Properties

Degree 60
Conductor $ 619^{30} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s + 3-s + 27·4-s + 21·5-s + 9·6-s + 2·7-s + 3·8-s − 23·9-s + 189·10-s + 23·11-s + 27·12-s + 9·13-s + 18·14-s + 21·15-s − 163·16-s + 4·17-s − 207·18-s − 19-s + 567·20-s + 2·21-s + 207·22-s + 4·23-s + 3·24-s + 163·25-s + 81·26-s − 26·27-s + 54·28-s + ⋯
L(s)  = 1  + 6.36·2-s + 0.577·3-s + 27/2·4-s + 9.39·5-s + 3.67·6-s + 0.755·7-s + 1.06·8-s − 7.66·9-s + 59.7·10-s + 6.93·11-s + 7.79·12-s + 2.49·13-s + 4.81·14-s + 5.42·15-s − 40.7·16-s + 0.970·17-s − 48.7·18-s − 0.229·19-s + 126.·20-s + 0.436·21-s + 44.1·22-s + 0.834·23-s + 0.612·24-s + 32.5·25-s + 15.8·26-s − 5.00·27-s + 10.2·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(619^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{30} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(619^{30}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{30} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(60\)
\( N \)  =  \(619^{30}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{619} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(60,\ 619^{30} ,\ ( \ : [1/2]^{30} ),\ 1 )$
$L(1)$  $\approx$  $18935.9$
$L(\frac12)$  $\approx$  $18935.9$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 619$, \(F_p(T)\) is a polynomial of degree 60. If $p = 619$, then $F_p(T)$ is a polynomial of degree at most 59.
$p$$F_p(T)$
bad619 \( ( 1 - T )^{30} \)
good2 \( 1 - 9 T + 27 p T^{2} - 123 p T^{3} + 473 p T^{4} - 1591 p T^{5} + 9667 T^{6} - 26963 T^{7} + 70063 T^{8} - 42795 p^{2} T^{9} + 396509 T^{10} - 875895 T^{11} + 927529 p T^{12} - 1891125 p T^{13} + 3725647 p T^{14} - 14226449 T^{15} + 6598481 p^{2} T^{16} - 47688801 T^{17} + 21019809 p^{2} T^{18} - 72445281 p T^{19} + 61100305 p^{2} T^{20} - 404030905 T^{21} + 327650781 p T^{22} - 1043710633 T^{23} + 25526593 p^{6} T^{24} - 2514820331 T^{25} + 1904551281 p T^{26} - 2839781441 p T^{27} + 1042423567 p^{3} T^{28} - 1507644413 p^{3} T^{29} + 1074046423 p^{4} T^{30} - 1507644413 p^{4} T^{31} + 1042423567 p^{5} T^{32} - 2839781441 p^{4} T^{33} + 1904551281 p^{5} T^{34} - 2514820331 p^{5} T^{35} + 25526593 p^{12} T^{36} - 1043710633 p^{7} T^{37} + 327650781 p^{9} T^{38} - 404030905 p^{9} T^{39} + 61100305 p^{12} T^{40} - 72445281 p^{12} T^{41} + 21019809 p^{14} T^{42} - 47688801 p^{13} T^{43} + 6598481 p^{16} T^{44} - 14226449 p^{15} T^{45} + 3725647 p^{17} T^{46} - 1891125 p^{18} T^{47} + 927529 p^{19} T^{48} - 875895 p^{19} T^{49} + 396509 p^{20} T^{50} - 42795 p^{23} T^{51} + 70063 p^{22} T^{52} - 26963 p^{23} T^{53} + 9667 p^{24} T^{54} - 1591 p^{26} T^{55} + 473 p^{27} T^{56} - 123 p^{28} T^{57} + 27 p^{29} T^{58} - 9 p^{29} T^{59} + p^{30} T^{60} \)
3 \( 1 - T + 8 p T^{2} - 7 p T^{3} + 310 T^{4} - 77 p T^{5} + 2852 T^{6} - 1817 T^{7} + 20963 T^{8} - 3889 p T^{9} + 130886 T^{10} - 21854 p T^{11} + 239900 p T^{12} - 111496 p T^{13} + 3567380 T^{14} - 1579124 T^{15} + 16208674 T^{16} - 6972109 T^{17} + 7596635 p^{2} T^{18} - 28944256 T^{19} + 270300338 T^{20} - 113215942 T^{21} + 336195464 p T^{22} - 417794126 T^{23} + 1189896053 p T^{24} - 485982434 p T^{25} + 12025783559 T^{26} - 1608851612 p T^{27} + 38647874311 T^{28} - 15199706906 T^{29} + 118621307344 T^{30} - 15199706906 p T^{31} + 38647874311 p^{2} T^{32} - 1608851612 p^{4} T^{33} + 12025783559 p^{4} T^{34} - 485982434 p^{6} T^{35} + 1189896053 p^{7} T^{36} - 417794126 p^{7} T^{37} + 336195464 p^{9} T^{38} - 113215942 p^{9} T^{39} + 270300338 p^{10} T^{40} - 28944256 p^{11} T^{41} + 7596635 p^{14} T^{42} - 6972109 p^{13} T^{43} + 16208674 p^{14} T^{44} - 1579124 p^{15} T^{45} + 3567380 p^{16} T^{46} - 111496 p^{18} T^{47} + 239900 p^{19} T^{48} - 21854 p^{20} T^{49} + 130886 p^{20} T^{50} - 3889 p^{22} T^{51} + 20963 p^{22} T^{52} - 1817 p^{23} T^{53} + 2852 p^{24} T^{54} - 77 p^{26} T^{55} + 310 p^{26} T^{56} - 7 p^{28} T^{57} + 8 p^{29} T^{58} - p^{29} T^{59} + p^{30} T^{60} \)
5 \( 1 - 21 T + 278 T^{2} - 2741 T^{3} + 22291 T^{4} - 156043 T^{5} + 970717 T^{6} - 5468936 T^{7} + 28326368 T^{8} - 27259357 p T^{9} + 614473266 T^{10} - 2612498216 T^{11} + 10532087139 T^{12} - 40436132412 T^{13} + 148404916881 T^{14} - 522270545073 T^{15} + 1767207079133 T^{16} - 5762625437033 T^{17} + 3629161313318 p T^{18} - 55273125437584 T^{19} + 6524842811754 p^{2} T^{20} - 467036140126606 T^{21} + 1298845545146308 T^{22} - 3512218811836043 T^{23} + 1848627316874511 p T^{24} - 23692222125287404 T^{25} + 59187059931528014 T^{26} - 144181980867405643 T^{27} + 342641412796337884 T^{28} - 794582888657395032 T^{29} + 1798406232223799877 T^{30} - 794582888657395032 p T^{31} + 342641412796337884 p^{2} T^{32} - 144181980867405643 p^{3} T^{33} + 59187059931528014 p^{4} T^{34} - 23692222125287404 p^{5} T^{35} + 1848627316874511 p^{7} T^{36} - 3512218811836043 p^{7} T^{37} + 1298845545146308 p^{8} T^{38} - 467036140126606 p^{9} T^{39} + 6524842811754 p^{12} T^{40} - 55273125437584 p^{11} T^{41} + 3629161313318 p^{13} T^{42} - 5762625437033 p^{13} T^{43} + 1767207079133 p^{14} T^{44} - 522270545073 p^{15} T^{45} + 148404916881 p^{16} T^{46} - 40436132412 p^{17} T^{47} + 10532087139 p^{18} T^{48} - 2612498216 p^{19} T^{49} + 614473266 p^{20} T^{50} - 27259357 p^{22} T^{51} + 28326368 p^{22} T^{52} - 5468936 p^{23} T^{53} + 970717 p^{24} T^{54} - 156043 p^{25} T^{55} + 22291 p^{26} T^{56} - 2741 p^{27} T^{57} + 278 p^{28} T^{58} - 21 p^{29} T^{59} + p^{30} T^{60} \)
