Properties

Degree 60
Conductor $ 2^{30} \cdot 19^{30} \cdot 211^{30} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 30

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 30·2-s − 10·3-s + 465·4-s − 12·5-s − 300·6-s − 15·7-s + 4.96e3·8-s + 10·9-s − 360·10-s − 17·11-s − 4.65e3·12-s − 19·13-s − 450·14-s + 120·15-s + 4.09e4·16-s − 18·17-s + 300·18-s + 30·19-s − 5.58e3·20-s + 150·21-s − 510·22-s − 15·23-s − 4.96e4·24-s − 5·25-s − 570·26-s + 231·27-s − 6.97e3·28-s + ⋯
L(s)  = 1  + 21.2·2-s − 5.77·3-s + 232.5·4-s − 5.36·5-s − 122.·6-s − 5.66·7-s + 1.75e3·8-s + 10/3·9-s − 113.·10-s − 5.12·11-s − 1.34e3·12-s − 5.26·13-s − 120.·14-s + 30.9·15-s + 1.02e4·16-s − 4.36·17-s + 70.7·18-s + 6.88·19-s − 1.24e3·20-s + 32.7·21-s − 108.·22-s − 3.12·23-s − 1.01e4·24-s − 25-s − 111.·26-s + 44.4·27-s − 1.31e3·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{30} \cdot 211^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{30} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{30} \cdot 211^{30}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{30} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(60\)
\( N \)  =  \(2^{30} \cdot 19^{30} \cdot 211^{30}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8018} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  30
Selberg data  =  $(60,\ 2^{30} \cdot 19^{30} \cdot 211^{30} ,\ ( \ : [1/2]^{30} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \(F_p\) is a polynomial of degree 60. If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 59.
$p$$F_p$
bad2 \( ( 1 - T )^{30} \)
19 \( ( 1 - T )^{30} \)
211 \( ( 1 + T )^{30} \)
good3 \( 1 + 10 T + 10 p^{2} T^{2} + 569 T^{3} + 3257 T^{4} + 5284 p T^{5} + 71273 T^{6} + 289765 T^{7} + 1104997 T^{8} + 3919678 T^{9} + 13183991 T^{10} + 41903945 T^{11} + 127283212 T^{12} + 368956607 T^{13} + 1028013392 T^{14} + 2751679909 T^{15} + 2370207110 p T^{16} + 17735471434 T^{17} + 42848106451 T^{18} + 100264253129 T^{19} + 227835268300 T^{20} + 502729450532 T^{21} + 1079323370635 T^{22} + 83496210958 p^{3} T^{23} + 1529401947478 p T^{24} + 9097258928513 T^{25} + 1954867807630 p^{2} T^{26} + 33179231330666 T^{27} + 2261929023152 p^{3} T^{28} + 109675364145292 T^{29} + 192303233680996 T^{30} + 109675364145292 p T^{31} + 2261929023152 p^{5} T^{32} + 33179231330666 p^{3} T^{33} + 1954867807630 p^{6} T^{34} + 9097258928513 p^{5} T^{35} + 1529401947478 p^{7} T^{36} + 83496210958 p^{10} T^{37} + 1079323370635 p^{8} T^{38} + 502729450532 p^{9} T^{39} + 227835268300 p^{10} T^{40} + 100264253129 p^{11} T^{41} + 42848106451 p^{12} T^{42} + 17735471434 p^{13} T^{43} + 2370207110 p^{15} T^{44} + 2751679909 p^{15} T^{45} + 1028013392 p^{16} T^{46} + 368956607 p^{17} T^{47} + 127283212 p^{18} T^{48} + 41903945 p^{19} T^{49} + 13183991 p^{20} T^{50} + 3919678 p^{21} T^{51} + 1104997 p^{22} T^{52} + 289765 p^{23} T^{53} + 71273 p^{24} T^{54} + 5284 p^{26} T^{55} + 3257 p^{26} T^{56} + 569 p^{27} T^{57} + 10 p^{30} T^{58} + 10 p^{29} T^{59} + p^{30} T^{60} \)
5 \( 1 + 12 T + 149 T^{2} + 1194 T^{3} + 9164 T^{4} + 57317 T^{5} + 68117 p T^{6} + 1786898 T^{7} + 8931886 T^{8} + 40940961 T^{9} + 179606952 T^{10} + 737939603 T^{11} + 582881147 p T^{12} + 10921243067 T^{13} + 7896980724 p T^{14} + 136625226427 T^{15} + 457498752052 T^{16} + 1474997135729 T^{17} + 4613416505034 T^{18} + 13952684358058 T^{19} + 41018408080166 T^{20} + 116965034325847 T^{21} + 324706079011991 T^{22} + 876321176749557 T^{23} + 461035252269761 p T^{24} + 5904473375697156 T^{25} + 14753485699349012 T^{26} + 35934447270761621 T^{27} + 85428474241610239 T^{28} + 7923653348563063 p^{2} T^{29} + 448457325372714579 T^{30} + 7923653348563063 p^{3} T^{31} + 85428474241610239 p^{2} T^{32} + 35934447270761621 p^{3} T^{33} + 14753485699349012 p^{4} T^{34} + 5904473375697156 p^{5} T^{35} + 461035252269761 p^{7} T^{36} + 876321176749557 p^{7} T^{37} + 324706079011991 p^{8} T^{38} + 116965034325847 p^{9} T^{39} + 41018408080166 p^{10} T^{40} + 13952684358058 p^{11} T^{41} + 4613416505034 p^{12} T^{42} + 1474997135729 p^{13} T^{43} + 457498752052 p^{14} T^{44} + 136625226427 p^{15} T^{45} + 7896980724 p^{17} T^{46} + 10921243067 p^{17} T^{47} + 582881147 p^{19} T^{48} + 737939603 p^{19} T^{49} + 179606952 p^{20} T^{50} + 40940961 p^{21} T^{51} + 8931886 p^{22} T^{52} + 1786898 p^{23} T^{53} + 68117 p^{25} T^{54} + 57317 p^{25} T^{55} + 9164 p^{26} T^{56} + 1194 p^{27} T^{57} + 149 p^{28} T^{58} + 12 p^{29} T^{59} + p^{30} T^{60} \)
7 \( 1 + 15 T + 218 T^{2} + 2122 T^{3} + 19255 T^{4} + 144901 T^{5} + 1021770 T^{6} + 6417354 T^{7} + 38095867 T^{8} + 29749192 p T^{9} + 154870407 p T^{10} + 5294024409 T^{11} + 24770333179 T^{12} + 109985128114 T^{13} + 470167569619 T^{14} + 1922347320518 T^{15} + 7596387327030 T^{16} + 4124660033190 p T^{17} + 106397068387201 T^{18} + 378709937435331 T^{19} + 1310309510815174 T^{20} + 4393208022897693 T^{21} + 14348458607697091 T^{22} + 45525985248695529 T^{23} + 140953178666431359 T^{24} + 424764534696758600 T^{25} + 1250701074180659840 T^{26} + 3589255503957415328 T^{27} + 10073402574185070115 T^{28} + 3939574853415585430 p T^{29} + 73864799365989259248 T^{30} + 3939574853415585430 p^{2} T^{31} + 10073402574185070115 p^{2} T^{32} + 3589255503957415328 p^{3} T^{33} + 1250701074180659840 p^{4} T^{34} + 424764534696758600 p^{5} T^{35} + 140953178666431359 p^{6} T^{36} + 45525985248695529 p^{7} T^{37} + 14348458607697091 p^{8} T^{38} + 4393208022897693 p^{9} T^{39} + 1310309510815174 p^{10} T^{40} + 378709937435331 p^{11} T^{41} + 106397068387201 p^{12} T^{42} + 4124660033190 p^{14} T^{43} + 7596387327030 p^{14} T^{44} + 1922347320518 p^{15} T^{45} + 470167569619 p^{16} T^{46} + 109985128114 p^{17} T^{47} + 24770333179 p^{18} T^{48} + 5294024409 p^{19} T^{49} + 154870407 p^{21} T^{50} + 29749192 p^{22} T^{51} + 38095867 p^{22} T^{52} + 6417354 p^{23} T^{53} + 1021770 p^{24} T^{54} + 144901 p^{25} T^{55} + 19255 p^{26} T^{56} + 2122 p^{27} T^{57} + 218 p^{28} T^{58} + 15 p^{29} T^{59} + p^{30} T^{60} \)
