Properties

Degree 54
Conductor $ 2^{54} \cdot 19^{27} \cdot 79^{27} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 27

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 10·5-s − 8·7-s − 23·9-s + 3·11-s − 5·13-s + 40·15-s − 17·17-s + 27·19-s + 32·21-s − 11·23-s − 11·25-s + 115·27-s − 39·29-s − 27·31-s − 12·33-s + 80·35-s − 37-s + 20·39-s − 36·41-s − 2·43-s + 230·45-s − 12·47-s − 55·49-s + 68·51-s − 28·53-s − 30·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 4.47·5-s − 3.02·7-s − 7.66·9-s + 0.904·11-s − 1.38·13-s + 10.3·15-s − 4.12·17-s + 6.19·19-s + 6.98·21-s − 2.29·23-s − 2.19·25-s + 22.1·27-s − 7.24·29-s − 4.84·31-s − 2.08·33-s + 13.5·35-s − 0.164·37-s + 3.20·39-s − 5.62·41-s − 0.304·43-s + 34.2·45-s − 1.75·47-s − 7.85·49-s + 9.52·51-s − 3.84·53-s − 4.04·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54} \cdot 19^{27} \cdot 79^{27}\right)^{s/2} \, \Gamma_{\C}(s)^{27} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54} \cdot 19^{27} \cdot 79^{27}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{27} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(54\)
\( N \)  =  \(2^{54} \cdot 19^{27} \cdot 79^{27}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  27
Selberg data  =  $(54,\ 2^{54} \cdot 19^{27} \cdot 79^{27} ,\ ( \ : [1/2]^{27} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;79\}$, \(F_p\) is a polynomial of degree 54. If $p \in \{2,\;19,\;79\}$, then $F_p$ is a polynomial of degree at most 53.
$p$$F_p$
bad2 \( 1 \)
19 \( ( 1 - T )^{27} \)
79 \( ( 1 + T )^{27} \)
good3 \( 1 + 4 T + 13 p T^{2} + 133 T^{3} + 748 T^{4} + 2257 T^{5} + 9520 T^{6} + 26068 T^{7} + 30377 p T^{8} + 230521 T^{9} + 234316 p T^{10} + 554582 p T^{11} + 4561393 T^{12} + 10196255 T^{13} + 2848099 p^{2} T^{14} + 18166198 p T^{15} + 127375039 T^{16} + 259006514 T^{17} + 568642865 T^{18} + 1110701366 T^{19} + 256594210 p^{2} T^{20} + 1449108287 p T^{21} + 8616726823 T^{22} + 15668561453 T^{23} + 29764840310 T^{24} + 17445820424 p T^{25} + 95687670326 T^{26} + 162664684088 T^{27} + 95687670326 p T^{28} + 17445820424 p^{3} T^{29} + 29764840310 p^{3} T^{30} + 15668561453 p^{4} T^{31} + 8616726823 p^{5} T^{32} + 1449108287 p^{7} T^{33} + 256594210 p^{9} T^{34} + 1110701366 p^{8} T^{35} + 568642865 p^{9} T^{36} + 259006514 p^{10} T^{37} + 127375039 p^{11} T^{38} + 18166198 p^{13} T^{39} + 2848099 p^{15} T^{40} + 10196255 p^{14} T^{41} + 4561393 p^{15} T^{42} + 554582 p^{17} T^{43} + 234316 p^{18} T^{44} + 230521 p^{18} T^{45} + 30377 p^{20} T^{46} + 26068 p^{20} T^{47} + 9520 p^{21} T^{48} + 2257 p^{22} T^{49} + 748 p^{23} T^{50} + 133 p^{24} T^{51} + 13 p^{26} T^{52} + 4 p^{26} T^{53} + p^{27} T^{54} \)