7 \( 1 - 2 T + 92 T^{2} - 153 T^{3} + 4359 T^{4} - 6008 T^{5} + 140900 T^{6} - 159108 T^{7} + 3482582 T^{8} - 3163716 T^{9} + 70020663 T^{10} - 49883786 T^{11} + 1190529390 T^{12} - 642571316 T^{13} + 17578545940 T^{14} - 6855842867 T^{15} + 229788281559 T^{16} - 60476038765 T^{17} + 2698261407045 T^{18} - 429050233032 T^{19} + 28781415820261 T^{20} - 2209967181326 T^{21} + 281309343879335 T^{22} - 4278499703946 T^{23} + 2536296799557894 T^{24} + 9964615496167 p T^{25} + 21199209124324068 T^{26} + 1131541289837402 T^{27} + 164832135790678012 T^{28} + 10612225966252194 T^{29} + 1194677185398803334 T^{30} + 10612225966252194 p T^{31} + 164832135790678012 p^{2} T^{32} + 1131541289837402 p^{3} T^{33} + 21199209124324068 p^{4} T^{34} + 9964615496167 p^{6} T^{35} + 2536296799557894 p^{6} T^{36} - 4278499703946 p^{7} T^{37} + 281309343879335 p^{8} T^{38} - 2209967181326 p^{9} T^{39} + 28781415820261 p^{10} T^{40} - 429050233032 p^{11} T^{41} + 2698261407045 p^{12} T^{42} - 60476038765 p^{13} T^{43} + 229788281559 p^{14} T^{44} - 6855842867 p^{15} T^{45} + 17578545940 p^{16} T^{46} - 642571316 p^{17} T^{47} + 1190529390 p^{18} T^{48} - 49883786 p^{19} T^{49} + 70020663 p^{20} T^{50} - 3163716 p^{21} T^{51} + 3482582 p^{22} T^{52} - 159108 p^{23} T^{53} + 140900 p^{24} T^{54} - 6008 p^{25} T^{55} + 4359 p^{26} T^{56} - 153 p^{27} T^{57} + 92 p^{28} T^{58} - 2 p^{29} T^{59} + p^{30} T^{60} \)
11 \( 1 - 23 T + 417 T^{2} - 5406 T^{3} + 60864 T^{4} - 581923 T^{5} + 5044038 T^{6} - 39346950 T^{7} + 284958715 T^{8} - 1910206568 T^{9} + 12064881042 T^{10} - 71688005412 T^{11} + 405325326493 T^{12} - 2178631492750 T^{13} + 11221651606018 T^{14} - 55350835682740 T^{15} + 263028348184128 T^{16} - 1203526951822814 T^{17} + 5328362989750426 T^{18} - 22815398334963428 T^{19} + 94878824297060786 T^{20} - 383064823324194221 T^{21} + 1507211682135660910 T^{22} - 5777779181057783344 T^{23} + 21654977351231973575 T^{24} - 7211909623319238430 p T^{25} + \)\(28\!\cdots\!07\)\( T^{26} - \)\(10\!\cdots\!85\)\( T^{27} + \)\(34\!\cdots\!90\)\( T^{28} - \)\(11\!\cdots\!76\)\( T^{29} + \)\(39\!\cdots\!40\)\( T^{30} - \)\(11\!\cdots\!76\)\( p T^{31} + \)\(34\!\cdots\!90\)\( p^{2} T^{32} - \)\(10\!\cdots\!85\)\( p^{3} T^{33} + \)\(28\!\cdots\!07\)\( p^{4} T^{34} - 7211909623319238430 p^{6} T^{35} + 21654977351231973575 p^{6} T^{36} - 5777779181057783344 p^{7} T^{37} + 1507211682135660910 p^{8} T^{38} - 383064823324194221 p^{9} T^{39} + 94878824297060786 p^{10} T^{40} - 22815398334963428 p^{11} T^{41} + 5328362989750426 p^{12} T^{42} - 1203526951822814 p^{13} T^{43} + 263028348184128 p^{14} T^{44} - 55350835682740 p^{15} T^{45} + 11221651606018 p^{16} T^{46} - 2178631492750 p^{17} T^{47} + 405325326493 p^{18} T^{48} - 71688005412 p^{19} T^{49} + 12064881042 p^{20} T^{50} - 1910206568 p^{21} T^{51} + 284958715 p^{22} T^{52} - 39346950 p^{23} T^{53} + 5044038 p^{24} T^{54} - 581923 p^{25} T^{55} + 60864 p^{26} T^{56} - 5406 p^{27} T^{57} + 417 p^{28} T^{58} - 23 p^{29} T^{59} + p^{30} T^{60} \)
13 \( 1 - 9 T + 230 T^{2} - 1753 T^{3} + 25058 T^{4} - 166005 T^{5} + 1740019 T^{6} - 10200829 T^{7} + 87198401 T^{8} - 458237691 T^{9} + 3382407624 T^{10} - 16087373425 T^{11} + 106387608297 T^{12} - 461447308057 T^{13} + 2809654186071 T^{14} - 11187570981445 T^{15} + 64140979200907 T^{16} - 236049257178399 T^{17} + 1299167572692133 T^{18} - 4453670473724057 T^{19} + 1838953079065561 p T^{20} - 77051063052576792 T^{21} + 31354528789110029 p T^{22} - 1247402355417714228 T^{23} + 6525337577309794600 T^{24} - 19125073519639054711 T^{25} + 98619268660968747733 T^{26} - \)\(27\!\cdots\!79\)\( T^{27} + \)\(14\!\cdots\!55\)\( T^{28} - \)\(38\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!30\)\( T^{30} - \)\(38\!\cdots\!76\)\( p T^{31} + \)\(14\!\cdots\!55\)\( p^{2} T^{32} - \)\(27\!\cdots\!79\)\( p^{3} T^{33} + 98619268660968747733 p^{4} T^{34} - 19125073519639054711 p^{5} T^{35} + 6525337577309794600 p^{6} T^{36} - 1247402355417714228 p^{7} T^{37} + 31354528789110029 p^{9} T^{38} - 77051063052576792 p^{9} T^{39} + 1838953079065561 p^{11} T^{40} - 4453670473724057 p^{11} T^{41} + 1299167572692133 p^{12} T^{42} - 236049257178399 p^{13} T^{43} + 64140979200907 p^{14} T^{44} - 11187570981445 p^{15} T^{45} + 2809654186071 p^{16} T^{46} - 461447308057 p^{17} T^{47} + 106387608297 p^{18} T^{48} - 16087373425 p^{19} T^{49} + 3382407624 p^{20} T^{50} - 458237691 p^{21} T^{51} + 87198401 p^{22} T^{52} - 10200829 p^{23} T^{53} + 1740019 p^{24} T^{54} - 166005 p^{25} T^{55} + 25058 p^{26} T^{56} - 1753 p^{27} T^{57} + 230 p^{28} T^{58} - 9 p^{29} T^{59} + p^{30} T^{60} \)
17 \( 1 - 4 T + 274 T^{2} - 1149 T^{3} + 37474 T^{4} - 159925 T^{5} + 3396929 T^{6} - 14402007 T^{7} + 228492593 T^{8} - 943071392 T^{9} + 12101729171 T^{10} - 47743198925 T^{11} + 522811336278 T^{12} - 1935567915616 T^{13} + 18838549194905 T^{14} - 64061486238193 T^{15} + 574296387593546 T^{16} - 1741874063748629 T^{17} + 878911259844931 p T^{18} - 38558089182049577 T^{19} + 333369419401902599 T^{20} - 668958686515401317 T^{21} + 6397986708260835339 T^{22} - 7995784155997884377 T^{23} + \)\(10\!\cdots\!02\)\( T^{24} - 23707362906635006004 T^{25} + \)\(15\!\cdots\!40\)\( T^{26} + \)\(17\!\cdots\!44\)\( T^{27} + \)\(21\!\cdots\!95\)\( T^{28} + \)\(54\!\cdots\!79\)\( T^{29} + \)\(33\!\cdots\!10\)\( T^{30} + \)\(54\!\cdots\!79\)\( p T^{31} + \)\(21\!\cdots\!95\)\( p^{2} T^{32} + \)\(17\!\cdots\!44\)\( p^{3} T^{33} + \)\(15\!\cdots\!40\)\( p^{4} T^{34} - 23707362906635006004 p^{5} T^{35} + \)\(10\!\cdots\!02\)\( p^{6} T^{36} - 7995784155997884377 p^{7} T^{37} + 6397986708260835339 p^{8} T^{38} - 668958686515401317 p^{9} T^{39} + 333369419401902599 p^{10} T^{40} - 38558089182049577 p^{11} T^{41} + 878911259844931 p^{13} T^{42} - 1741874063748629 p^{13} T^{43} + 574296387593546 p^{14} T^{44} - 64061486238193 p^{15} T^{45} + 18838549194905 p^{16} T^{46} - 1935567915616 p^{17} T^{47} + 522811336278 p^{18} T^{48} - 47743198925 p^{19} T^{49} + 12101729171 p^{20} T^{50} - 943071392 p^{21} T^{51} + 228492593 p^{22} T^{52} - 14402007 p^{23} T^{53} + 3396929 p^{24} T^{54} - 159925 p^{25} T^{55} + 37474 p^{26} T^{56} - 1149 p^{27} T^{57} + 274 p^{28} T^{58} - 4 p^{29} T^{59} + p^{30} T^{60} \)