11 \( 1 + 17 T + 326 T^{2} + 3852 T^{3} + 45340 T^{4} + 424114 T^{5} + 3852308 T^{6} + 30398252 T^{7} + 231687655 T^{8} + 1600269521 T^{9} + 10678380215 T^{10} + 66111105004 T^{11} + 396140736135 T^{12} + 2234549677769 T^{13} + 12225434659004 T^{14} + 63574330801030 T^{15} + 321307266332022 T^{16} + 141259008459499 p T^{17} + 7316536217917614 T^{18} + 33123901912667388 T^{19} + 146236799866429612 T^{20} + 622932777726225315 T^{21} + 2590946276634031595 T^{22} + 10424780517434675229 T^{23} + 40996541086737342659 T^{24} + \)\(15\!\cdots\!18\)\( T^{25} + \)\(58\!\cdots\!19\)\( T^{26} + \)\(21\!\cdots\!84\)\( T^{27} + \)\(74\!\cdots\!50\)\( T^{28} + \)\(25\!\cdots\!07\)\( T^{29} + \)\(86\!\cdots\!10\)\( T^{30} + \)\(25\!\cdots\!07\)\( p T^{31} + \)\(74\!\cdots\!50\)\( p^{2} T^{32} + \)\(21\!\cdots\!84\)\( p^{3} T^{33} + \)\(58\!\cdots\!19\)\( p^{4} T^{34} + \)\(15\!\cdots\!18\)\( p^{5} T^{35} + 40996541086737342659 p^{6} T^{36} + 10424780517434675229 p^{7} T^{37} + 2590946276634031595 p^{8} T^{38} + 622932777726225315 p^{9} T^{39} + 146236799866429612 p^{10} T^{40} + 33123901912667388 p^{11} T^{41} + 7316536217917614 p^{12} T^{42} + 141259008459499 p^{14} T^{43} + 321307266332022 p^{14} T^{44} + 63574330801030 p^{15} T^{45} + 12225434659004 p^{16} T^{46} + 2234549677769 p^{17} T^{47} + 396140736135 p^{18} T^{48} + 66111105004 p^{19} T^{49} + 10678380215 p^{20} T^{50} + 1600269521 p^{21} T^{51} + 231687655 p^{22} T^{52} + 30398252 p^{23} T^{53} + 3852308 p^{24} T^{54} + 424114 p^{25} T^{55} + 45340 p^{26} T^{56} + 3852 p^{27} T^{57} + 326 p^{28} T^{58} + 17 p^{29} T^{59} + p^{30} T^{60} \)
13 \( 1 + 19 T + 383 T^{2} + 4955 T^{3} + 62160 T^{4} + 633755 T^{5} + 6184344 T^{6} + 53163722 T^{7} + 437570868 T^{8} + 3294582958 T^{9} + 23824433976 T^{10} + 160919110085 T^{11} + 1047491500810 T^{12} + 6450759075757 T^{13} + 38403424182081 T^{14} + 218128789682288 T^{15} + 92373820606042 p T^{16} + 6344525653171689 T^{17} + 32559790100408045 T^{18} + 161033070782598239 T^{19} + 774971684389156590 T^{20} + 3605268142776232496 T^{21} + 16342999283492545895 T^{22} + 71775732533189854170 T^{23} + \)\(30\!\cdots\!35\)\( T^{24} + \)\(12\!\cdots\!58\)\( T^{25} + \)\(51\!\cdots\!08\)\( T^{26} + \)\(20\!\cdots\!64\)\( T^{27} + \)\(78\!\cdots\!77\)\( T^{28} + \)\(29\!\cdots\!48\)\( T^{29} + \)\(10\!\cdots\!14\)\( T^{30} + \)\(29\!\cdots\!48\)\( p T^{31} + \)\(78\!\cdots\!77\)\( p^{2} T^{32} + \)\(20\!\cdots\!64\)\( p^{3} T^{33} + \)\(51\!\cdots\!08\)\( p^{4} T^{34} + \)\(12\!\cdots\!58\)\( p^{5} T^{35} + \)\(30\!\cdots\!35\)\( p^{6} T^{36} + 71775732533189854170 p^{7} T^{37} + 16342999283492545895 p^{8} T^{38} + 3605268142776232496 p^{9} T^{39} + 774971684389156590 p^{10} T^{40} + 161033070782598239 p^{11} T^{41} + 32559790100408045 p^{12} T^{42} + 6344525653171689 p^{13} T^{43} + 92373820606042 p^{15} T^{44} + 218128789682288 p^{15} T^{45} + 38403424182081 p^{16} T^{46} + 6450759075757 p^{17} T^{47} + 1047491500810 p^{18} T^{48} + 160919110085 p^{19} T^{49} + 23824433976 p^{20} T^{50} + 3294582958 p^{21} T^{51} + 437570868 p^{22} T^{52} + 53163722 p^{23} T^{53} + 6184344 p^{24} T^{54} + 633755 p^{25} T^{55} + 62160 p^{26} T^{56} + 4955 p^{27} T^{57} + 383 p^{28} T^{58} + 19 p^{29} T^{59} + p^{30} T^{60} \)
17 \( 1 + 18 T + 451 T^{2} + 5963 T^{3} + 89534 T^{4} + 960757 T^{5} + 10992895 T^{6} + 100876534 T^{7} + 961467409 T^{8} + 7787780347 T^{9} + 64698052536 T^{10} + 472361760094 T^{11} + 3514474273942 T^{12} + 23472878384759 T^{13} + 159235686840698 T^{14} + 983583460343115 T^{15} + 6161351062217123 T^{16} + 35493179091870666 T^{17} + 207244989441471466 T^{18} + 1120754722586976596 T^{19} + 6144074354695440490 T^{20} + 31356507933851618832 T^{21} + \)\(16\!\cdots\!52\)\( T^{22} + \)\(78\!\cdots\!33\)\( T^{23} + \)\(38\!\cdots\!19\)\( T^{24} + \)\(17\!\cdots\!50\)\( T^{25} + \)\(82\!\cdots\!61\)\( T^{26} + \)\(36\!\cdots\!08\)\( T^{27} + \)\(16\!\cdots\!66\)\( T^{28} + \)\(67\!\cdots\!71\)\( T^{29} + \)\(28\!\cdots\!66\)\( T^{30} + \)\(67\!\cdots\!71\)\( p T^{31} + \)\(16\!\cdots\!66\)\( p^{2} T^{32} + \)\(36\!\cdots\!08\)\( p^{3} T^{33} + \)\(82\!\cdots\!61\)\( p^{4} T^{34} + \)\(17\!\cdots\!50\)\( p^{5} T^{35} + \)\(38\!\cdots\!19\)\( p^{6} T^{36} + \)\(78\!\cdots\!33\)\( p^{7} T^{37} + \)\(16\!\cdots\!52\)\( p^{8} T^{38} + 31356507933851618832 p^{9} T^{39} + 6144074354695440490 p^{10} T^{40} + 1120754722586976596 p^{11} T^{41} + 207244989441471466 p^{12} T^{42} + 35493179091870666 p^{13} T^{43} + 6161351062217123 p^{14} T^{44} + 983583460343115 p^{15} T^{45} + 159235686840698 p^{16} T^{46} + 23472878384759 p^{17} T^{47} + 3514474273942 p^{18} T^{48} + 472361760094 p^{19} T^{49} + 64698052536 p^{20} T^{50} + 7787780347 p^{21} T^{51} + 961467409 p^{22} T^{52} + 100876534 p^{23} T^{53} + 10992895 p^{24} T^{54} + 960757 p^{25} T^{55} + 89534 p^{26} T^{56} + 5963 p^{27} T^{57} + 451 p^{28} T^{58} + 18 p^{29} T^{59} + p^{30} T^{60} \)