5 \( 1 + 2 p T + 111 T^{2} + 777 T^{3} + 1067 p T^{4} + 29606 T^{5} + 31521 p T^{6} + 737804 T^{7} + 3304943 T^{8} + 13539353 T^{9} + 53205657 T^{10} + 195419386 T^{11} + 138214659 p T^{12} + 2315497827 T^{13} + 7499056691 T^{14} + 23226835103 T^{15} + 69795996113 T^{16} + 201954082592 T^{17} + 568844179156 T^{18} + 1550729147982 T^{19} + 825422017817 p T^{20} + 10670955294347 T^{21} + 26996884286312 T^{22} + 66531035983613 T^{23} + 160687505959512 T^{24} + 378670172322903 T^{25} + 175073801415967 p T^{26} + 1975990965903571 T^{27} + 175073801415967 p^{2} T^{28} + 378670172322903 p^{2} T^{29} + 160687505959512 p^{3} T^{30} + 66531035983613 p^{4} T^{31} + 26996884286312 p^{5} T^{32} + 10670955294347 p^{6} T^{33} + 825422017817 p^{8} T^{34} + 1550729147982 p^{8} T^{35} + 568844179156 p^{9} T^{36} + 201954082592 p^{10} T^{37} + 69795996113 p^{11} T^{38} + 23226835103 p^{12} T^{39} + 7499056691 p^{13} T^{40} + 2315497827 p^{14} T^{41} + 138214659 p^{16} T^{42} + 195419386 p^{16} T^{43} + 53205657 p^{17} T^{44} + 13539353 p^{18} T^{45} + 3304943 p^{19} T^{46} + 737804 p^{20} T^{47} + 31521 p^{22} T^{48} + 29606 p^{22} T^{49} + 1067 p^{24} T^{50} + 777 p^{24} T^{51} + 111 p^{25} T^{52} + 2 p^{27} T^{53} + p^{27} T^{54} \)
7 \( 1 + 8 T + 17 p T^{2} + 786 T^{3} + 6883 T^{4} + 39169 T^{5} + 260518 T^{6} + 1314382 T^{7} + 7300291 T^{8} + 33314950 T^{9} + 162064240 T^{10} + 678706612 T^{11} + 2973513553 T^{12} + 11550316405 T^{13} + 46395287060 T^{14} + 168494321986 T^{15} + 89724519995 p T^{16} + 2145336945844 T^{17} + 7484193913281 T^{18} + 24150440971618 T^{19} + 79327518262304 T^{20} + 34656649236184 p T^{21} + 753528765585221 T^{22} + 2188714148761949 T^{23} + 6447876856612529 T^{24} + 2544514579474372 p T^{25} + 49863655536664629 T^{26} + 131066268684784589 T^{27} + 49863655536664629 p T^{28} + 2544514579474372 p^{3} T^{29} + 6447876856612529 p^{3} T^{30} + 2188714148761949 p^{4} T^{31} + 753528765585221 p^{5} T^{32} + 34656649236184 p^{7} T^{33} + 79327518262304 p^{7} T^{34} + 24150440971618 p^{8} T^{35} + 7484193913281 p^{9} T^{36} + 2145336945844 p^{10} T^{37} + 89724519995 p^{12} T^{38} + 168494321986 p^{12} T^{39} + 46395287060 p^{13} T^{40} + 11550316405 p^{14} T^{41} + 2973513553 p^{15} T^{42} + 678706612 p^{16} T^{43} + 162064240 p^{17} T^{44} + 33314950 p^{18} T^{45} + 7300291 p^{19} T^{46} + 1314382 p^{20} T^{47} + 260518 p^{21} T^{48} + 39169 p^{22} T^{49} + 6883 p^{23} T^{50} + 786 p^{24} T^{51} + 17 p^{26} T^{52} + 8 p^{26} T^{53} + p^{27} T^{54} \)
11 \( 1 - 3 T + 14 p T^{2} - 442 T^{3} + 11901 T^{4} - 33120 T^{5} + 615840 T^{6} - 1676677 T^{7} + 24012093 T^{8} - 64316066 T^{9} + 752523133 T^{10} - 1988773134 T^{11} + 19743039237 T^{12} - 51514505849 T^{13} + 445838144520 T^{14} - 1147038105028 T^{15} + 8839285858044 T^{16} - 22359961566003 T^{17} + 156096991731394 T^{18} - 386717746909575 T^{19} + 225562249110326 p T^{20} - 49524479925710 p^{2} T^{21} + 35764521106034400 