19 \( 1 + T + 299 T^{2} + 364 T^{3} + 44785 T^{4} + 60228 T^{5} + 4475186 T^{6} + 6217725 T^{7} + 334984069 T^{8} + 455483465 T^{9} + 19992328576 T^{10} + 25284906309 T^{11} + 988990820287 T^{12} + 1100533281113 T^{13} + 41646909395312 T^{14} + 37973150404832 T^{15} + 1522776657538006 T^{16} + 1020347274584634 T^{17} + 49118284403988693 T^{18} + 19543910820176217 T^{19} + 1416901962342862126 T^{20} + 155481168968878177 T^{21} + 37014318468367588964 T^{22} - 6429477447635855710 T^{23} + \)\(88\!\cdots\!68\)\( T^{24} - \)\(37\!\cdots\!74\)\( T^{25} + \)\(19\!\cdots\!20\)\( T^{26} - \)\(11\!\cdots\!52\)\( T^{27} + \)\(40\!\cdots\!70\)\( T^{28} - \)\(28\!\cdots\!11\)\( T^{29} + \)\(79\!\cdots\!96\)\( T^{30} - \)\(28\!\cdots\!11\)\( p T^{31} + \)\(40\!\cdots\!70\)\( p^{2} T^{32} - \)\(11\!\cdots\!52\)\( p^{3} T^{33} + \)\(19\!\cdots\!20\)\( p^{4} T^{34} - \)\(37\!\cdots\!74\)\( p^{5} T^{35} + \)\(88\!\cdots\!68\)\( p^{6} T^{36} - 6429477447635855710 p^{7} T^{37} + 37014318468367588964 p^{8} T^{38} + 155481168968878177 p^{9} T^{39} + 1416901962342862126 p^{10} T^{40} + 19543910820176217 p^{11} T^{41} + 49118284403988693 p^{12} T^{42} + 1020347274584634 p^{13} T^{43} + 1522776657538006 p^{14} T^{44} + 37973150404832 p^{15} T^{45} + 41646909395312 p^{16} T^{46} + 1100533281113 p^{17} T^{47} + 988990820287 p^{18} T^{48} + 25284906309 p^{19} T^{49} + 19992328576 p^{20} T^{50} + 455483465 p^{21} T^{51} + 334984069 p^{22} T^{52} + 6217725 p^{23} T^{53} + 4475186 p^{24} T^{54} + 60228 p^{25} T^{55} + 44785 p^{26} T^{56} + 364 p^{27} T^{57} + 299 p^{28} T^{58} + p^{29} T^{59} + p^{30} T^{60} \)
23 \( 1 - 4 T + 393 T^{2} - 1551 T^{3} + 76233 T^{4} - 296441 T^{5} + 9721382 T^{6} - 37218489 T^{7} + 915943887 T^{8} - 3449866302 T^{9} + 67949985366 T^{10} - 251485324555 T^{11} + 4131039191504 T^{12} - 14994887076808 T^{13} + 211570143218677 T^{14} - 751062451179800 T^{15} + 9316915931348688 T^{16} - 32226618502036128 T^{17} + 358641366498286617 T^{18} - 1203432350443318255 T^{19} + 12244050650885226417 T^{20} - 1725286048636054336 p T^{21} + \)\(37\!\cdots\!60\)\( T^{22} - \)\(11\!\cdots\!75\)\( T^{23} + \)\(10\!\cdots\!89\)\( T^{24} - \)\(31\!\cdots\!39\)\( T^{25} + \)\(27\!\cdots\!12\)\( T^{26} - \)\(34\!\cdots\!41\)\( p T^{27} + \)\(67\!\cdots\!96\)\( T^{28} - \)\(19\!\cdots\!52\)\( T^{29} + \)\(15\!\cdots\!23\)\( T^{30} - \)\(19\!\cdots\!52\)\( p T^{31} + \)\(67\!\cdots\!96\)\( p^{2} T^{32} - \)\(34\!\cdots\!41\)\( p^{4} T^{33} + \)\(27\!\cdots\!12\)\( p^{4} T^{34} - \)\(31\!\cdots\!39\)\( p^{5} T^{35} + \)\(10\!\cdots\!89\)\( p^{6} T^{36} - \)\(11\!\cdots\!75\)\( p^{7} T^{37} + \)\(37\!\cdots\!60\)\( p^{8} T^{38} - 1725286048636054336 p^{10} T^{39} + 12244050650885226417 p^{10} T^{40} - 1203432350443318255 p^{11} T^{41} + 358641366498286617 p^{12} T^{42} - 32226618502036128 p^{13} T^{43} + 9316915931348688 p^{14} T^{44} - 751062451179800 p^{15} T^{45} + 211570143218677 p^{16} T^{46} - 14994887076808 p^{17} T^{47} + 4131039191504 p^{18} T^{48} - 251485324555 p^{19} T^{49} + 67949985366 p^{20} T^{50} - 3449866302 p^{21} T^{51} + 915943887 p^{22} T^{52} - 37218489 p^{23} T^{53} + 9721382 p^{24} T^{54} - 296441 p^{25} T^{55} + 76233 p^{26} T^{56} - 1551 p^{27} T^{57} + 393 p^{28} T^{58} - 4 p^{29} T^{59} + p^{30} T^{60} \)
29 \( 1 - 90 T + 4352 T^{2} - 148548 T^{3} + 3990530 T^{4} - 89421834 T^{5} + 1732761454 T^{6} - 29749973014 T^{7} + 460549714199 T^{8} - 6513268715532 T^{9} + 85008446716026 T^{10} - 1032210221937566 T^{11} + 11736983174707927 T^{12} - 125651055102605697 T^{13} + 1272182238288786456 T^{14} - 12227915630802629496 T^{15} + \)\(11\!\cdots\!36\)\( T^{16} - \)\(97\!\cdots\!08\)\( T^{17} + \)\(81\!\cdots\!58\)\( T^{18} - \)\(65\!\cdots\!58\)\( T^{19} + \)\(50\!\cdots\!35\)\( T^{20} - \)\(37\!\cdots\!52\)\( T^{21} + \)\(26\!\cdots\!56\)\( T^{22} - \)\(18\!\cdots\!98\)\( T^{23} + \)\(12\!\cdots\!90\)\( T^{24} - \)\(79\!\cdots\!70\)\( T^{25} + \)\(49\!\cdots\!96\)\( T^{26} - \)\(30\!\cdots\!40\)\( T^{27} + \)\(17\!\cdots\!34\)\( T^{28} - \)\(98\!\cdots\!29\)\( T^{29} + \)\(54\!\cdots\!80\)\( T^{30} - \)\(98\!\cdots\!29\)\( p T^{31} + \)\(17\!\cdots\!34\)\( p^{2} T^{32} - \)\(30\!\cdots\!40\)\( p^{3} T^{33} + \)\(49\!\cdots\!96\)\( p^{4} T^{34} - \)\(79\!\cdots\!70\)\( p^{5} T^{35} + \)\(12\!\cdots\!90\)\( p^{6} T^{36} - \)\(18\!\cdots\!98\)\( p^{7} T^{37} + \)\(26\!\cdots\!56\)\( p^{8} T^{38} - \)\(37\!\cdots\!52\)\( p^{9} T^{39} + \)\(50\!\cdots\!35\)\( p^{10} T^{40} - \)\(65\!\cdots\!58\)\( p^{11} T^{41} + \)\(81\!\cdots\!58\)\( p^{12} T^{42} - \)\(97\!\cdots\!08\)\( p^{13} T^{43} + \)\(11\!\cdots\!36\)\( p^{14} T^{44} - 12227915630802629496 p^{15} T^{45} + 1272182238288786456 p^{16} T^{46} - 125651055102605697 p^{17} T^{47} + 11736983174707927 p^{18} T^{48} - 1032210221937566 p^{19} T^{49} + 85008446716026 p^{20} T^{50} - 6513268715532 p^{21} T^{51} + 460549714199 p^{22} T^{52} - 29749973014 p^{23} T^{53} + 1732761454 p^{24} T^{54} - 89421834 p^{25} T^{55} + 3990530 p^{26} T^{56} - 148548 p^{27} T^{57} + 4352 p^{28} T^{58} - 90 p^{29} T^{59} + p^{30} T^{60} \)