23 \( 1 + 15 T + 454 T^{2} + 5626 T^{3} + 97509 T^{4} + 1038895 T^{5} + 13381997 T^{6} + 125799490 T^{7} + 1329066347 T^{8} + 11223368873 T^{9} + 102262470167 T^{10} + 785882266740 T^{11} + 6362162464556 T^{12} + 44931907786693 T^{13} + 329545618846632 T^{14} + 2155175689321634 T^{15} + 14519352021217764 T^{16} + 88480618696530525 T^{17} + 553390450724288551 T^{18} + 3160188392133537964 T^{19} + 18517998919177508246 T^{20} + 99679462250137871798 T^{21} + \)\(55\!\cdots\!01\)\( T^{22} + \)\(28\!\cdots\!14\)\( T^{23} + \)\(14\!\cdots\!79\)\( T^{24} + \)\(73\!\cdots\!46\)\( T^{25} + \)\(37\!\cdots\!16\)\( T^{26} + \)\(17\!\cdots\!44\)\( T^{27} + \)\(89\!\cdots\!97\)\( T^{28} + \)\(41\!\cdots\!02\)\( T^{29} + \)\(20\!\cdots\!37\)\( T^{30} + \)\(41\!\cdots\!02\)\( p T^{31} + \)\(89\!\cdots\!97\)\( p^{2} T^{32} + \)\(17\!\cdots\!44\)\( p^{3} T^{33} + \)\(37\!\cdots\!16\)\( p^{4} T^{34} + \)\(73\!\cdots\!46\)\( p^{5} T^{35} + \)\(14\!\cdots\!79\)\( p^{6} T^{36} + \)\(28\!\cdots\!14\)\( p^{7} T^{37} + \)\(55\!\cdots\!01\)\( p^{8} T^{38} + 99679462250137871798 p^{9} T^{39} + 18517998919177508246 p^{10} T^{40} + 3160188392133537964 p^{11} T^{41} + 553390450724288551 p^{12} T^{42} + 88480618696530525 p^{13} T^{43} + 14519352021217764 p^{14} T^{44} + 2155175689321634 p^{15} T^{45} + 329545618846632 p^{16} T^{46} + 44931907786693 p^{17} T^{47} + 6362162464556 p^{18} T^{48} + 785882266740 p^{19} T^{49} + 102262470167 p^{20} T^{50} + 11223368873 p^{21} T^{51} + 1329066347 p^{22} T^{52} + 125799490 p^{23} T^{53} + 13381997 p^{24} T^{54} + 1038895 p^{25} T^{55} + 97509 p^{26} T^{56} + 5626 p^{27} T^{57} + 454 p^{28} T^{58} + 15 p^{29} T^{59} + p^{30} T^{60} \)
29 \( 1 + 37 T + 1161 T^{2} + 25974 T^{3} + 513267 T^{4} + 8650504 T^{5} + 132781208 T^{6} + 1841294086 T^{7} + 23716668096 T^{8} + 283461629078 T^{9} + 3188376671858 T^{10} + 33783999564506 T^{11} + 339978385605993 T^{12} + 3254047853043151 T^{13} + 29776003453233216 T^{14} + 260857464203507627 T^{15} + 2195607249741462182 T^{16} + 17777537331666120825 T^{17} + \)\(13\!\cdots\!13\)\( T^{18} + \)\(10\!\cdots\!84\)\( T^{19} + \)\(76\!\cdots\!70\)\( T^{20} + \)\(53\!\cdots\!46\)\( T^{21} + \)\(36\!\cdots\!63\)\( T^{22} + \)\(24\!\cdots\!42\)\( T^{23} + \)\(15\!\cdots\!15\)\( T^{24} + \)\(98\!\cdots\!28\)\( T^{25} + \)\(60\!\cdots\!42\)\( T^{26} + \)\(35\!\cdots\!46\)\( T^{27} + \)\(20\!\cdots\!65\)\( T^{28} + \)\(11\!\cdots\!12\)\( T^{29} + \)\(62\!\cdots\!49\)\( T^{30} + \)\(11\!\cdots\!12\)\( p T^{31} + \)\(20\!\cdots\!65\)\( p^{2} T^{32} + \)\(35\!\cdots\!46\)\( p^{3} T^{33} + \)\(60\!\cdots\!42\)\( p^{4} T^{34} + \)\(98\!\cdots\!28\)\( p^{5} T^{35} + \)\(15\!\cdots\!15\)\( p^{6} T^{36} + \)\(24\!\cdots\!42\)\( p^{7} T^{37} + \)\(36\!\cdots\!63\)\( p^{8} T^{38} + \)\(53\!\cdots\!46\)\( p^{9} T^{39} + \)\(76\!\cdots\!70\)\( p^{10} T^{40} + \)\(10\!\cdots\!84\)\( p^{11} T^{41} + \)\(13\!\cdots\!13\)\( p^{12} T^{42} + 17777537331666120825 p^{13} T^{43} + 2195607249741462182 p^{14} T^{44} + 260857464203507627 p^{15} T^{45} + 29776003453233216 p^{16} T^{46} + 3254047853043151 p^{17} T^{47} + 339978385605993 p^{18} T^{48} + 33783999564506 p^{19} T^{49} + 3188376671858 p^{20} T^{50} + 283461629078 p^{21} T^{51} + 23716668096 p^{22} T^{52} + 1841294086 p^{23} T^{53} + 132781208 p^{24} T^{54} + 8650504 p^{25} T^{55} + 513267 p^{26} T^{56} + 25974 p^{27} T^{57} + 1161 p^{28} T^{58} + 37 p^{29} T^{59} + p^{30} T^{60} \)
31 \( 1 + 11 T + 484 T^{2} + 4771 T^{3} + 116072 T^{4} + 1040286 T^{5} + 18402472 T^{6} + 151577354 T^{7} + 2170785863 T^{8} + 16569566409 T^{9} + 203315087607 T^{10} + 1447753902836 T^{11} + 15760863991956 T^{12} + 3396381888937 p T^{13} + 1041390427152090 T^{14} + 6559366973416871 T^{15} + 59982736686661304 T^{16} + 11545430977977142 p T^{17} + 3067148206554220719 T^{18} + 17417071110416450616 T^{19} + \)\(14\!\cdots\!73\)\( T^{20} + \)\(76\!\cdots\!53\)\( T^{21} + \)\(59\!\cdots\!75\)\( T^{22} + \)\(31\!\cdots\!75\)\( T^{23} + \)\(23\!\cdots\!32\)\( T^{24} + \)\(11\!\cdots\!11\)\( T^{25} + \)\(83\!\cdots\!24\)\( T^{26} + \)\(40\!\cdots\!02\)\( T^{27} + \)\(28\!\cdots\!36\)\( T^{28} + \)\(13\!\cdots\!07\)\( T^{29} + \)\(90\!\cdots\!08\)\( T^{30} + \)\(13\!\cdots\!07\)\( p T^{31} + \)\(28\!\cdots\!36\)\( p^{2} T^{32} + \)\(40\!\cdots\!02\)\( p^{3} T^{33} + \)\(83\!\cdots\!24\)\( p^{4} T^{34} + \)\(11\!\cdots\!11\)\( p^{5} T^{35} + \)\(23\!\cdots\!32\)\( p^{6} T^{36} + \)\(31\!\cdots\!75\)\( p^{7} T^{37} + \)\(59\!\cdots\!75\)\( p^{8} T^{38} + \)\(76\!\cdots\!53\)\( p^{9} T^{39} + \)\(14\!\cdots\!73\)\( p^{10} T^{40} + 17417071110416450616 p^{11} T^{41} + 3067148206554220719 p^{12} T^{42} + 11545430977977142 p^{14} T^{43} + 59982736686661304 p^{14} T^{44} + 6559366973416871 p^{15} T^{45} + 1041390427152090 p^{16} T^{46} + 3396381888937 p^{18} T^{47} + 15760863991956 p^{18} T^{48} + 1447753902836 p^{19} T^{49} + 203315087607 p^{20} T^{50} + 16569566409 p^{21} T^{51} + 2170785863 p^{22} T^{52} + 151577354 p^{23} T^{53} + 18402472 p^{24} T^{54} + 1040286 p^{25} T^{55} + 116072 p^{26} T^{56} + 4771 p^{27} T^{57} + 484 p^{28} T^{58} + 11 p^{29} T^{59} + p^{30} T^{60} \)