T^{22} - 7617302656160959 p T^{23} + 469875584883225567 T^{24} - 1062371789977915678 T^{25} + 5644405267409282981 T^{26} - 12248860957495242664 T^{27} + 5644405267409282981 p T^{28} - 1062371789977915678 p^{2} T^{29} + 469875584883225567 p^{3} T^{30} - 7617302656160959 p^{5} T^{31} + 35764521106034400 p^{5} T^{32} - 49524479925710 p^{8} T^{33} + 225562249110326 p^{8} T^{34} - 386717746909575 p^{8} T^{35} + 156096991731394 p^{9} T^{36} - 22359961566003 p^{10} T^{37} + 8839285858044 p^{11} T^{38} - 1147038105028 p^{12} T^{39} + 445838144520 p^{13} T^{40} - 51514505849 p^{14} T^{41} + 19743039237 p^{15} T^{42} - 1988773134 p^{16} T^{43} + 752523133 p^{17} T^{44} - 64316066 p^{18} T^{45} + 24012093 p^{19} T^{46} - 1676677 p^{20} T^{47} + 615840 p^{21} T^{48} - 33120 p^{22} T^{49} + 11901 p^{23} T^{50} - 442 p^{24} T^{51} + 14 p^{26} T^{52} - 3 p^{26} T^{53} + p^{27} T^{54} \)
13 \( 1 + 5 T + 215 T^{2} + 75 p T^{3} + 22798 T^{4} + 94851 T^{5} + 1594130 T^{6} + 6135630 T^{7} + 82817447 T^{8} + 296698554 T^{9} + 3411928320 T^{10} + 11430438680 T^{11} + 116121887124 T^{12} + 365096170949 T^{13} + 3356947771384 T^{14} + 9934182203889 T^{15} + 84090403964031 T^{16} + 234806243609876 T^{17} + 1852340194558900 T^{18} + 4891423538899815 T^{19} + 36280013315815979 T^{20} + 90790752870909595 T^{21} + 637029422530693500 T^{22} + 1513664757748213457 T^{23} + 10086853179991473738 T^{24} + 22794826614577692990 T^{25} + 11121741636123672825 p T^{26} + \)\(31\!\cdots\!62\)\( T^{27} + 11121741636123672825 p^{2} T^{28} + 22794826614577692990 p^{2} T^{29} + 10086853179991473738 p^{3} T^{30} + 1513664757748213457 p^{4} T^{31} + 637029422530693500 p^{5} T^{32} + 90790752870909595 p^{6} T^{33} + 36280013315815979 p^{7} T^{34} + 4891423538899815 p^{8} T^{35} + 1852340194558900 p^{9} T^{36} + 234806243609876 p^{10} T^{37} + 84090403964031 p^{11} T^{38} + 9934182203889 p^{12} T^{39} + 3356947771384 p^{13} T^{40} + 365096170949 p^{14} T^{41} + 116121887124 p^{15} T^{42} + 11430438680 p^{16} T^{43} + 3411928320 p^{17} T^{44} + 296698554 p^{18} T^{45} + 82817447 p^{19} T^{46} + 6135630 p^{20} T^{47} + 1594130 p^{21} T^{48} + 94851 p^{22} T^{49} + 22798 p^{23} T^{50} + 75 p^{25} T^{51} + 215 p^{25} T^{52} + 5 p^{26} T^{53} + p^{27} T^{54} \)
17 \( 1 + p T + 383 T^{2} + 4895 T^{3} + 66148 T^{4} + 688906 T^{5} + 7122055 T^{6} + 63352176 T^{7} + 548047758 T^{8} + 252549233 p T^{9} + 32518991657 T^{10} + 229196738663 T^{11} + 91843384038 p T^{12} + 10055292427971 T^{13} + 62691980402592 T^{14} + 373188976724172 T^{15} + 2155164392085423 T^{16} + 11960348182054063 T^{17} + 64521802776257671 T^{18} + 335957258217686445 T^{19} + 1703242623030453509 T^{20} + 8359563811588831583 T^{21} + 39998909952332826454 T^{22} + \)\(18\!\cdots\!55\)\( T^{23} + \)\(84\!\cdots\!96\)\( T^{24} + \)\(36\!