31 \( 1 - 2 T + 447 T^{2} - 882 T^{3} + 101112 T^{4} - 191257 T^{5} + 15419952 T^{6} - 27299265 T^{7} + 1781644764 T^{8} - 2895164888 T^{9} + 166190841591 T^{10} - 244098538188 T^{11} + 13026761278769 T^{12} - 17100796479877 T^{13} + 882052408252468 T^{14} - 1027962714610463 T^{15} + 52638815146756317 T^{16} - 54372601467646263 T^{17} + 2810773259974449189 T^{18} - 2582980634700917962 T^{19} + \)\(13\!\cdots\!42\)\( T^{20} - \)\(11\!\cdots\!67\)\( T^{21} + \)\(59\!\cdots\!58\)\( T^{22} - \)\(44\!\cdots\!34\)\( T^{23} + \)\(24\!\cdots\!61\)\( T^{24} - \)\(16\!\cdots\!41\)\( T^{25} + \)\(91\!\cdots\!98\)\( T^{26} - \)\(58\!\cdots\!89\)\( T^{27} + \)\(31\!\cdots\!85\)\( T^{28} - \)\(19\!\cdots\!99\)\( T^{29} + \)\(10\!\cdots\!56\)\( T^{30} - \)\(19\!\cdots\!99\)\( p T^{31} + \)\(31\!\cdots\!85\)\( p^{2} T^{32} - \)\(58\!\cdots\!89\)\( p^{3} T^{33} + \)\(91\!\cdots\!98\)\( p^{4} T^{34} - \)\(16\!\cdots\!41\)\( p^{5} T^{35} + \)\(24\!\cdots\!61\)\( p^{6} T^{36} - \)\(44\!\cdots\!34\)\( p^{7} T^{37} + \)\(59\!\cdots\!58\)\( p^{8} T^{38} - \)\(11\!\cdots\!67\)\( p^{9} T^{39} + \)\(13\!\cdots\!42\)\( p^{10} T^{40} - 2582980634700917962 p^{11} T^{41} + 2810773259974449189 p^{12} T^{42} - 54372601467646263 p^{13} T^{43} + 52638815146756317 p^{14} T^{44} - 1027962714610463 p^{15} T^{45} + 882052408252468 p^{16} T^{46} - 17100796479877 p^{17} T^{47} + 13026761278769 p^{18} T^{48} - 244098538188 p^{19} T^{49} + 166190841591 p^{20} T^{50} - 2895164888 p^{21} T^{51} + 1781644764 p^{22} T^{52} - 27299265 p^{23} T^{53} + 15419952 p^{24} T^{54} - 191257 p^{25} T^{55} + 101112 p^{26} T^{56} - 882 p^{27} T^{57} + 447 p^{28} T^{58} - 2 p^{29} T^{59} + p^{30} T^{60} \)
37 \( 1 - 19 T + 827 T^{2} - 13119 T^{3} + 322185 T^{4} - 4419171 T^{5} + 79704785 T^{6} - 967868002 T^{7} + 14174936071 T^{8} - 154986654974 T^{9} + 1940589972944 T^{10} - 19349421692339 T^{11} + 213600251758575 T^{12} - 1961709031545811 T^{13} + 19483869611152708 T^{14} - 166181780161093581 T^{15} + 1506684001689047801 T^{16} - 12020578479733803124 T^{17} + \)\(10\!\cdots\!61\)\( T^{18} - \)\(75\!\cdots\!56\)\( T^{19} + \)\(58\!\cdots\!99\)\( T^{20} - \)\(41\!\cdots\!33\)\( T^{21} + \)\(30\!\cdots\!64\)\( T^{22} - \)\(20\!\cdots\!56\)\( T^{23} + \)\(14\!\cdots\!35\)\( T^{24} - \)\(94\!\cdots\!47\)\( T^{25} + \)\(62\!\cdots\!79\)\( T^{26} - \)\(39\!\cdots\!32\)\( T^{27} + \)\(25\!\cdots\!65\)\( T^{28} - \)\(15\!\cdots\!45\)\( T^{29} + \)\(96\!\cdots\!37\)\( T^{30} - \)\(15\!\cdots\!45\)\( p T^{31} + \)\(25\!\cdots\!65\)\( p^{2} T^{32} - \)\(39\!\cdots\!32\)\( p^{3} T^{33} + \)\(62\!\cdots\!79\)\( p^{4} T^{34} - \)\(94\!\cdots\!47\)\( p^{5} T^{35} + \)\(14\!\cdots\!35\)\( p^{6} T^{36} - \)\(20\!\cdots\!56\)\( p^{7} T^{37} + \)\(30\!\cdots\!64\)\( p^{8} T^{38} - \)\(41\!\cdots\!33\)\( p^{9} T^{39} + \)\(58\!\cdots\!99\)\( p^{10} T^{40} - \)\(75\!\cdots\!56\)\( p^{11} T^{41} + \)\(10\!\cdots\!61\)\( p^{12} T^{42} - 12020578479733803124 p^{13} T^{43} + 1506684001689047801 p^{14} T^{44} - 166181780161093581 p^{15} T^{45} + 19483869611152708 p^{16} T^{46} - 1961709031545811 p^{17} T^{47} + 213600251758575 p^{18} T^{48} - 19349421692339 p^{19} T^{49} + 1940589972944 p^{20} T^{50} - 154986654974 p^{21} T^{51} + 14174936071 p^{22} T^{52} - 967868002 p^{23} T^{53} + 79704785 p^{24} T^{54} - 4419171 p^{25} T^{55} + 322185 p^{26} T^{56} - 13119 p^{27} T^{57} + 827 p^{28} T^{58} - 19 p^{29} T^{59} + p^{30} T^{60} \)
41 \( 1 - 59 T + 2426 T^{2} - 73454 T^{3} + 1857601 T^{4} - 40158554 T^{5} + 770809457 T^{6} - 13310933620 T^{7} + 210425033757 T^{8} - 3071248213495 T^{9} + 41797697118575 T^{10} - 533566516992332 T^{11} + 6429917949274107 T^{12} - 73467369700123603 T^{13} + 799471242821592820 T^{14} - 8312988173874365158 T^{15} + 2021123414786559148 p T^{16} - \)\(79\!\cdots\!77\)\( T^{17} + \)\(73\!\cdots\!23\)\( T^{18} - \)\(65\!\cdots\!40\)\( T^{19} + \)\(56\!\cdots\!92\)\( T^{20} - \)\(46\!\cdots\!58\)\( T^{21} + \)\(37\!\cdots\!37\)\( T^{22} - \)\(29\!\cdots\!11\)\( T^{23} + \)\(22\!\cdots\!27\)\( T^{24} - \)\(16\!\cdots\!94\)\( T^{25} + \)\(12\!\cdots\!90\)\( T^{26} - \)\(85\!\cdots\!41\)\( T^{27} + \)\(58\!\cdots\!77\)\( T^{28} - \)\(38\!\cdots\!66\)\( T^{29} + \)\(61\!\cdots\!09\)\( p T^{30} - \)\(38\!\cdots\!66\)\( p T^{31} + \)\(58\!\cdots\!77\)\( p^{2} T^{32} - \)\(85\!\cdots\!41\)\( p^{3} T^{33} + \)\(12\!\cdots\!90\)\( p^{4} T^{34} - \)\(16\!\cdots\!94\)\( p^{5} T^{35} + \)\(22\!\cdots\!27\)\( p^{6} T^{36} - \)\(29\!\cdots\!11\)\( p^{7} T^{37} + \)\(37\!\cdots\!37\)\( p^{8} T^{38} - \)\(46\!\cdots\!58\)\( p^{9} T^{39} + \)\(56\!\cdots\!92\)\( p^{10} T^{40} - \)\(65\!\cdots\!40\)\( p^{11} T^{41} + \)\(73\!\cdots\!23\)\( p^{12} T^{42} - \)\(79\!\cdots\!77\)\( p^{13} T^{43} + 2021123414786559148 p^{15} T^{44} - 8312988173874365158 p^{15} T^{45} + 799471242821592820 p^{16} T^{46} - 73467369700123603 p^{17} T^{47} + 6429917949274107 p^{18} T^{48} - 533566516992332 p^{19} T^{49} + 41797697118575 p^{20} T^{50} - 3071248213495 p^{21} T^{51} + 210425033757 p^{22} T^{52} - 13310933620 p^{23} T^{53} + 770809457 p^{24} T^{54} - 40158554 p^{25} T^{55} + 1857601 p^{26} T^{56} - 73454 p^{27} T^{57} + 2426 p^{28} T^{58} - 59 p^{29} T^{59} + p^{30} T^{60} \)