37 \( 1 + 46 T + 1724 T^{2} + 46548 T^{3} + 1097230 T^{4} + 22099052 T^{5} + 403278301 T^{6} + 6644414802 T^{7} + 101368273898 T^{8} + 1432951497687 T^{9} + 19018141623091 T^{10} + 237361416078140 T^{11} + 2807661911715904 T^{12} + 31527674646630812 T^{13} + 337787143777627783 T^{14} + 3458079149609429892 T^{15} + 33945288252346202429 T^{16} + \)\(31\!\cdots\!86\)\( T^{17} + \)\(29\!\cdots\!39\)\( T^{18} + \)\(25\!\cdots\!79\)\( T^{19} + \)\(21\!\cdots\!07\)\( T^{20} + \)\(17\!\cdots\!36\)\( T^{21} + \)\(13\!\cdots\!58\)\( T^{22} + \)\(10\!\cdots\!50\)\( T^{23} + \)\(77\!\cdots\!62\)\( T^{24} + \)\(55\!\cdots\!15\)\( T^{25} + \)\(38\!\cdots\!95\)\( T^{26} + \)\(70\!\cdots\!20\)\( p T^{27} + \)\(17\!\cdots\!79\)\( T^{28} + \)\(10\!\cdots\!39\)\( T^{29} + \)\(66\!\cdots\!78\)\( T^{30} + \)\(10\!\cdots\!39\)\( p T^{31} + \)\(17\!\cdots\!79\)\( p^{2} T^{32} + \)\(70\!\cdots\!20\)\( p^{4} T^{33} + \)\(38\!\cdots\!95\)\( p^{4} T^{34} + \)\(55\!\cdots\!15\)\( p^{5} T^{35} + \)\(77\!\cdots\!62\)\( p^{6} T^{36} + \)\(10\!\cdots\!50\)\( p^{7} T^{37} + \)\(13\!\cdots\!58\)\( p^{8} T^{38} + \)\(17\!\cdots\!36\)\( p^{9} T^{39} + \)\(21\!\cdots\!07\)\( p^{10} T^{40} + \)\(25\!\cdots\!79\)\( p^{11} T^{41} + \)\(29\!\cdots\!39\)\( p^{12} T^{42} + \)\(31\!\cdots\!86\)\( p^{13} T^{43} + 33945288252346202429 p^{14} T^{44} + 3458079149609429892 p^{15} T^{45} + 337787143777627783 p^{16} T^{46} + 31527674646630812 p^{17} T^{47} + 2807661911715904 p^{18} T^{48} + 237361416078140 p^{19} T^{49} + 19018141623091 p^{20} T^{50} + 1432951497687 p^{21} T^{51} + 101368273898 p^{22} T^{52} + 6644414802 p^{23} T^{53} + 403278301 p^{24} T^{54} + 22099052 p^{25} T^{55} + 1097230 p^{26} T^{56} + 46548 p^{27} T^{57} + 1724 p^{28} T^{58} + 46 p^{29} T^{59} + p^{30} T^{60} \)
41 \( 1 + 28 T + 1017 T^{2} + 19959 T^{3} + 432605 T^{4} + 6702305 T^{5} + 109942056 T^{6} + 1427662094 T^{7} + 19381964356 T^{8} + 218619810167 T^{9} + 2574651971481 T^{10} + 25828427104216 T^{11} + 271819619556295 T^{12} + 2466756567065039 T^{13} + 23701901974275438 T^{14} + 197186804997353797 T^{15} + 1760209241424444512 T^{16} + 13579858043468354045 T^{17} + \)\(11\!\cdots\!67\)\( T^{18} + \)\(82\!\cdots\!40\)\( T^{19} + \)\(66\!\cdots\!28\)\( T^{20} + \)\(45\!\cdots\!95\)\( T^{21} + \)\(35\!\cdots\!00\)\( T^{22} + \)\(23\!\cdots\!07\)\( T^{23} + \)\(17\!\cdots\!10\)\( T^{24} + \)\(11\!\cdots\!15\)\( T^{25} + \)\(83\!\cdots\!11\)\( T^{26} + \)\(50\!\cdots\!53\)\( T^{27} + \)\(37\!\cdots\!44\)\( T^{28} + \)\(22\!\cdots\!95\)\( T^{29} + \)\(15\!\cdots\!20\)\( T^{30} + \)\(22\!\cdots\!95\)\( p T^{31} + \)\(37\!\cdots\!44\)\( p^{2} T^{32} + \)\(50\!\cdots\!53\)\( p^{3} T^{33} + \)\(83\!\cdots\!11\)\( p^{4} T^{34} + \)\(11\!\cdots\!15\)\( p^{5} T^{35} + \)\(17\!\cdots\!10\)\( p^{6} T^{36} + \)\(23\!\cdots\!07\)\( p^{7} T^{37} + \)\(35\!\cdots\!00\)\( p^{8} T^{38} + \)\(45\!\cdots\!95\)\( p^{9} T^{39} + \)\(66\!\cdots\!28\)\( p^{10} T^{40} + \)\(82\!\cdots\!40\)\( p^{11} T^{41} + \)\(11\!\cdots\!67\)\( p^{12} T^{42} + 13579858043468354045 p^{13} T^{43} + 1760209241424444512 p^{14} T^{44} + 197186804997353797 p^{15} T^{45} + 23701901974275438 p^{16} T^{46} + 2466756567065039 p^{17} T^{47} + 271819619556295 p^{18} T^{48} + 25828427104216 p^{19} T^{49} + 2574651971481 p^{20} T^{50} + 218619810167 p^{21} T^{51} + 19381964356 p^{22} T^{52} + 1427662094 p^{23} T^{53} + 109942056 p^{24} T^{54} + 6702305 p^{25} T^{55} + 432605 p^{26} T^{56} + 19959 p^{27} T^{57} + 1017 p^{28} T^{58} + 28 p^{29} T^{59} + p^{30} T^{60} \)
43 \( 1 + 61 T + 2574 T^{2} + 80064 T^{3} + 2077309 T^{4} + 46067335 T^{5} + 906259259 T^{6} + 16026411111 T^{7} + 259125429584 T^{8} + 3862406352171 T^{9} + 53586349422179 T^{10} + 695901969906651 T^{11} + 8512505350598574 T^{12} + 98489439479059129 T^{13} + 1082691499645333807 T^{14} + 11345631007889325437 T^{15} + \)\(11\!\cdots\!66\)\( T^{16} + \)\(10\!\cdots\!67\)\( T^{17} + \)\(23\!\cdots\!88\)\( p T^{18} + \)\(90\!\cdots\!72\)\( T^{19} + \)\(77\!\cdots\!95\)\( T^{20} + \)\(64\!\cdots\!60\)\( T^{21} + \)\(52\!\cdots\!63\)\( T^{22} + \)\(41\!\cdots\!49\)\( T^{23} + \)\(31\!\cdots\!51\)\( T^{24} + \)\(23\!\cdots\!00\)\( T^{25} + \)\(17\!\cdots\!96\)\( T^{26} + \)\(12\!\cdots\!31\)\( T^{27} + \)\(85\!\cdots\!72\)\( T^{28} + \)\(58\!\cdots\!45\)\( T^{29} + \)\(38\!\cdots\!02\)\( T^{30} + \)\(58\!\cdots\!45\)\( p T^{31} + \)\(85\!\cdots\!72\)\( p^{2} T^{32} + \)\(12\!\cdots\!31\)\( p^{3} T^{33} + \)\(17\!\cdots\!96\)\( p^{4} T^{34} + \)\(23\!\cdots\!00\)\( p^{5} T^{35} + \)\(31\!\cdots\!51\)\( p^{6} T^{36} + \)\(41\!\cdots\!49\)\( p^{7} T^{37} + \)\(52\!\cdots\!63\)\( p^{8} T^{38} + \)\(64\!\cdots\!60\)\( p^{9} T^{39} + \)\(77\!\cdots\!95\)\( p^{10} T^{40} + \)\(90\!\cdots\!72\)\( p^{11} T^{41} + \)\(23\!\cdots\!88\)\( p^{13} T^{42} + \)\(10\!\cdots\!67\)\( p^{13} T^{43} + \)\(11\!\cdots\!66\)\( p^{14} T^{44} + 11345631007889325437 p^{15} T^{45} + 1082691499645333807 p^{16} T^{46} + 98489439479059129 p^{17} T^{47} + 8512505350598574 p^{18} T^{48} + 695901969906651 p^{19} T^{49} + 53586349422179 p^{20} T^{50} + 3862406352171 p^{21} T^{51} + 259125429584 p^{22} T^{52} + 16026411111 p^{23} T^{53} + 906259259 p^{24} T^{54} + 46067335 p^{25} T^{55} + 2077309 p^{26} T^{56} + 80064 p^{27} T^{57} + 2574 p^{28} T^{58} + 61 p^{29} T^{59} + p^{30} T^{60} \)