\cdots\!42\)\( T^{25} + \)\(15\!\cdots\!38\)\( T^{26} + \)\(66\!\cdots\!86\)\( T^{27} + \)\(15\!\cdots\!38\)\( p T^{28} + \)\(36\!\cdots\!42\)\( p^{2} T^{29} + \)\(84\!\cdots\!96\)\( p^{3} T^{30} + \)\(18\!\cdots\!55\)\( p^{4} T^{31} + 39998909952332826454 p^{5} T^{32} + 8359563811588831583 p^{6} T^{33} + 1703242623030453509 p^{7} T^{34} + 335957258217686445 p^{8} T^{35} + 64521802776257671 p^{9} T^{36} + 11960348182054063 p^{10} T^{37} + 2155164392085423 p^{11} T^{38} + 373188976724172 p^{12} T^{39} + 62691980402592 p^{13} T^{40} + 10055292427971 p^{14} T^{41} + 91843384038 p^{16} T^{42} + 229196738663 p^{16} T^{43} + 32518991657 p^{17} T^{44} + 252549233 p^{19} T^{45} + 548047758 p^{19} T^{46} + 63352176 p^{20} T^{47} + 7122055 p^{21} T^{48} + 688906 p^{22} T^{49} + 66148 p^{23} T^{50} + 4895 p^{24} T^{51} + 383 p^{25} T^{52} + p^{27} T^{53} + p^{27} T^{54} \)
23 \( 1 + 11 T + 362 T^{2} + 3409 T^{3} + 62978 T^{4} + 516583 T^{5} + 7023361 T^{6} + 50780767 T^{7} + 564491778 T^{8} + 3619584074 T^{9} + 34792511662 T^{10} + 197713105469 T^{11} + 1704280303219 T^{12} + 8495263780264 T^{13} + 67627578900943 T^{14} + 287669241849282 T^{15} + 2184689019328438 T^{16} + 7407645815359140 T^{17} + 56728390234874866 T^{18} + 124154156792566754 T^{19} + 1129295048191310548 T^{20} + 54355141170494959 T^{21} + 14654682343723691068 T^{22} - 87055726469260385458 T^{23} + 15835981630808486985 T^{24} - \)\(37\!\cdots\!39\)\( T^{25} - \)\(47\!\cdots\!03\)\( T^{26} - \)\(10\!\cdots\!82\)\( T^{27} - \)\(47\!\cdots\!03\)\( p T^{28} - \)\(37\!\cdots\!39\)\( p^{2} T^{29} + 15835981630808486985 p^{3} T^{30} - 87055726469260385458 p^{4} T^{31} + 14654682343723691068 p^{5} T^{32} + 54355141170494959 p^{6} T^{33} + 1129295048191310548 p^{7} T^{34} + 124154156792566754 p^{8} T^{35} + 56728390234874866 p^{9} T^{36} + 7407645815359140 p^{10} T^{37} + 2184689019328438 p^{11} T^{38} + 287669241849282 p^{12} T^{39} + 67627578900943 p^{13} T^{40} + 8495263780264 p^{14} T^{41} + 1704280303219 p^{15} T^{42} + 197713105469 p^{16} T^{43} + 34792511662 p^{17} T^{44} + 3619584074 p^{18} T^{45} + 564491778 p^{19} T^{46} + 50780767 p^{20} T^{47} + 7023361 p^{21} T^{48} + 516583 p^{22} T^{49} + 62978 p^{23} T^{50} + 3409 p^{24} T^{51} + 362 p^{25} T^{52} + 11 p^{26} T^{53} + p^{27} T^{54} \)
29 \( 1 + 39 T + 1119 T^{2} + 23208 T^{3} + 408388 T^{4} + 6093251 T^{5} + 81284293 T^{6} + 970711324 T^{7} + 10652524677 T^{8} + 107545286098 T^{9} + 1015097095561 T^{10} + 8965587884707 T^{11} + 2585081201956 p T^{12} + 593865846170098 T^{13} + 4499525854851999 T^{14} + 32624500325834078 T^{15} + 228278274302754322 T^{16} + 1541781084810024805 T^{17} + 10126873588473131625 T^{18} + 64653912889703995895 T^{19} + \)\(40\!\cdots\!86\)\( T^{20} + \)\(24\!\cdots\!88\)\( T^{21} + \)\(14\!