43 \( 1 + 4 T + 663 T^{2} + 2960 T^{3} + 218829 T^{4} + 1044706 T^{5} + 48007367 T^{6} + 236761990 T^{7} + 7872192777 T^{8} + 38982329665 T^{9} + 1027519969169 T^{10} + 4987067692928 T^{11} + 110956376738033 T^{12} + 516518726019784 T^{13} + 10172223965091866 T^{14} + 44475046381146890 T^{15} + 806693658887715321 T^{16} + 3240205067778251322 T^{17} + 56162658871929824055 T^{18} + \)\(20\!\cdots\!01\)\( T^{19} + \)\(34\!\cdots\!82\)\( T^{20} + \)\(10\!\cdots\!39\)\( T^{21} + \)\(19\!\cdots\!23\)\( T^{22} + \)\(51\!\cdots\!49\)\( T^{23} + \)\(99\!\cdots\!22\)\( T^{24} + \)\(21\!\cdots\!36\)\( T^{25} + \)\(47\!\cdots\!33\)\( T^{26} + \)\(83\!\cdots\!62\)\( T^{27} + \)\(21\!\cdots\!03\)\( T^{28} + \)\(32\!\cdots\!08\)\( T^{29} + \)\(93\!\cdots\!24\)\( T^{30} + \)\(32\!\cdots\!08\)\( p T^{31} + \)\(21\!\cdots\!03\)\( p^{2} T^{32} + \)\(83\!\cdots\!62\)\( p^{3} T^{33} + \)\(47\!\cdots\!33\)\( p^{4} T^{34} + \)\(21\!\cdots\!36\)\( p^{5} T^{35} + \)\(99\!\cdots\!22\)\( p^{6} T^{36} + \)\(51\!\cdots\!49\)\( p^{7} T^{37} + \)\(19\!\cdots\!23\)\( p^{8} T^{38} + \)\(10\!\cdots\!39\)\( p^{9} T^{39} + \)\(34\!\cdots\!82\)\( p^{10} T^{40} + \)\(20\!\cdots\!01\)\( p^{11} T^{41} + 56162658871929824055 p^{12} T^{42} + 3240205067778251322 p^{13} T^{43} + 806693658887715321 p^{14} T^{44} + 44475046381146890 p^{15} T^{45} + 10172223965091866 p^{16} T^{46} + 516518726019784 p^{17} T^{47} + 110956376738033 p^{18} T^{48} + 4987067692928 p^{19} T^{49} + 1027519969169 p^{20} T^{50} + 38982329665 p^{21} T^{51} + 7872192777 p^{22} T^{52} + 236761990 p^{23} T^{53} + 48007367 p^{24} T^{54} + 1044706 p^{25} T^{55} + 218829 p^{26} T^{56} + 2960 p^{27} T^{57} + 663 p^{28} T^{58} + 4 p^{29} T^{59} + p^{30} T^{60} \)
47 \( 1 - 4 T + 16 p T^{2} - 3562 T^{3} + 286836 T^{4} - 1527600 T^{5} + 73827974 T^{6} - 425532020 T^{7} + 14385153692 T^{8} - 87161908434 T^{9} + 2256151095495 T^{10} - 14052269016410 T^{11} + 295740094994387 T^{12} - 39598637180816 p T^{13} + 33225057222230658 T^{14} - 208513311865820675 T^{15} + 3257247765938279416 T^{16} - 20185382974927578725 T^{17} + \)\(28\!\cdots\!96\)\( T^{18} - \)\(17\!\cdots\!00\)\( T^{19} + \)\(21\!\cdots\!83\)\( T^{20} - \)\(12\!\cdots\!50\)\( T^{21} + \)\(15\!\cdots\!52\)\( T^{22} - \)\(88\!\cdots\!24\)\( T^{23} + \)\(97\!\cdots\!29\)\( T^{24} - \)\(54\!\cdots\!22\)\( T^{25} + \)\(56\!\cdots\!53\)\( T^{26} - \)\(30\!\cdots\!61\)\( T^{27} + \)\(30\!\cdots\!40\)\( T^{28} - \)\(15\!\cdots\!69\)\( T^{29} + \)\(14\!\cdots\!64\)\( T^{30} - \)\(15\!\cdots\!69\)\( p T^{31} + \)\(30\!\cdots\!40\)\( p^{2} T^{32} - \)\(30\!\cdots\!61\)\( p^{3} T^{33} + \)\(56\!\cdots\!53\)\( p^{4} T^{34} - \)\(54\!\cdots\!22\)\( p^{5} T^{35} + \)\(97\!\cdots\!29\)\( p^{6} T^{36} - \)\(88\!\cdots\!24\)\( p^{7} T^{37} + \)\(15\!\cdots\!52\)\( p^{8} T^{38} - \)\(12\!\cdots\!50\)\( p^{9} T^{39} + \)\(21\!\cdots\!83\)\( p^{10} T^{40} - \)\(17\!\cdots\!00\)\( p^{11} T^{41} + \)\(28\!\cdots\!96\)\( p^{12} T^{42} - 20185382974927578725 p^{13} T^{43} + 3257247765938279416 p^{14} T^{44} - 208513311865820675 p^{15} T^{45} + 33225057222230658 p^{16} T^{46} - 39598637180816 p^{18} T^{47} + 295740094994387 p^{18} T^{48} - 14052269016410 p^{19} T^{49} + 2256151095495 p^{20} T^{50} - 87161908434 p^{21} T^{51} + 14385153692 p^{22} T^{52} - 425532020 p^{23} T^{53} + 73827974 p^{24} T^{54} - 1527600 p^{25} T^{55} + 286836 p^{26} T^{56} - 3562 p^{27} T^{57} + 16 p^{29} T^{58} - 4 p^{29} T^{59} + p^{30} T^{60} \)
53 \( 1 - 34 T + 1471 T^{2} - 36401 T^{3} + 952246 T^{4} - 19021035 T^{5} + 381481040 T^{6} - 6494230706 T^{7} + 108972496934 T^{8} - 1633856330805 T^{9} + 23974945439715 T^{10} - 6104280380469 p T^{11} + 4261872831921753 T^{12} - 52550624419061396 T^{13} + 632176206327032573 T^{14} - 7201141246370869569 T^{15} + 80050782093835862903 T^{16} - 16024188599051228686 p T^{17} + \)\(87\!\cdots\!55\)\( T^{18} - \)\(87\!\cdots\!48\)\( T^{19} + \)\(84\!\cdots\!42\)\( T^{20} - \)\(79\!\cdots\!17\)\( T^{21} + \)\(72\!\cdots\!74\)\( T^{22} - \)\(64\!\cdots\!40\)\( T^{23} + \)\(55\!\cdots\!11\)\( T^{24} - \)\(46\!\cdots\!11\)\( T^{25} + \)\(38\!\cdots\!57\)\( T^{26} - \)\(30\!\cdots\!89\)\( T^{27} + \)\(23\!\cdots\!75\)\( T^{28} - \)\(17\!\cdots\!97\)\( T^{29} + \)\(13\!\cdots\!78\)\( T^{30} - \)\(17\!\cdots\!97\)\( p T^{31} + \)\(23\!\cdots\!75\)\( p^{2} T^{32} - \)\(30\!\cdots\!89\)\( p^{3} T^{33} + \)\(38\!\cdots\!57\)\( p^{4} T^{34} - \)\(46\!\cdots\!11\)\( p^{5} T^{35} + \)\(55\!\cdots\!11\)\( p^{6} T^{36} - \)\(64\!\cdots\!40\)\( p^{7} T^{37} + \)\(72\!\cdots\!74\)\( p^{8} T^{38} - \)\(79\!\cdots\!17\)\( p^{9} T^{39} + \)\(84\!\cdots\!42\)\( p^{10} T^{40} - \)\(87\!\cdots\!48\)\( p^{11} T^{41} + \)\(87\!\cdots\!55\)\( p^{12} T^{42} - 16024188599051228686 p^{14} T^{43} + 80050782093835862903 p^{14} T^{44} - 7201141246370869569 p^{15} T^{45} + 632176206327032573 p^{16} T^{46} - 52550624419061396 p^{17} T^{47} + 4261872831921753 p^{18} T^{48} - 6104280380469 p^{20} T^{49} + 23974945439715 p^{20} T^{50} - 1633856330805 p^{21} T^{51} + 108972496934 p^{22} T^{52} - 6494230706 p^{23} T^{53} + 381481040 p^{24} T^{54} - 19021035 p^{25} T^{55} + 952246 p^{26} T^{56} - 36401 p^{27} T^{57} + 1471 p^{28} T^{58} - 34 p^{29} T^{59} + p^{30} T^{60} \)
59 \( 1 - 13 T + 887 T^{2} - 9832 T^{3} + 6554 p T^{4} - 3776072 T^{5} + 112087766 T^{6} - 984203758 T^{7} + 24469911811 T^{8} - 195812176184 T^{9} + 4304788741632 T^{10} - 31690828190954 T^{11} + 636536459537504 T^{12} - 4341657166114684 T^{13} + 81404241895797897 T^{14} - 517378143126114585 T^{15} + 9189152233410898885 T^{16} - 54689785635228592808 T^{17} + \)\(92\!\cdots\!25\)\( T^{18} - \)\(52\!\cdots\!82\)\( T^{19} + \)\(85\!\cdots\!96\)\( T^{20} - \)\(45\!\cdots\!76\)\( T^{21} + \)\(71\!\cdots\!02\)\( T^{22} - \)\(35\!\cdots\!04\)\( T^{23} + \)\(54\!\cdots\!79\)\( T^{24} - \)\(26\!\cdots\!74\)\( T^{25} + \)\(39\!\cdots\!56\)\( T^{26} - \)\(17\!\cdots\!39\)\( T^{27} + \)\(25\!\cdots\!42\)\( T^{28} - \)\(11\!\cdots\!39\)\( T^{29} + \)\(15\!\cdots\!22\)\( T^{30} - \)\(11\!\cdots\!39\)\( p T^{31} + \)\(25\!\cdots\!42\)\( p^{2} T^{32} - \)\(17\!\cdots\!39\)\( p^{3} T^{33} + \)\(39\!\cdots\!56\)\( p^{4} T^{34} - \)\(26\!\cdots\!74\)\( p^{5} T^{35} + \)\(54\!\cdots\!79\)\( p^{6} T^{36} - \)\(35\!\cdots\!04\)\( p^{7} T^{37} + \)\(71\!\cdots\!02\)\( p^{8} T^{38} - \)\(45\!\cdots\!76\)\( p^{9} T^{39} + \)\(85\!\cdots\!96\)\( p^{10} T^{40} - \)\(52\!\cdots\!82\)\( p^{11} T^{41} + \)\(92\!\cdots\!25\)\( p^{12} T^{42} - 54689785635228592808 p^{13} T^{43} + 9189152233410898885 p^{14} T^{44} - 517378143126114585 p^{15} T^{45} + 81404241895797897 p^{16} T^{46} - 4341657166114684 p^{17} T^{47} + 636536459537504 p^{18} T^{48} - 31690828190954 p^{19} T^{49} + 4304788741632 p^{20} T^{50} - 195812176184 p^{21} T^{51} + 24469911811 p^{22} T^{52} - 984203758 p^{23} T^{53} + 112087766 p^{24} T^{54} - 3776072 p^{25} T^{55} + 6554 p^{27} T^{56} - 9832 p^{27} T^{57} + 887 p^{28} T^{58} - 13 p^{29} T^{59} + p^{30} T^{60} \)