47 \( 1 + 4 T + 849 T^{2} + 2854 T^{3} + 355614 T^{4} + 993171 T^{5} + 98143760 T^{6} + 224979855 T^{7} + 20106554312 T^{8} + 37398767834 T^{9} + 3265356471573 T^{10} + 4884201226401 T^{11} + 9324378158399 p T^{12} + 525170852038337 T^{13} + 50018129560638783 T^{14} + 48249955598256319 T^{15} + 4956419781176065347 T^{16} + 3911598047443129352 T^{17} + \)\(43\!\cdots\!16\)\( T^{18} + \)\(28\!\cdots\!03\)\( T^{19} + \)\(33\!\cdots\!34\)\( T^{20} + \)\(19\!\cdots\!56\)\( T^{21} + \)\(23\!\cdots\!91\)\( T^{22} + \)\(12\!\cdots\!62\)\( T^{23} + \)\(15\!\cdots\!67\)\( T^{24} + \)\(74\!\cdots\!95\)\( T^{25} + \)\(88\!\cdots\!95\)\( T^{26} + \)\(41\!\cdots\!78\)\( T^{27} + \)\(47\!\cdots\!44\)\( T^{28} + \)\(21\!\cdots\!58\)\( T^{29} + \)\(23\!\cdots\!38\)\( T^{30} + \)\(21\!\cdots\!58\)\( p T^{31} + \)\(47\!\cdots\!44\)\( p^{2} T^{32} + \)\(41\!\cdots\!78\)\( p^{3} T^{33} + \)\(88\!\cdots\!95\)\( p^{4} T^{34} + \)\(74\!\cdots\!95\)\( p^{5} T^{35} + \)\(15\!\cdots\!67\)\( p^{6} T^{36} + \)\(12\!\cdots\!62\)\( p^{7} T^{37} + \)\(23\!\cdots\!91\)\( p^{8} T^{38} + \)\(19\!\cdots\!56\)\( p^{9} T^{39} + \)\(33\!\cdots\!34\)\( p^{10} T^{40} + \)\(28\!\cdots\!03\)\( p^{11} T^{41} + \)\(43\!\cdots\!16\)\( p^{12} T^{42} + 3911598047443129352 p^{13} T^{43} + 4956419781176065347 p^{14} T^{44} + 48249955598256319 p^{15} T^{45} + 50018129560638783 p^{16} T^{46} + 525170852038337 p^{17} T^{47} + 9324378158399 p^{19} T^{48} + 4884201226401 p^{19} T^{49} + 3265356471573 p^{20} T^{50} + 37398767834 p^{21} T^{51} + 20106554312 p^{22} T^{52} + 224979855 p^{23} T^{53} + 98143760 p^{24} T^{54} + 993171 p^{25} T^{55} + 355614 p^{26} T^{56} + 2854 p^{27} T^{57} + 849 p^{28} T^{58} + 4 p^{29} T^{59} + p^{30} T^{60} \)
53 \( 1 + 19 T + 920 T^{2} + 14954 T^{3} + 403963 T^{4} + 5749764 T^{5} + 113489141 T^{6} + 1439573654 T^{7} + 23050629756 T^{8} + 264269363109 T^{9} + 3626921245727 T^{10} + 38038339499742 T^{11} + 463125793008887 T^{12} + 4492634400624360 T^{13} + 49724759834041013 T^{14} + 450938873637431039 T^{15} + 4624113296784275194 T^{16} + 39609231736276801843 T^{17} + \)\(38\!\cdots\!81\)\( T^{18} + \)\(31\!\cdots\!08\)\( T^{19} + \)\(28\!\cdots\!24\)\( T^{20} + \)\(22\!\cdots\!23\)\( T^{21} + \)\(19\!\cdots\!25\)\( T^{22} + \)\(15\!\cdots\!75\)\( T^{23} + \)\(12\!\cdots\!68\)\( T^{24} + \)\(94\!\cdots\!79\)\( T^{25} + \)\(78\!\cdots\!56\)\( T^{26} + \)\(55\!\cdots\!91\)\( T^{27} + \)\(44\!\cdots\!93\)\( T^{28} + \)\(31\!\cdots\!88\)\( T^{29} + \)\(24\!\cdots\!54\)\( T^{30} + \)\(31\!\cdots\!88\)\( p T^{31} + \)\(44\!\cdots\!93\)\( p^{2} T^{32} + \)\(55\!\cdots\!91\)\( p^{3} T^{33} + \)\(78\!\cdots\!56\)\( p^{4} T^{34} + \)\(94\!\cdots\!79\)\( p^{5} T^{35} + \)\(12\!\cdots\!68\)\( p^{6} T^{36} + \)\(15\!\cdots\!75\)\( p^{7} T^{37} + \)\(19\!\cdots\!25\)\( p^{8} T^{38} + \)\(22\!\cdots\!23\)\( p^{9} T^{39} + \)\(28\!\cdots\!24\)\( p^{10} T^{40} + \)\(31\!\cdots\!08\)\( p^{11} T^{41} + \)\(38\!\cdots\!81\)\( p^{12} T^{42} + 39609231736276801843 p^{13} T^{43} + 4624113296784275194 p^{14} T^{44} + 450938873637431039 p^{15} T^{45} + 49724759834041013 p^{16} T^{46} + 4492634400624360 p^{17} T^{47} + 463125793008887 p^{18} T^{48} + 38038339499742 p^{19} T^{49} + 3626921245727 p^{20} T^{50} + 264269363109 p^{21} T^{51} + 23050629756 p^{22} T^{52} + 1439573654 p^{23} T^{53} + 113489141 p^{24} T^{54} + 5749764 p^{25} T^{55} + 403963 p^{26} T^{56} + 14954 p^{27} T^{57} + 920 p^{28} T^{58} + 19 p^{29} T^{59} + p^{30} T^{60} \)
59 \( 1 + 6 T + 739 T^{2} + 3324 T^{3} + 274151 T^{4} + 820028 T^{5} + 68113590 T^{6} + 98063081 T^{7} + 12806185361 T^{8} - 2128146339 T^{9} + 1957419203221 T^{10} - 3505763443354 T^{11} + 255567740851894 T^{12} - 855496555286909 T^{13} + 29544797289717295 T^{14} - 139271557038731293 T^{15} + 3099363888707652112 T^{16} - 17923597491779811026 T^{17} + \)\(29\!\cdots\!18\)\( T^{18} - \)\(19\!\cdots\!37\)\( T^{19} + \)\(26\!\cdots\!62\)\( T^{20} - \)\(18\!\cdots\!17\)\( T^{21} + \)\(22\!\cdots\!02\)\( T^{22} - \)\(15\!\cdots\!49\)\( T^{23} + \)\(16\!\cdots\!67\)\( T^{24} - \)\(12\!\cdots\!01\)\( T^{25} + \)\(12\!\cdots\!31\)\( T^{26} - \)\(85\!\cdots\!80\)\( T^{27} + \)\(79\!\cdots\!50\)\( T^{28} - \)\(55\!\cdots\!46\)\( T^{29} + \)\(48\!\cdots\!67\)\( T^{30} - \)\(55\!\cdots\!46\)\( p T^{31} + \)\(79\!\cdots\!50\)\( p^{2} T^{32} - \)\(85\!\cdots\!80\)\( p^{3} T^{33} + \)\(12\!\cdots\!31\)\( p^{4} T^{34} - \)\(12\!\cdots\!01\)\( p^{5} T^{35} + \)\(16\!\cdots\!67\)\( p^{6} T^{36} - \)\(15\!\cdots\!49\)\( p^{7} T^{37} + \)\(22\!\cdots\!02\)\( p^{8} T^{38} - \)\(18\!\cdots\!17\)\( p^{9} T^{39} + \)\(26\!\cdots\!62\)\( p^{10} T^{40} - \)\(19\!\cdots\!37\)\( p^{11} T^{41} + \)\(29\!\cdots\!18\)\( p^{12} T^{42} - 17923597491779811026 p^{13} T^{43} + 3099363888707652112 p^{14} T^{44} - 139271557038731293 p^{15} T^{45} + 29544797289717295 p^{16} T^{46} - 855496555286909 p^{17} T^{47} + 255567740851894 p^{18} T^{48} - 3505763443354 p^{19} T^{49} + 1957419203221 p^{20} T^{50} - 2128146339 p^{21} T^{51} + 12806185361 p^{22} T^{52} + 98063081 p^{23} T^{53} + 68113590 p^{24} T^{54} + 820028 p^{25} T^{55} + 274151 p^{26} T^{56} + 3324 p^{27} T^{57} + 739 p^{28} T^{58} + 6 p^{29} T^{59} + p^{30} T^{60} \)
