\!\cdots\!93\)\( p^{9} T^{35} + \)\(25\!\cdots\!29\)\( p^{10} T^{36} - 19329398787216040573 p^{11} T^{37} + 1462674238614125863 p^{12} T^{38} - 105023542465769154 p^{13} T^{39} + 7454524451434571 p^{14} T^{40} - 494128651625603 p^{15} T^{41} + 32671533011495 p^{16} T^{42} - 1965995891697 p^{17} T^{43} + 120043498963 p^{18} T^{44} - 6384561206 p^{19} T^{45} + 356168790 p^{20} T^{46} - 15974866 p^{21} T^{47} + 803977 p^{22} T^{48} - 27619 p^{23} T^{49} + 1239 p^{24} T^{50} - 25 p^{25} T^{51} + p^{26} T^{52} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{52} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.05002211511824257549310848198, −1.00614588856557401035785748015, −0.987414549523421277293288386132, −0.983002249500871371813378093838, −0.973057917069258482134063118319, −0.911381265721873903418492593212, −0.876292476308589115466420585488, −0.828195383406123268083611027347, −0.806258213282553919356179206094, −0.75485184158611352864356976433, −0.71097408928670303933176293006, −0.62608620390795535423288063813, −0.60077304548996441154510847183, −0.58491676842469093239322489245, −0.57799063690462197429439185325, −0.56562823562890893731280143581, −0.56266378842928157673989980928, −0.53047566190904852154934483446, −0.49854455193298883356982225223, −0.48860877890919594995270846709, −0.29657642886246120847937450251, −0.27944394733648782595891829883, −0.25799222473851706727821619090, −0.18253035624982413178492872370, −0.06798201048333432450694230362, 0.06798201048333432450694230362, 0.18253035624982413178492872370, 0.25799222473851706727821619090, 0.27944394733648782595891829883, 0.29657642886246120847937450251, 0.48860877890919594995270846709, 0.49854455193298883356982225223, 0.53047566190904852154934483446, 0.56266378842928157673989980928, 0.56562823562890893731280143581, 0.57799063690462197429439185325, 0.58491676842469093239322489245, 0.60077304548996441154510847183, 0.62608620390795535423288063813, 0.71097408928670303933176293006, 0.75485184158611352864356976433, 0.806258213282553919356179206094, 0.828195383406123268083611027347, 0.876292476308589115466420585488, 0.911381265721873903418492593212, 0.973057917069258482134063118319, 0.983002249500871371813378093838, 0.987414549523421277293288386132, 1.00614588856557401035785748015, 1.05002211511824257549310848198

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

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