61 \( 1 - 16 T + 944 T^{2} - 12636 T^{3} + 424447 T^{4} - 4882872 T^{5} + 122295415 T^{6} - 1225631465 T^{7} + 25515747750 T^{8} - 223701126115 T^{9} + 4122981145028 T^{10} - 31487743626423 T^{11} + 538959655680344 T^{12} - 3537516453242321 T^{13} + 58907762383574934 T^{14} - 324131892940428286 T^{15} + 5547436325776916060 T^{16} - 24622439190592177723 T^{17} + \)\(46\!\cdots\!17\)\( T^{18} - \)\(15\!\cdots\!83\)\( T^{19} + \)\(35\!\cdots\!38\)\( T^{20} - \)\(90\!\cdots\!01\)\( T^{21} + \)\(26\!\cdots\!51\)\( T^{22} - \)\(50\!\cdots\!86\)\( T^{23} + \)\(19\!\cdots\!41\)\( T^{24} - \)\(30\!\cdots\!34\)\( T^{25} + \)\(13\!\cdots\!67\)\( T^{26} - \)\(20\!\cdots\!87\)\( T^{27} + \)\(92\!\cdots\!52\)\( T^{28} - \)\(13\!\cdots\!78\)\( T^{29} + \)\(58\!\cdots\!74\)\( T^{30} - \)\(13\!\cdots\!78\)\( p T^{31} + \)\(92\!\cdots\!52\)\( p^{2} T^{32} - \)\(20\!\cdots\!87\)\( p^{3} T^{33} + \)\(13\!\cdots\!67\)\( p^{4} T^{34} - \)\(30\!\cdots\!34\)\( p^{5} T^{35} + \)\(19\!\cdots\!41\)\( p^{6} T^{36} - \)\(50\!\cdots\!86\)\( p^{7} T^{37} + \)\(26\!\cdots\!51\)\( p^{8} T^{38} - \)\(90\!\cdots\!01\)\( p^{9} T^{39} + \)\(35\!\cdots\!38\)\( p^{10} T^{40} - \)\(15\!\cdots\!83\)\( p^{11} T^{41} + \)\(46\!\cdots\!17\)\( p^{12} T^{42} - 24622439190592177723 p^{13} T^{43} + 5547436325776916060 p^{14} T^{44} - 324131892940428286 p^{15} T^{45} + 58907762383574934 p^{16} T^{46} - 3537516453242321 p^{17} T^{47} + 538959655680344 p^{18} T^{48} - 31487743626423 p^{19} T^{49} + 4122981145028 p^{20} T^{50} - 223701126115 p^{21} T^{51} + 25515747750 p^{22} T^{52} - 1225631465 p^{23} T^{53} + 122295415 p^{24} T^{54} - 4882872 p^{25} T^{55} + 424447 p^{26} T^{56} - 12636 p^{27} T^{57} + 944 p^{28} T^{58} - 16 p^{29} T^{59} + p^{30} T^{60} \)
67 \( 1 + 11 T + 846 T^{2} + 8288 T^{3} + 351603 T^{4} + 3110979 T^{5} + 96629581 T^{6} + 779777717 T^{7} + 19911050034 T^{8} + 147375625581 T^{9} + 3301704343853 T^{10} + 22459426473930 T^{11} + 461264442122808 T^{12} + 2881680453962508 T^{13} + 56088011180301037 T^{14} + 321254510493364346 T^{15} + 6085534278278239734 T^{16} + 31930518674711185518 T^{17} + \)\(60\!\cdots\!11\)\( T^{18} + \)\(28\!\cdots\!12\)\( T^{19} + \)\(54\!\cdots\!52\)\( T^{20} + \)\(24\!\cdots\!74\)\( T^{21} + \)\(46\!\cdots\!43\)\( T^{22} + \)\(19\!\cdots\!90\)\( T^{23} + \)\(37\!\cdots\!75\)\( T^{24} + \)\(14\!\cdots\!17\)\( T^{25} + \)\(28\!\cdots\!02\)\( T^{26} + \)\(10\!\cdots\!53\)\( T^{27} + \)\(20\!\cdots\!01\)\( T^{28} + \)\(75\!\cdots\!12\)\( T^{29} + \)\(14\!\cdots\!38\)\( T^{30} + \)\(75\!\cdots\!12\)\( p T^{31} + \)\(20\!\cdots\!01\)\( p^{2} T^{32} + \)\(10\!\cdots\!53\)\( p^{3} T^{33} + \)\(28\!\cdots\!02\)\( p^{4} T^{34} + \)\(14\!\cdots\!17\)\( p^{5} T^{35} + \)\(37\!\cdots\!75\)\( p^{6} T^{36} + \)\(19\!\cdots\!90\)\( p^{7} T^{37} + \)\(46\!\cdots\!43\)\( p^{8} T^{38} + \)\(24\!\cdots\!74\)\( p^{9} T^{39} + \)\(54\!\cdots\!52\)\( p^{10} T^{40} + \)\(28\!\cdots\!12\)\( p^{11} T^{41} + \)\(60\!\cdots\!11\)\( p^{12} T^{42} + 31930518674711185518 p^{13} T^{43} + 6085534278278239734 p^{14} T^{44} + 321254510493364346 p^{15} T^{45} + 56088011180301037 p^{16} T^{46} + 2881680453962508 p^{17} T^{47} + 461264442122808 p^{18} T^{48} + 22459426473930 p^{19} T^{49} + 3301704343853 p^{20} T^{50} + 147375625581 p^{21} T^{51} + 19911050034 p^{22} T^{52} + 779777717 p^{23} T^{53} + 96629581 p^{24} T^{54} + 3110979 p^{25} T^{55} + 351603 p^{26} T^{56} + 8288 p^{27} T^{57} + 846 p^{28} T^{58} + 11 p^{29} T^{59} + p^{30} T^{60} \)
71 \( 1 - 42 T + 1971 T^{2} - 57804 T^{3} + 1668458 T^{4} - 38705165 T^{5} + 862501020 T^{6} - 16835312434 T^{7} + 314500821030 T^{8} - 5355614201242 T^{9} + 87431161066239 T^{10} - 1329807181180718 T^{11} + 19449467465878964 T^{12} - 268599645353191582 T^{13} + 3578610832910637606 T^{14} - 45425097892176447364 T^{15} + \)\(55\!\cdots\!41\)\( T^{16} - \)\(65\!\cdots\!29\)\( T^{17} + \)\(75\!\cdots\!89\)\( T^{18} - \)\(82\!\cdots\!48\)\( T^{19} + \)\(88\!\cdots\!82\)\( T^{20} - \)\(92\!\cdots\!00\)\( T^{21} + \)\(93\!\cdots\!12\)\( T^{22} - \)\(92\!\cdots\!31\)\( T^{23} + \)\(88\!\cdots\!17\)\( T^{24} - \)\(83\!\cdots\!63\)\( T^{25} + \)\(76\!\cdots\!91\)\( T^{26} - \)\(68\!\cdots\!40\)\( T^{27} + \)\(60\!\cdots\!93\)\( T^{28} - \)\(52\!\cdots\!58\)\( T^{29} + \)\(44\!\cdots\!21\)\( T^{30} - \)\(52\!\cdots\!58\)\( p T^{31} + \)\(60\!\cdots\!93\)\( p^{2} T^{32} - \)\(68\!\cdots\!40\)\( p^{3} T^{33} + \)\(76\!\cdots\!91\)\( p^{4} T^{34} - \)\(83\!\cdots\!63\)\( p^{5} T^{35} + \)\(88\!\cdots\!17\)\( p^{6} T^{36} - \)\(92\!\cdots\!31\)\( p^{7} T^{37} + \)\(93\!\cdots\!12\)\( p^{8} T^{38} - \)\(92\!\cdots\!00\)\( p^{9} T^{39} + \)\(88\!\cdots\!82\)\( p^{10} T^{40} - \)\(82\!\cdots\!48\)\( p^{11} T^{41} + \)\(75\!\cdots\!89\)\( p^{12} T^{42} - \)\(65\!\cdots\!29\)\( p^{13} T^{43} + \)\(55\!\cdots\!41\)\( p^{14} T^{44} - 45425097892176447364 p^{15} T^{45} + 3578610832910637606 p^{16} T^{46} - 268599645353191582 p^{17} T^{47} + 19449467465878964 p^{18} T^{48} - 1329807181180718 p^{19} T^{49} + 87431161066239 p^{20} T^{50} - 5355614201242 p^{21} T^{51} + 314500821030 p^{22} T^{52} - 16835312434 p^{23} T^{53} + 862501020 p^{24} T^{54} - 38705165 p^{25} T^{55} + 1668458 p^{26} T^{56} - 57804 p^{27} T^{57} + 1971 p^{28} T^{58} - 42 p^{29} T^{59} + p^{30} T^{60} \)