61 \( 1 + 32 T + 1573 T^{2} + 37847 T^{3} + 1093787 T^{4} + 21507964 T^{5} + 467832337 T^{6} + 7880954438 T^{7} + 141590593555 T^{8} + 2106080939636 T^{9} + 32796135933796 T^{10} + 439913767772847 T^{11} + 6114863455448528 T^{12} + 75124114155145638 T^{13} + 950789039017233244 T^{14} + 10826072583438646150 T^{15} + \)\(12\!\cdots\!46\)\( T^{16} + \)\(13\!\cdots\!44\)\( T^{17} + \)\(14\!\cdots\!01\)\( T^{18} + \)\(14\!\cdots\!84\)\( T^{19} + \)\(15\!\cdots\!76\)\( T^{20} + \)\(14\!\cdots\!82\)\( T^{21} + \)\(13\!\cdots\!37\)\( T^{22} + \)\(12\!\cdots\!98\)\( T^{23} + \)\(11\!\cdots\!08\)\( T^{24} + \)\(10\!\cdots\!39\)\( T^{25} + \)\(88\!\cdots\!04\)\( T^{26} + \)\(73\!\cdots\!36\)\( T^{27} + \)\(61\!\cdots\!58\)\( T^{28} + \)\(48\!\cdots\!02\)\( T^{29} + \)\(39\!\cdots\!26\)\( T^{30} + \)\(48\!\cdots\!02\)\( p T^{31} + \)\(61\!\cdots\!58\)\( p^{2} T^{32} + \)\(73\!\cdots\!36\)\( p^{3} T^{33} + \)\(88\!\cdots\!04\)\( p^{4} T^{34} + \)\(10\!\cdots\!39\)\( p^{5} T^{35} + \)\(11\!\cdots\!08\)\( p^{6} T^{36} + \)\(12\!\cdots\!98\)\( p^{7} T^{37} + \)\(13\!\cdots\!37\)\( p^{8} T^{38} + \)\(14\!\cdots\!82\)\( p^{9} T^{39} + \)\(15\!\cdots\!76\)\( p^{10} T^{40} + \)\(14\!\cdots\!84\)\( p^{11} T^{41} + \)\(14\!\cdots\!01\)\( p^{12} T^{42} + \)\(13\!\cdots\!44\)\( p^{13} T^{43} + \)\(12\!\cdots\!46\)\( p^{14} T^{44} + 10826072583438646150 p^{15} T^{45} + 950789039017233244 p^{16} T^{46} + 75124114155145638 p^{17} T^{47} + 6114863455448528 p^{18} T^{48} + 439913767772847 p^{19} T^{49} + 32796135933796 p^{20} T^{50} + 2106080939636 p^{21} T^{51} + 141590593555 p^{22} T^{52} + 7880954438 p^{23} T^{53} + 467832337 p^{24} T^{54} + 21507964 p^{25} T^{55} + 1093787 p^{26} T^{56} + 37847 p^{27} T^{57} + 1573 p^{28} T^{58} + 32 p^{29} T^{59} + p^{30} T^{60} \)
67 \( 1 + 44 T + 2126 T^{2} + 63659 T^{3} + 1879790 T^{4} + 44046032 T^{5} + 997586809 T^{6} + 19546462137 T^{7} + 369036635920 T^{8} + 6280928488024 T^{9} + 103132252402829 T^{10} + 1562268033999744 T^{11} + 22885933767217843 T^{12} + 313789894387402531 T^{13} + 4171545542908179946 T^{14} + 52413682653305748148 T^{15} + \)\(64\!\cdots\!73\)\( T^{16} + \)\(74\!\cdots\!14\)\( T^{17} + \)\(84\!\cdots\!93\)\( T^{18} + \)\(91\!\cdots\!36\)\( T^{19} + \)\(96\!\cdots\!81\)\( T^{20} + \)\(98\!\cdots\!60\)\( T^{21} + \)\(98\!\cdots\!96\)\( T^{22} + \)\(94\!\cdots\!41\)\( T^{23} + \)\(89\!\cdots\!87\)\( T^{24} + \)\(82\!\cdots\!51\)\( T^{25} + \)\(73\!\cdots\!80\)\( T^{26} + \)\(64\!\cdots\!36\)\( T^{27} + \)\(55\!\cdots\!29\)\( T^{28} + \)\(46\!\cdots\!92\)\( T^{29} + \)\(38\!\cdots\!48\)\( T^{30} + \)\(46\!\cdots\!92\)\( p T^{31} + \)\(55\!\cdots\!29\)\( p^{2} T^{32} + \)\(64\!\cdots\!36\)\( p^{3} T^{33} + \)\(73\!\cdots\!80\)\( p^{4} T^{34} + \)\(82\!\cdots\!51\)\( p^{5} T^{35} + \)\(89\!\cdots\!87\)\( p^{6} T^{36} + \)\(94\!\cdots\!41\)\( p^{7} T^{37} + \)\(98\!\cdots\!96\)\( p^{8} T^{38} + \)\(98\!\cdots\!60\)\( p^{9} T^{39} + \)\(96\!\cdots\!81\)\( p^{10} T^{40} + \)\(91\!\cdots\!36\)\( p^{11} T^{41} + \)\(84\!\cdots\!93\)\( p^{12} T^{42} + \)\(74\!\cdots\!14\)\( p^{13} T^{43} + \)\(64\!\cdots\!73\)\( p^{14} T^{44} + 52413682653305748148 p^{15} T^{45} + 4171545542908179946 p^{16} T^{46} + 313789894387402531 p^{17} T^{47} + 22885933767217843 p^{18} T^{48} + 1562268033999744 p^{19} T^{49} + 103132252402829 p^{20} T^{50} + 6280928488024 p^{21} T^{51} + 369036635920 p^{22} T^{52} + 19546462137 p^{23} T^{53} + 997586809 p^{24} T^{54} + 44046032 p^{25} T^{55} + 1879790 p^{26} T^{56} + 63659 p^{27} T^{57} + 2126 p^{28} T^{58} + 44 p^{29} T^{59} + p^{30} T^{60} \)
71 \( 1 - 10 T + 1391 T^{2} - 13101 T^{3} + 949796 T^{4} - 8456662 T^{5} + 424563504 T^{6} - 3582935818 T^{7} + 139779680611 T^{8} - 1119892171519 T^{9} + 36154130388142 T^{10} - 275206247366577 T^{11} + 7652665350251776 T^{12} - 779549265256999 p T^{13} + 1363604960238633399 T^{14} - 9366096001136979369 T^{15} + \)\(20\!\cdots\!37\)\( T^{16} - \)\(13\!\cdots\!96\)\( T^{17} + \)\(27\!\cdots\!64\)\( T^{18} - \)\(17\!\cdots\!51\)\( T^{19} + \)\(33\!\cdots\!91\)\( T^{20} - \)\(19\!\cdots\!86\)\( T^{21} + \)\(35\!\cdots\!88\)\( T^{22} - \)\(19\!\cdots\!08\)\( T^{23} + \)\(33\!\cdots\!44\)\( T^{24} - \)\(17\!\cdots\!24\)\( T^{25} + \)\(29\!\cdots\!26\)\( T^{26} - \)\(14\!\cdots\!46\)\( T^{27} + \)\(23\!\cdots\!79\)\( T^{28} - \)\(11\!\cdots\!52\)\( T^{29} + \)\(17\!\cdots\!73\)\( T^{30} - \)\(11\!\cdots\!52\)\( p T^{31} + \)\(23\!\cdots\!79\)\( p^{2} T^{32} - \)\(14\!\cdots\!46\)\( p^{3} T^{33} + \)\(29\!\cdots\!26\)\( p^{4} T^{34} - \)\(17\!\cdots\!24\)\( p^{5} T^{35} + \)\(33\!\cdots\!44\)\( p^{6} T^{36} - \)\(19\!\cdots\!08\)\( p^{7} T^{37} + \)\(35\!\cdots\!88\)\( p^{8} T^{38} - \)\(19\!\cdots\!86\)\( p^{9} T^{39} + \)\(33\!\cdots\!91\)\( p^{10} T^{40} - \)\(17\!\cdots\!51\)\( p^{11} T^{41} + \)\(27\!\cdots\!64\)\( p^{12} T^{42} - \)\(13\!\cdots\!96\)\( p^{13} T^{43} + \)\(20\!\cdots\!37\)\( p^{14} T^{44} - 9366096001136979369 p^{15} T^{45} + 1363604960238633399 p^{16} T^{46} - 779549265256999 p^{18} T^{47} + 7652665350251776 p^{18} T^{48} - 275206247366577 p^{19} T^{49} + 36154130388142 p^{20} T^{50} - 1119892171519 p^{21} T^{51} + 139779680611 p^{22} T^{52} - 3582935818 p^{23} T^{53} + 424563504 p^{24} T^{54} - 8456662 p^{25} T^{55} + 949796 p^{26} T^{56} - 13101 p^{27} T^{57} + 1391 p^{28} T^{58} - 10 p^{29} T^{59} + p^{30} T^{60} \)