\cdots\!09\)\( T^{22} + \)\(86\!\cdots\!53\)\( T^{23} + \)\(49\!\cdots\!77\)\( T^{24} + \)\(28\!\cdots\!29\)\( T^{25} + \)\(15\!\cdots\!75\)\( T^{26} + \)\(84\!\cdots\!86\)\( T^{27} + \)\(15\!\cdots\!75\)\( p T^{28} + \)\(28\!\cdots\!29\)\( p^{2} T^{29} + \)\(49\!\cdots\!77\)\( p^{3} T^{30} + \)\(86\!\cdots\!53\)\( p^{4} T^{31} + \)\(14\!\cdots\!09\)\( p^{5} T^{32} + \)\(24\!\cdots\!88\)\( p^{6} T^{33} + \)\(40\!\cdots\!86\)\( p^{7} T^{34} + 64653912889703995895 p^{8} T^{35} + 10126873588473131625 p^{9} T^{36} + 1541781084810024805 p^{10} T^{37} + 228278274302754322 p^{11} T^{38} + 32624500325834078 p^{12} T^{39} + 4499525854851999 p^{13} T^{40} + 593865846170098 p^{14} T^{41} + 2585081201956 p^{16} T^{42} + 8965587884707 p^{16} T^{43} + 1015097095561 p^{17} T^{44} + 107545286098 p^{18} T^{45} + 10652524677 p^{19} T^{46} + 970711324 p^{20} T^{47} + 81284293 p^{21} T^{48} + 6093251 p^{22} T^{49} + 408388 p^{23} T^{50} + 23208 p^{24} T^{51} + 1119 p^{25} T^{52} + 39 p^{26} T^{53} + p^{27} T^{54} \)
31 \( 1 + 27 T + 843 T^{2} + 15922 T^{3} + 302734 T^{4} + 4519799 T^{5} + 2125738 p T^{6} + 826994545 T^{7} + 10061880439 T^{8} + 110090998531 T^{9} + 1167175954704 T^{10} + 11405115196125 T^{11} + 108160054577999 T^{12} + 960022047440857 T^{13} + 8287954352442147 T^{14} + 67658922175940815 T^{15} + 17367670388181538 p T^{16} + 4080489142922348057 T^{17} + 30203950136778682068 T^{18} + \)\(21\!\cdots\!73\)\( T^{19} + \)\(14\!\cdots\!38\)\( T^{20} + \)\(98\!\cdots\!29\)\( T^{21} + \)\(64\!\cdots\!10\)\( T^{22} + \)\(40\!\cdots\!59\)\( T^{23} + \)\(24\!\cdots\!05\)\( T^{24} + \)\(14\!\cdots\!48\)\( T^{25} + \)\(86\!\cdots\!60\)\( T^{26} + \)\(48\!\cdots\!74\)\( T^{27} + \)\(86\!\cdots\!60\)\( p T^{28} + \)\(14\!\cdots\!48\)\( p^{2} T^{29} + \)\(24\!\cdots\!05\)\( p^{3} T^{30} + \)\(40\!\cdots\!59\)\( p^{4} T^{31} + \)\(64\!\cdots\!10\)\( p^{5} T^{32} + \)\(98\!\cdots\!29\)\( p^{6} T^{33} + \)\(14\!\cdots\!38\)\( p^{7} T^{34} + \)\(21\!\cdots\!73\)\( p^{8} T^{35} + 30203950136778682068 p^{9} T^{36} + 4080489142922348057 p^{10} T^{37} + 17367670388181538 p^{12} T^{38} + 67658922175940815 p^{12} T^{39} + 8287954352442147 p^{13} T^{40} + 960022047440857 p^{14} T^{41} + 108160054577999 p^{15} T^{42} + 11405115196125 p^{16} T^{43} + 1167175954704 p^{17} T^{44} + 110090998531 p^{18} T^{45} + 10061880439 p^{19} T^{46} + 826994545 p^{20} T^{47} + 2125738 p^{22} T^{48} + 4519799 p^{22} T^{49} + 302734 p^{23} T^{50} + 15922 p^{24} T^{51} + 843 p^{25} T^{52} + 27 p^{26} T^{53} + p^{27} T^{54} \)