73 \( 1 + 4 T + 722 T^{2} + 2362 T^{3} + 276270 T^{4} + 10735 p T^{5} + 74110400 T^{6} + 191819681 T^{7} + 15554758549 T^{8} + 38394265493 T^{9} + 2707438462779 T^{10} + 6607337868968 T^{11} + 405213832538147 T^{12} + 1004949913207266 T^{13} + 53466590846342682 T^{14} + 137326746336877797 T^{15} + 6335779327548177533 T^{16} + 17042347978968403800 T^{17} + \)\(68\!\cdots\!10\)\( T^{18} + \)\(19\!\cdots\!16\)\( T^{19} + \)\(68\!\cdots\!16\)\( T^{20} + \)\(20\!\cdots\!00\)\( T^{21} + \)\(63\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!64\)\( T^{23} + \)\(55\!\cdots\!41\)\( T^{24} + \)\(17\!\cdots\!88\)\( T^{25} + \)\(62\!\cdots\!06\)\( p T^{26} + \)\(14\!\cdots\!39\)\( T^{27} + \)\(35\!\cdots\!24\)\( T^{28} + \)\(11\!\cdots\!59\)\( T^{29} + \)\(26\!\cdots\!48\)\( T^{30} + \)\(11\!\cdots\!59\)\( p T^{31} + \)\(35\!\cdots\!24\)\( p^{2} T^{32} + \)\(14\!\cdots\!39\)\( p^{3} T^{33} + \)\(62\!\cdots\!06\)\( p^{5} T^{34} + \)\(17\!\cdots\!88\)\( p^{5} T^{35} + \)\(55\!\cdots\!41\)\( p^{6} T^{36} + \)\(19\!\cdots\!64\)\( p^{7} T^{37} + \)\(63\!\cdots\!52\)\( p^{8} T^{38} + \)\(20\!\cdots\!00\)\( p^{9} T^{39} + \)\(68\!\cdots\!16\)\( p^{10} T^{40} + \)\(19\!\cdots\!16\)\( p^{11} T^{41} + \)\(68\!\cdots\!10\)\( p^{12} T^{42} + 17042347978968403800 p^{13} T^{43} + 6335779327548177533 p^{14} T^{44} + 137326746336877797 p^{15} T^{45} + 53466590846342682 p^{16} T^{46} + 1004949913207266 p^{17} T^{47} + 405213832538147 p^{18} T^{48} + 6607337868968 p^{19} T^{49} + 2707438462779 p^{20} T^{50} + 38394265493 p^{21} T^{51} + 15554758549 p^{22} T^{52} + 191819681 p^{23} T^{53} + 74110400 p^{24} T^{54} + 10735 p^{26} T^{55} + 276270 p^{26} T^{56} + 2362 p^{27} T^{57} + 722 p^{28} T^{58} + 4 p^{29} T^{59} + p^{30} T^{60} \)
79 \( 1 - 3 T + 1136 T^{2} - 2022 T^{3} + 639332 T^{4} - 390675 T^{5} + 238799691 T^{6} + 127657089 T^{7} + 66861034175 T^{8} + 111635876355 T^{9} + 15024051852733 T^{10} + 41907036765757 T^{11} + 2832944182428813 T^{12} + 10960504956008917 T^{13} + 462799375040891776 T^{14} + 2251557272870794061 T^{15} + 67090062664617096608 T^{16} + \)\(38\!\cdots\!67\)\( T^{17} + \)\(87\!\cdots\!32\)\( T^{18} + \)\(56\!\cdots\!25\)\( T^{19} + \)\(10\!\cdots\!90\)\( T^{20} + \)\(71\!\cdots\!35\)\( T^{21} + \)\(11\!\cdots\!70\)\( T^{22} + \)\(82\!\cdots\!00\)\( T^{23} + \)\(11\!\cdots\!01\)\( T^{24} + \)\(84\!\cdots\!94\)\( T^{25} + \)\(11\!\cdots\!60\)\( T^{26} + \)\(79\!\cdots\!29\)\( T^{27} + \)\(10\!\cdots\!12\)\( T^{28} + \)\(68\!\cdots\!45\)\( T^{29} + \)\(82\!\cdots\!92\)\( T^{30} + \)\(68\!\cdots\!45\)\( p T^{31} + \)\(10\!\cdots\!12\)\( p^{2} T^{32} + \)\(79\!\cdots\!29\)\( p^{3} T^{33} + \)\(11\!\cdots\!60\)\( p^{4} T^{34} + \)\(84\!\cdots\!94\)\( p^{5} T^{35} + \)\(11\!\cdots\!01\)\( p^{6} T^{36} + \)\(82\!\cdots\!00\)\( p^{7} T^{37} + \)\(11\!\cdots\!70\)\( p^{8} T^{38} + \)\(71\!\cdots\!35\)\( p^{9} T^{39} + \)\(10\!\cdots\!90\)\( p^{10} T^{40} + \)\(56\!\cdots\!25\)\( p^{11} T^{41} + \)\(87\!\cdots\!32\)\( p^{12} T^{42} + \)\(38\!\cdots\!67\)\( p^{13} T^{43} + 67090062664617096608 p^{14} T^{44} + 2251557272870794061 p^{15} T^{45} + 462799375040891776 p^{16} T^{46} + 10960504956008917 p^{17} T^{47} + 2832944182428813 p^{18} T^{48} + 41907036765757 p^{19} T^{49} + 15024051852733 p^{20} T^{50} + 111635876355 p^{21} T^{51} + 66861034175 p^{22} T^{52} + 127657089 p^{23} T^{53} + 238799691 p^{24} T^{54} - 390675 p^{25} T^{55} + 639332 p^{26} T^{56} - 2022 p^{27} T^{57} + 1136 p^{28} T^{58} - 3 p^{29} T^{59} + p^{30} T^{60} \)
83 \( 1 + 11 T + 1450 T^{2} + 16917 T^{3} + 1045573 T^{4} + 12743629 T^{5} + 501123752 T^{6} + 6278621097 T^{7} + 179856430739 T^{8} + 2279600876143 T^{9} + 51580045799908 T^{10} + 651616461067156 T^{11} + 12304132251607085 T^{12} + 152987064880418652 T^{13} + 2507544542239681747 T^{14} + 30385605885144729703 T^{15} + \)\(44\!\cdots\!27\)\( T^{16} + \)\(52\!\cdots\!13\)\( T^{17} + \)\(69\!\cdots\!32\)\( T^{18} + \)\(78\!\cdots\!18\)\( T^{19} + \)\(97\!\cdots\!93\)\( T^{20} + \)\(10\!\cdots\!80\)\( T^{21} + \)\(12\!\cdots\!96\)\( T^{22} + \)\(12\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!67\)\( T^{24} + \)\(13\!\cdots\!40\)\( T^{25} + \)\(14\!\cdots\!34\)\( T^{26} + \)\(13\!\cdots\!04\)\( T^{27} + \)\(13\!\cdots\!83\)\( T^{28} + \)\(12\!\cdots\!87\)\( T^{29} + \)\(11\!\cdots\!86\)\( T^{30} + \)\(12\!\cdots\!87\)\( p T^{31} + \)\(13\!\cdots\!83\)\( p^{2} T^{32} + \)\(13\!\cdots\!04\)\( p^{3} T^{33} + \)\(14\!\cdots\!34\)\( p^{4} T^{34} + \)\(13\!\cdots\!40\)\( p^{5} T^{35} + \)\(13\!\cdots\!67\)\( p^{6} T^{36} + \)\(12\!\cdots\!08\)\( p^{7} T^{37} + \)\(12\!\cdots\!96\)\( p^{8} T^{38} + \)\(10\!\cdots\!80\)\( p^{9} T^{39} + \)\(97\!\cdots\!93\)\( p^{10} T^{40} + \)\(78\!\cdots\!18\)\( p^{11} T^{41} + \)\(69\!\cdots\!32\)\( p^{12} T^{42} + \)\(52\!\cdots\!13\)\( p^{13} T^{43} + \)\(44\!\cdots\!27\)\( p^{14} T^{44} + 30385605885144729703 p^{15} T^{45} + 2507544542239681747 p^{16} T^{46} + 152987064880418652 p^{17} T^{47} + 12304132251607085 p^{18} T^{48} + 651616461067156 p^{19} T^{49} + 51580045799908 p^{20} T^{50} + 2279600876143 p^{21} T^{51} + 179856430739 p^{22} T^{52} + 6278621097 p^{23} T^{53} + 501123752 p^{24} T^{54} + 12743629 p^{25} T^{55} + 1045573 p^{26} T^{56} + 16917 p^{27} T^{57} + 1450 p^{28} T^{58} + 11 p^{29} T^{59} + p^{30} T^{60} \)