73 \( 1 + 58 T + 2609 T^{2} + 84537 T^{3} + 2362735 T^{4} + 56266335 T^{5} + 1210385460 T^{6} + 23486020115 T^{7} + 422391967036 T^{8} + 7048341709165 T^{9} + 110805861893717 T^{10} + 1643641595953068 T^{11} + 23228133949213628 T^{12} + 313206803421850727 T^{13} + 4056192766749844854 T^{14} + 50516946250838671330 T^{15} + \)\(60\!\cdots\!20\)\( T^{16} + \)\(70\!\cdots\!34\)\( T^{17} + \)\(79\!\cdots\!06\)\( T^{18} + \)\(87\!\cdots\!37\)\( T^{19} + \)\(93\!\cdots\!66\)\( T^{20} + \)\(97\!\cdots\!92\)\( T^{21} + \)\(99\!\cdots\!89\)\( T^{22} + \)\(98\!\cdots\!07\)\( T^{23} + \)\(96\!\cdots\!89\)\( T^{24} + \)\(91\!\cdots\!93\)\( T^{25} + \)\(85\!\cdots\!19\)\( T^{26} + \)\(78\!\cdots\!50\)\( T^{27} + \)\(70\!\cdots\!60\)\( T^{28} + \)\(61\!\cdots\!26\)\( T^{29} + \)\(53\!\cdots\!33\)\( T^{30} + \)\(61\!\cdots\!26\)\( p T^{31} + \)\(70\!\cdots\!60\)\( p^{2} T^{32} + \)\(78\!\cdots\!50\)\( p^{3} T^{33} + \)\(85\!\cdots\!19\)\( p^{4} T^{34} + \)\(91\!\cdots\!93\)\( p^{5} T^{35} + \)\(96\!\cdots\!89\)\( p^{6} T^{36} + \)\(98\!\cdots\!07\)\( p^{7} T^{37} + \)\(99\!\cdots\!89\)\( p^{8} T^{38} + \)\(97\!\cdots\!92\)\( p^{9} T^{39} + \)\(93\!\cdots\!66\)\( p^{10} T^{40} + \)\(87\!\cdots\!37\)\( p^{11} T^{41} + \)\(79\!\cdots\!06\)\( p^{12} T^{42} + \)\(70\!\cdots\!34\)\( p^{13} T^{43} + \)\(60\!\cdots\!20\)\( p^{14} T^{44} + 50516946250838671330 p^{15} T^{45} + 4056192766749844854 p^{16} T^{46} + 313206803421850727 p^{17} T^{47} + 23228133949213628 p^{18} T^{48} + 1643641595953068 p^{19} T^{49} + 110805861893717 p^{20} T^{50} + 7048341709165 p^{21} T^{51} + 422391967036 p^{22} T^{52} + 23486020115 p^{23} T^{53} + 1210385460 p^{24} T^{54} + 56266335 p^{25} T^{55} + 2362735 p^{26} T^{56} + 84537 p^{27} T^{57} + 2609 p^{28} T^{58} + 58 p^{29} T^{59} + p^{30} T^{60} \)
79 \( 1 + 42 T + 2151 T^{2} + 63954 T^{3} + 1982483 T^{4} + 46972174 T^{5} + 1114387216 T^{6} + 22341373099 T^{7} + 442497170607 T^{8} + 7779817339221 T^{9} + 134507925094220 T^{10} + 2123438932489659 T^{11} + 32924713455018701 T^{12} + 474542590734418790 T^{13} + 6719107152609143208 T^{14} + 89511040514324581850 T^{15} + \)\(11\!\cdots\!15\)\( T^{16} + \)\(14\!\cdots\!58\)\( T^{17} + \)\(17\!\cdots\!29\)\( T^{18} + \)\(20\!\cdots\!06\)\( T^{19} + \)\(23\!\cdots\!09\)\( T^{20} + \)\(26\!\cdots\!75\)\( T^{21} + \)\(28\!\cdots\!53\)\( T^{22} + \)\(30\!\cdots\!25\)\( T^{23} + \)\(31\!\cdots\!66\)\( T^{24} + \)\(31\!\cdots\!92\)\( T^{25} + \)\(30\!\cdots\!62\)\( T^{26} + \)\(29\!\cdots\!09\)\( T^{27} + \)\(27\!\cdots\!91\)\( T^{28} + \)\(25\!\cdots\!08\)\( T^{29} + \)\(23\!\cdots\!44\)\( T^{30} + \)\(25\!\cdots\!08\)\( p T^{31} + \)\(27\!\cdots\!91\)\( p^{2} T^{32} + \)\(29\!\cdots\!09\)\( p^{3} T^{33} + \)\(30\!\cdots\!62\)\( p^{4} T^{34} + \)\(31\!\cdots\!92\)\( p^{5} T^{35} + \)\(31\!\cdots\!66\)\( p^{6} T^{36} + \)\(30\!\cdots\!25\)\( p^{7} T^{37} + \)\(28\!\cdots\!53\)\( p^{8} T^{38} + \)\(26\!\cdots\!75\)\( p^{9} T^{39} + \)\(23\!\cdots\!09\)\( p^{10} T^{40} + \)\(20\!\cdots\!06\)\( p^{11} T^{41} + \)\(17\!\cdots\!29\)\( p^{12} T^{42} + \)\(14\!\cdots\!58\)\( p^{13} T^{43} + \)\(11\!\cdots\!15\)\( p^{14} T^{44} + 89511040514324581850 p^{15} T^{45} + 6719107152609143208 p^{16} T^{46} + 474542590734418790 p^{17} T^{47} + 32924713455018701 p^{18} T^{48} + 2123438932489659 p^{19} T^{49} + 134507925094220 p^{20} T^{50} + 7779817339221 p^{21} T^{51} + 442497170607 p^{22} T^{52} + 22341373099 p^{23} T^{53} + 1114387216 p^{24} T^{54} + 46972174 p^{25} T^{55} + 1982483 p^{26} T^{56} + 63954 p^{27} T^{57} + 2151 p^{28} T^{58} + 42 p^{29} T^{59} + p^{30} T^{60} \)
83 \( 1 + 25 T + 17 p T^{2} + 29541 T^{3} + 957590 T^{4} + 17643819 T^{5} + 427394892 T^{6} + 7142073760 T^{7} + 142951895712 T^{8} + 2207170921539 T^{9} + 38429509412326 T^{10} + 554708797119660 T^{11} + 8661684547228414 T^{12} + 117794331792339519 T^{13} + 1682559634863045928 T^{14} + 21674145475157645090 T^{15} + 3458850957892555585 p T^{16} + \)\(35\!\cdots\!48\)\( T^{17} + \)\(43\!\cdots\!69\)\( T^{18} + \)\(50\!\cdots\!11\)\( T^{19} + \)\(59\!\cdots\!90\)\( T^{20} + \)\(66\!\cdots\!04\)\( T^{21} + \)\(73\!\cdots\!54\)\( T^{22} + \)\(78\!\cdots\!66\)\( T^{23} + \)\(83\!\cdots\!70\)\( T^{24} + \)\(84\!\cdots\!60\)\( T^{25} + \)\(85\!\cdots\!42\)\( T^{26} + \)\(83\!\cdots\!29\)\( T^{27} + \)\(80\!\cdots\!13\)\( T^{28} + \)\(75\!\cdots\!52\)\( T^{29} + \)\(69\!\cdots\!14\)\( T^{30} + \)\(75\!\cdots\!52\)\( p T^{31} + \)\(80\!\cdots\!13\)\( p^{2} T^{32} + \)\(83\!\cdots\!29\)\( p^{3} T^{33} + \)\(85\!\cdots\!42\)\( p^{4} T^{34} + \)\(84\!\cdots\!60\)\( p^{5} T^{35} + \)\(83\!\cdots\!70\)\( p^{6} T^{36} + \)\(78\!\cdots\!66\)\( p^{7} T^{37} + \)\(73\!\cdots\!54\)\( p^{8} T^{38} + \)\(66\!\cdots\!04\)\( p^{9} T^{39} + \)\(59\!\cdots\!90\)\( p^{10} T^{40} + \)\(50\!\cdots\!11\)\( p^{11} T^{41} + \)\(43\!\cdots\!69\)\( p^{12} T^{42} + \)\(35\!\cdots\!48\)\( p^{13} T^{43} + 3458850957892555585 p^{15} T^{44} + 21674145475157645090 p^{15} T^{45} + 1682559634863045928 p^{16} T^{46} + 117794331792339519 p^{17} T^{47} + 8661684547228414 p^{18} T^{48} + 554708797119660 p^{19} T^{49} + 38429509412326 p^{20} T^{50} + 2207170921539 p^{21} T^{51} + 142951895712 p^{22} T^{52} + 7142073760 p^{23} T^{53} + 427394892 p^{24} T^{54} + 17643819 p^{25} T^{55} + 957590 p^{26} T^{56} + 29541 p^{27} T^{57} + 17 p^{29} T^{58} + 25 p^{29} T^{59} + p^{30} T^{60} \)