37 \( 1 + T + 461 T^{2} + 1220 T^{3} + 108951 T^{4} + 444278 T^{5} + 17752340 T^{6} + 93143993 T^{7} + 2247191957 T^{8} + 13740340400 T^{9} + 234984229717 T^{10} + 1569051271539 T^{11} + 21014921190780 T^{12} + 146464594033543 T^{13} + 1641020460596173 T^{14} + 11566146932005139 T^{15} + 113356794030664392 T^{16} + 790601792918764265 T^{17} + 6986492567225472326 T^{18} + 47523077087469123096 T^{19} + \)\(38\!\cdots\!34\)\( T^{20} + \)\(25\!\cdots\!97\)\( T^{21} + \)\(19\!\cdots\!59\)\( T^{22} + \)\(12\!\cdots\!09\)\( T^{23} + \)\(86\!\cdots\!70\)\( T^{24} + \)\(52\!\cdots\!22\)\( T^{25} + \)\(35\!\cdots\!13\)\( T^{26} + \)\(20\!\cdots\!96\)\( T^{27} + \)\(35\!\cdots\!13\)\( p T^{28} + \)\(52\!\cdots\!22\)\( p^{2} T^{29} + \)\(86\!\cdots\!70\)\( p^{3} T^{30} + \)\(12\!\cdots\!09\)\( p^{4} T^{31} + \)\(19\!\cdots\!59\)\( p^{5} T^{32} + \)\(25\!\cdots\!97\)\( p^{6} T^{33} + \)\(38\!\cdots\!34\)\( p^{7} T^{34} + 47523077087469123096 p^{8} T^{35} + 6986492567225472326 p^{9} T^{36} + 790601792918764265 p^{10} T^{37} + 113356794030664392 p^{11} T^{38} + 11566146932005139 p^{12} T^{39} + 1641020460596173 p^{13} T^{40} + 146464594033543 p^{14} T^{41} + 21014921190780 p^{15} T^{42} + 1569051271539 p^{16} T^{43} + 234984229717 p^{17} T^{44} + 13740340400 p^{18} T^{45} + 2247191957 p^{19} T^{46} + 93143993 p^{20} T^{47} + 17752340 p^{21} T^{48} + 444278 p^{22} T^{49} + 108951 p^{23} T^{50} + 1220 p^{24} T^{51} + 461 p^{25} T^{52} + p^{26} T^{53} + p^{27} T^{54} \)
41 \( 1 + 36 T + 1090 T^{2} + 24167 T^{3} + 472858 T^{4} + 8015739 T^{5} + 124144503 T^{6} + 1755543905 T^{7} + 23166253399 T^{8} + 285967237548 T^{9} + 3338694310173 T^{10} + 36962909637455 T^{11} + 390528655731415 T^{12} + 3946291106710286 T^{13} + 38293586329017357 T^{14} + 357449538103523941 T^{15} + 3218325222706516431 T^{16} + 27986337610677976587 T^{17} + \)\(23\!\cdots\!04\)\( T^{18} + \)\(19\!\cdots\!68\)\( T^{19} + \)\(15\!\cdots\!67\)\( T^{20} + \)\(28\!\cdots\!23\)\( p T^{21} + \)\(86\!\cdots\!53\)\( T^{22} + \)\(62\!\cdots\!09\)\( T^{23} + \)\(44\!\cdots\!30\)\( T^{24} + \)\(30\!\cdots\!49\)\( T^{25} + \)\(20\!\cdots\!25\)\( T^{26} + \)\(13\!\cdots\!18\)\( T^{27} + \)\(20\!\cdots\!25\)\( p T^{28} + \)\(30\!\cdots\!49\)\( p^{2} T^{29} + \)\(44\!\cdots\!30\)\( p^{3} T^{30} + \)\(62\!\cdots\!09\)\( p^{4} T^{31} + \)\(86\!\cdots\!53\)\( p^{5} T^{32} + \)\(28\!\cdots\!23\)\( p^{7} T^{33} + \)\(15\!\cdots\!67\)\( p^{7} T^{34} + \)\(19\!\cdots\!68\)\( p^{8} T^{35} + \)\(23\!\cdots\!04\)\( p^{9} T^{36} + 27986337610677976587 p^{10} T^{37} + 3218325222706516431 p^{11} T^{38} + 357449538103523941 p^{12} T^{39} + 38293586329017357 p^{13} T^{40} + 3946291106710286 p^{14} T^{41} + 390528655731415 p^{15} T^{42} + 36962909637455 p^{16} T^{43} + 3338694310173 p^{17} T^{44} + 285967237548 p^{18} T^{45} + 23166253399 p^{19} T^{46} + 1755543905 p^{20} T^{47} + 124144503 p^{21} T^{48} + 8015739 p^{22} T^{49} + 472858 p^{23} T^{50} + 24167 p^{24} T^{51} + 1090 p^{25} T^{52} + 36 p^{26} T^{53} + p^{27} T^{54} \)