89 \( 1 - 58 T + 2780 T^{2} - 95356 T^{3} + 2876226 T^{4} - 73934486 T^{5} + 1731571432 T^{6} - 36605848964 T^{7} + 720394916192 T^{8} - 13157237838442 T^{9} + 226923053636204 T^{10} - 3692370742708525 T^{11} + 57319346234147767 T^{12} - 848791577273762007 T^{13} + 12084486060566831535 T^{14} - \)\(16\!\cdots\!08\)\( T^{15} + \)\(21\!\cdots\!30\)\( T^{16} - \)\(28\!\cdots\!64\)\( T^{17} + \)\(34\!\cdots\!91\)\( T^{18} - \)\(42\!\cdots\!92\)\( T^{19} + \)\(49\!\cdots\!31\)\( T^{20} - \)\(57\!\cdots\!77\)\( T^{21} + \)\(64\!\cdots\!63\)\( T^{22} - \)\(70\!\cdots\!41\)\( T^{23} + \)\(75\!\cdots\!74\)\( T^{24} - \)\(79\!\cdots\!15\)\( T^{25} + \)\(82\!\cdots\!49\)\( T^{26} - \)\(82\!\cdots\!74\)\( T^{27} + \)\(82\!\cdots\!01\)\( T^{28} - \)\(79\!\cdots\!12\)\( T^{29} + \)\(75\!\cdots\!78\)\( T^{30} - \)\(79\!\cdots\!12\)\( p T^{31} + \)\(82\!\cdots\!01\)\( p^{2} T^{32} - \)\(82\!\cdots\!74\)\( p^{3} T^{33} + \)\(82\!\cdots\!49\)\( p^{4} T^{34} - \)\(79\!\cdots\!15\)\( p^{5} T^{35} + \)\(75\!\cdots\!74\)\( p^{6} T^{36} - \)\(70\!\cdots\!41\)\( p^{7} T^{37} + \)\(64\!\cdots\!63\)\( p^{8} T^{38} - \)\(57\!\cdots\!77\)\( p^{9} T^{39} + \)\(49\!\cdots\!31\)\( p^{10} T^{40} - \)\(42\!\cdots\!92\)\( p^{11} T^{41} + \)\(34\!\cdots\!91\)\( p^{12} T^{42} - \)\(28\!\cdots\!64\)\( p^{13} T^{43} + \)\(21\!\cdots\!30\)\( p^{14} T^{44} - \)\(16\!\cdots\!08\)\( p^{15} T^{45} + 12084486060566831535 p^{16} T^{46} - 848791577273762007 p^{17} T^{47} + 57319346234147767 p^{18} T^{48} - 3692370742708525 p^{19} T^{49} + 226923053636204 p^{20} T^{50} - 13157237838442 p^{21} T^{51} + 720394916192 p^{22} T^{52} - 36605848964 p^{23} T^{53} + 1731571432 p^{24} T^{54} - 73934486 p^{25} T^{55} + 2876226 p^{26} T^{56} - 95356 p^{27} T^{57} + 2780 p^{28} T^{58} - 58 p^{29} T^{59} + p^{30} T^{60} \)
97 \( 1 + 9 T + 1909 T^{2} + 16401 T^{3} + 1782756 T^{4} + 14651235 T^{5} + 1085814760 T^{6} + 8564647292 T^{7} + 485281682024 T^{8} + 3690912431369 T^{9} + 169810642893694 T^{10} + 1252661885234693 T^{11} + 48486322672964794 T^{12} + 349294011121954731 T^{13} + 11628348198809696655 T^{14} + 82415496716830117527 T^{15} + \)\(23\!\cdots\!05\)\( T^{16} + \)\(16\!\cdots\!85\)\( T^{17} + \)\(43\!\cdots\!87\)\( T^{18} + \)\(30\!\cdots\!05\)\( T^{19} + \)\(68\!\cdots\!65\)\( T^{20} + \)\(48\!\cdots\!77\)\( T^{21} + \)\(97\!\cdots\!76\)\( T^{22} + \)\(68\!\cdots\!82\)\( T^{23} + \)\(12\!\cdots\!75\)\( T^{24} + \)\(88\!\cdots\!05\)\( T^{25} + \)\(14\!\cdots\!02\)\( T^{26} + \)\(10\!\cdots\!97\)\( T^{27} + \)\(16\!\cdots\!28\)\( T^{28} + \)\(11\!\cdots\!04\)\( T^{29} + \)\(16\!\cdots\!86\)\( T^{30} + \)\(11\!\cdots\!04\)\( p T^{31} + \)\(16\!\cdots\!28\)\( p^{2} T^{32} + \)\(10\!\cdots\!97\)\( p^{3} T^{33} + \)\(14\!\cdots\!02\)\( p^{4} T^{34} + \)\(88\!\cdots\!05\)\( p^{5} T^{35} + \)\(12\!\cdots\!75\)\( p^{6} T^{36} + \)\(68\!\cdots\!82\)\( p^{7} T^{37} + \)\(97\!\cdots\!76\)\( p^{8} T^{38} + \)\(48\!\cdots\!77\)\( p^{9} T^{39} + \)\(68\!\cdots\!65\)\( p^{10} T^{40} + \)\(30\!\cdots\!05\)\( p^{11} T^{41} + \)\(43\!\cdots\!87\)\( p^{12} T^{42} + \)\(16\!\cdots\!85\)\( p^{13} T^{43} + \)\(23\!\cdots\!05\)\( p^{14} T^{44} + 82415496716830117527 p^{15} T^{45} + 11628348198809696655 p^{16} T^{46} + 349294011121954731 p^{17} T^{47} + 48486322672964794 p^{18} T^{48} + 1252661885234693 p^{19} T^{49} + 169810642893694 p^{20} T^{50} + 3690912431369 p^{21} T^{51} + 485281682024 p^{22} T^{52} + 8564647292 p^{23} T^{53} + 1085814760 p^{24} T^{54} + 14651235 p^{25} T^{55} + 1782756 p^{26} T^{56} + 16401 p^{27} T^{57} + 1909 p^{28} T^{58} + 9 p^{29} T^{59} + p^{30} T^{60} \)
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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

−1.96485365607532540361497522307, −1.95526282336867629364559990759, −1.85280751890730749370709431071, −1.85277470008660528151292015325, −1.72326303544247175209811883673, −1.56667067300482226653643235754, −1.53979467988529118867202299012, −1.37289052153298972985459062886, −1.36830780614071780582187215135, −1.35060995591817130036840513814, −1.34785198323514141736890921103, −1.32197642990595936093293754730, −1.24246628498784892666841127290, −1.14079133280932028941658238897, −1.06488908327438773038967481836, −1.01496464563590593437914778616, −0.843853602991687097253199822477, −0.834929733689204520348011313208, −0.806724464850011905546358673447, −0.78666677260535171729752320000, −0.73865660663455951245009933203, −0.57424308568506499240605936784, −0.48848415051343897176240104551, −0.41928144542520850998814378259, −0.22359698785891014393310962308, 0.22359698785891014393310962308, 0.41928144542520850998814378259, 0.48848415051343897176240104551, 0.57424308568506499240605936784, 0.73865660663455951245009933203, 0.78666677260535171729752320000, 0.806724464850011905546358673447, 0.834929733689204520348011313208, 0.843853602991687097253199822477, 1.01496464563590593437914778616, 1.06488908327438773038967481836, 1.14079133280932028941658238897, 1.24246628498784892666841127290, 1.32197642990595936093293754730, 1.34785198323514141736890921103, 1.35060995591817130036840513814, 1.36830780614071780582187215135, 1.37289052153298972985459062886, 1.53979467988529118867202299012, 1.56667067300482226653643235754, 1.72326303544247175209811883673, 1.85277470008660528151292015325, 1.85280751890730749370709431071, 1.95526282336867629364559990759, 1.96485365607532540361497522307

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

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