89 \( 1 + 39 T + 2553 T^{2} + 78922 T^{3} + 3011513 T^{4} + 78091859 T^{5} + 2237164793 T^{6} + 50380565444 T^{7} + 1190822268801 T^{8} + 23833943872156 T^{9} + 487494130698607 T^{10} + 8813962796445305 T^{11} + 160439381603799939 T^{12} + 2651768943498282375 T^{13} + 43742207711735656956 T^{14} + \)\(66\!\cdots\!64\)\( T^{15} + \)\(10\!\cdots\!86\)\( T^{16} + \)\(14\!\cdots\!04\)\( T^{17} + \)\(20\!\cdots\!03\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{19} + \)\(34\!\cdots\!82\)\( T^{20} + \)\(42\!\cdots\!56\)\( T^{21} + \)\(52\!\cdots\!32\)\( T^{22} + \)\(60\!\cdots\!03\)\( T^{23} + \)\(69\!\cdots\!08\)\( T^{24} + \)\(76\!\cdots\!34\)\( T^{25} + \)\(82\!\cdots\!14\)\( T^{26} + \)\(85\!\cdots\!13\)\( T^{27} + \)\(87\!\cdots\!45\)\( T^{28} + \)\(85\!\cdots\!74\)\( T^{29} + \)\(82\!\cdots\!65\)\( T^{30} + \)\(85\!\cdots\!74\)\( p T^{31} + \)\(87\!\cdots\!45\)\( p^{2} T^{32} + \)\(85\!\cdots\!13\)\( p^{3} T^{33} + \)\(82\!\cdots\!14\)\( p^{4} T^{34} + \)\(76\!\cdots\!34\)\( p^{5} T^{35} + \)\(69\!\cdots\!08\)\( p^{6} T^{36} + \)\(60\!\cdots\!03\)\( p^{7} T^{37} + \)\(52\!\cdots\!32\)\( p^{8} T^{38} + \)\(42\!\cdots\!56\)\( p^{9} T^{39} + \)\(34\!\cdots\!82\)\( p^{10} T^{40} + \)\(26\!\cdots\!49\)\( p^{11} T^{41} + \)\(20\!\cdots\!03\)\( p^{12} T^{42} + \)\(14\!\cdots\!04\)\( p^{13} T^{43} + \)\(10\!\cdots\!86\)\( p^{14} T^{44} + \)\(66\!\cdots\!64\)\( p^{15} T^{45} + 43742207711735656956 p^{16} T^{46} + 2651768943498282375 p^{17} T^{47} + 160439381603799939 p^{18} T^{48} + 8813962796445305 p^{19} T^{49} + 487494130698607 p^{20} T^{50} + 23833943872156 p^{21} T^{51} + 1190822268801 p^{22} T^{52} + 50380565444 p^{23} T^{53} + 2237164793 p^{24} T^{54} + 78091859 p^{25} T^{55} + 3011513 p^{26} T^{56} + 78922 p^{27} T^{57} + 2553 p^{28} T^{58} + 39 p^{29} T^{59} + p^{30} T^{60} \)
97 \( 1 + 33 T + 1974 T^{2} + 54505 T^{3} + 1887479 T^{4} + 44966208 T^{5} + 1173356452 T^{6} + 24690581180 T^{7} + 536022340409 T^{8} + 10145311972192 T^{9} + 192577538468617 T^{10} + 3325341875787166 T^{11} + 56806978422944290 T^{12} + 904968643304537993 T^{13} + 14171042950832535602 T^{14} + \)\(21\!\cdots\!70\)\( T^{15} + \)\(30\!\cdots\!47\)\( T^{16} + \)\(42\!\cdots\!63\)\( T^{17} + \)\(57\!\cdots\!22\)\( T^{18} + \)\(75\!\cdots\!17\)\( T^{19} + \)\(96\!\cdots\!21\)\( T^{20} + \)\(12\!\cdots\!77\)\( T^{21} + \)\(14\!\cdots\!16\)\( T^{22} + \)\(17\!\cdots\!98\)\( T^{23} + \)\(19\!\cdots\!41\)\( T^{24} + \)\(22\!\cdots\!71\)\( T^{25} + \)\(24\!\cdots\!81\)\( T^{26} + \)\(25\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!05\)\( T^{28} + \)\(27\!\cdots\!67\)\( T^{29} + \)\(27\!\cdots\!03\)\( T^{30} + \)\(27\!\cdots\!67\)\( p T^{31} + \)\(27\!\cdots\!05\)\( p^{2} T^{32} + \)\(25\!\cdots\!00\)\( p^{3} T^{33} + \)\(24\!\cdots\!81\)\( p^{4} T^{34} + \)\(22\!\cdots\!71\)\( p^{5} T^{35} + \)\(19\!\cdots\!41\)\( p^{6} T^{36} + \)\(17\!\cdots\!98\)\( p^{7} T^{37} + \)\(14\!\cdots\!16\)\( p^{8} T^{38} + \)\(12\!\cdots\!77\)\( p^{9} T^{39} + \)\(96\!\cdots\!21\)\( p^{10} T^{40} + \)\(75\!\cdots\!17\)\( p^{11} T^{41} + \)\(57\!\cdots\!22\)\( p^{12} T^{42} + \)\(42\!\cdots\!63\)\( p^{13} T^{43} + \)\(30\!\cdots\!47\)\( p^{14} T^{44} + \)\(21\!\cdots\!70\)\( p^{15} T^{45} + 14171042950832535602 p^{16} T^{46} + 904968643304537993 p^{17} T^{47} + 56806978422944290 p^{18} T^{48} + 3325341875787166 p^{19} T^{49} + 192577538468617 p^{20} T^{50} + 10145311972192 p^{21} T^{51} + 536022340409 p^{22} T^{52} + 24690581180 p^{23} T^{53} + 1173356452 p^{24} T^{54} + 44966208 p^{25} T^{55} + 1887479 p^{26} T^{56} + 54505 p^{27} T^{57} + 1974 p^{28} T^{58} + 33 p^{29} T^{59} + p^{30} T^{60} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{60} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.65104188744116694953331636561, −1.64077988517538465786514690808, −1.63425692909703502647080694375, −1.59409285758841356847971979027, −1.59257913817707638900088634492, −1.50942449095690714030855860479, −1.50487334306441104995165970202, −1.49426203223721646308967936084, −1.43508431479748758931349643959, −1.43410921545636161905737553360, −1.42912102220169143540537570011, −1.41933230479102129309596849358, −1.41293242036486651731074345387, −1.37546159903023060944484672814, −1.36859176921252077902887965310, −1.30359148376382467492414265007, −1.26706232446887645713970544660, −1.22160400724322805009456388189, −1.20991165283800632370276430635, −1.13775712449719920279470983473, −1.03808353825987832176935234206, −1.00179035525224963913414570963, −0.959177492867094051376393720103, −0.946147187977739536330686042913, −0.77475302560310758174189215675, 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, 0, 0.77475302560310758174189215675, 0.946147187977739536330686042913, 0.959177492867094051376393720103, 1.00179035525224963913414570963, 1.03808353825987832176935234206, 1.13775712449719920279470983473, 1.20991165283800632370276430635, 1.22160400724322805009456388189, 1.26706232446887645713970544660, 1.30359148376382467492414265007, 1.36859176921252077902887965310, 1.37546159903023060944484672814, 1.41293242036486651731074345387, 1.41933230479102129309596849358, 1.42912102220169143540537570011, 1.43410921545636161905737553360, 1.43508431479748758931349643959, 1.49426203223721646308967936084, 1.50487334306441104995165970202, 1.50942449095690714030855860479, 1.59257913817707638900088634492, 1.59409285758841356847971979027, 1.63425692909703502647080694375, 1.64077988517538465786514690808, 1.65104188744116694953331636561

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

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