43 \( 1 + 2 T + 433 T^{2} + 1100 T^{3} + 100239 T^{4} + 288287 T^{5} + 16215690 T^{6} + 50321403 T^{7} + 2038290812 T^{8} + 6641314001 T^{9} + 210746506825 T^{10} + 708768505151 T^{11} + 18571890826444 T^{12} + 63744075329678 T^{13} + 1429762399676925 T^{14} + 4969069060323935 T^{15} + 97952990705961628 T^{16} + 342721675160269946 T^{17} + 6061057624126892386 T^{18} + 21246579888536835229 T^{19} + \)\(34\!\cdots\!16\)\( T^{20} + \)\(11\!\cdots\!35\)\( T^{21} + \)\(17\!\cdots\!25\)\( T^{22} + \)\(62\!\cdots\!54\)\( T^{23} + \)\(87\!\cdots\!34\)\( T^{24} + \)\(29\!\cdots\!08\)\( T^{25} + \)\(39\!\cdots\!27\)\( T^{26} + \)\(13\!\cdots\!30\)\( T^{27} + \)\(39\!\cdots\!27\)\( p T^{28} + \)\(29\!\cdots\!08\)\( p^{2} T^{29} + \)\(87\!\cdots\!34\)\( p^{3} T^{30} + \)\(62\!\cdots\!54\)\( p^{4} T^{31} + \)\(17\!\cdots\!25\)\( p^{5} T^{32} + \)\(11\!\cdots\!35\)\( p^{6} T^{33} + \)\(34\!\cdots\!16\)\( p^{7} T^{34} + 21246579888536835229 p^{8} T^{35} + 6061057624126892386 p^{9} T^{36} + 342721675160269946 p^{10} T^{37} + 97952990705961628 p^{11} T^{38} + 4969069060323935 p^{12} T^{39} + 1429762399676925 p^{13} T^{40} + 63744075329678 p^{14} T^{41} + 18571890826444 p^{15} T^{42} + 708768505151 p^{16} T^{43} + 210746506825 p^{17} T^{44} + 6641314001 p^{18} T^{45} + 2038290812 p^{19} T^{46} + 50321403 p^{20} T^{47} + 16215690 p^{21} T^{48} + 288287 p^{22} T^{49} + 100239 p^{23} T^{50} + 1100 p^{24} T^{51} + 433 p^{25} T^{52} + 2 p^{26} T^{53} + p^{27} T^{54} \)
47 \( 1 + 12 T + 697 T^{2} + 7707 T^{3} + 235850 T^{4} + 2421856 T^{5} + 51807222 T^{6} + 497008533 T^{7} + 8325096369 T^{8} + 74934973158 T^{9} + 1044484553582 T^{10} + 8843777136241 T^{11} + 106523936006355 T^{12} + 849179944441324 T^{13} + 9072894795072283 T^{14} + 68044707679091451 T^{15} + 658046684382044717 T^{16} + 98592173116490185 p T^{17} + 41302606072078547168 T^{18} + \)\(27\!\cdots\!82\)\( T^{19} + \)\(22\!\cdots\!11\)\( T^{20} + \)\(14\!\cdots\!82\)\( T^{21} + \)\(11\!\cdots\!59\)\( T^{22} + \)\(66\!\cdots\!53\)\( T^{23} + \)\(53\!\cdots\!43\)\( T^{24} + \)\(29\!\cdots\!45\)\( T^{25} + \)\(24\!\cdots\!58\)\( T^{26} + \)\(13\!\cdots\!11\)\( T^{27} + \)\(24\!\cdots\!58\)\( p T^{28} + \)\(29\!\cdots\!45\)\( p^{2} T^{29} + \)\(53\!\cdots\!43\)\( p^{3} T^{30} + \)\(66\!\cdots\!53\)\( p^{4} T^{31} + \)\(11\!\cdots\!59\)\( p^{5} T^{32} + \)\(14\!\cdots\!82\)\( p^{6} T^{33} + \)\(22\!\cdots\!11\)\( p^{7} T^{34} + \)\(27\!\cdots\!82\)\( p^{8} T^{35} + 41302606072078547168 p^{9} T^{36} + 98592173116490185 p^{11} T^{37} + 658046684382044717 p^{11} T^{38} + 68044707679091451 p^{12} T^{39} + 9072894795072283 p^{13} T^{40} + 849179944441324 p^{14} T^{41} + 106523936006355 p^{15} T^{42} + 8843777136241 p^{16} T^{43} + 1044484553582 p^{17} T^{44} + 74934973158 p^{18} T^{45} + 8325096369 p^{19} T^{46} + 497008533 p^{20} T^{47} + 51807222 p^{21} T^{48} + 2421856 p^{22} T^{49} + 235850 p^{23} T^{50} + 7707 p^{24} T^{51} + 697 p^{25} T^{52} + 12 p^{26} T^{53} + p^{27} T^{54} \)
53 \( 1 + 28 T + 986 T^{2} + 20075 T^{3} + 429281 T^{4} + 7052815 T^{5} + 115943336 T^{6} + 1623507890 T^{7} + 22374531634 T^{8} + 276086744042 T^{9} + 3333858334325 T^{10} + 37079553118937 T^{11} + 403019366399650 T^{12} + 4107907566496949 T^{13} + 40939870619650182 T^{14} + 387467122225855030 T^{15} + 3590744845632832227 T^{16} + 31903026375675424322 T^{17} + \)\(27\!\cdots\!01\)\( T^{18} + \)\(23\!\cdots\!81\)\( T^{19} + \)\(19\!\cdots\!83\)\( T^{20} + \)\(15\!\cdots\!09\)\( T^{21} + \)\(12\!\cdots\!07\)\( T^{22} + \)\(96\!\cdots\!11\)\( T^{23} + \)\(73\!\cdots\!41\)\( T^{24} + \)\(55\!\cdots\!93\)\( T^{25} + \)\(41\!\cdots\!92\)\( T^{26} + \)\(30\!\cdots\!80\)\( T^{27} + \)\(41\!\cdots\!92\)\( p T^{28} + \)\(55\!\cdots\!93\)\( p^{2} T^{29} + \)\(73\!\cdots\!41\)\( p^{3} T^{30} + \)\(96\!\cdots\!11\)\( p^{4} T^{31} + \)\(12\!\cdots\!07\)\( p^{5} T^{32} + \)\(15\!\cdots\!09\)\( p^{6} T^{33} + \)\(19\!\cdots\!83\)\( p^{7} T^{34} + \)\(23\!\cdots\!81\)\( p^{8} T^{35} + \)\(27\!\cdots\!01\)\( p^{9} T^{36} + 31903026375675424322 p^{10} T^{37} + 3590744845632832227 p^{11} T^{38} + 387467122225855030 p^{12} T^{39} + 40939870619650182 p^{13} T^{40} + 4107907566496949 p^{14} T^{41} + 403019366399650 p^{15} T^{42} + 37079553118937 p^{16} T^{43} + 3333858334325 p^{17} T^{44} + 276086744042 p^{18} T^{45} + 22374531634 p^{19} T^{46} + 1623507890 p^{20} T^{47} + 115943336 p^{21} T^{48} + 7052815 p^{22} T^{49} + 429281 p^{23} T^{50} + 20075 p^{24} T^{51} + 986 p^{25} T^{52} + 28 p^{26} T^{53} + p^{27} T^{54} \)
59 \( 1 + 30 T + 1080 T^{2} + 23067 T^{3} + 516533 T^{4} + 8935104 T^{5} + 156416274 T^{6} + 2331109100 T^{7} + 34705009110 T^{8} + 461690332476 T^{9} + 6099602096846 T^{10} + 74062952942258 T^{11} + 890198037778932 T^{12} + 10011746505763742 T^{13} + 111228275602238208 T^{14} + 1170402867592359904 T^{15} + 12147787658623886764 T^{16} + \)\(12\!\cdots\!49\)\( T^{17} + \)\(11\!\cdots\!70\)\( T^{18} + \)\(11\!\cdots\!79\)\( T^{19} + \)\(10\!\cdots\!31\)\( T^{20} + \)\(91\!\cdots\!82\)\( T^{21} + \)\(80\!\cdots\!44\)\( T^{22} + \)\(68\!\cdots\!70\)\( T^{23} + \)\(56\!\cdots\!42\)\( T^{24} + \)\(46\!\cdots\!99\)\( T^{25} + \)\(36\!\cdots\!15\)\( T^{26} + \)\(28\!\cdots\!64\)\( T^{27} + \)\(36\!\cdots\!15\)\( p T^{28} + \)\(46\!\cdots\!99\)\( p^{2} T^{29} + \)\(56\!\cdots\!42\)\( p^{3} T^{30} + \)\(68\!\cdots\!70\)\( p^{4} T^{31} + \)\(80\!\cdots\!44\)\( p^{5} T^{32} + \)\(91\!\cdots\!82\)\( p^{6} T^{33} + \)\(10\!\cdots\!31\)\( p^{7} T^{34} + \)\(11\!\cdots\!79\)\( p^{8} T^{35} + \)\(11\!\cdots\!70\)\( p^{9} T^{36} + \)\(12\!\cdots\!49\)\( p^{10} T^{37} + 12147787658623886764 p^{11} T^{38} + 1170402867592359904 p^{12} T^{39} + 111228275602238208 p^{13} T^{40} + 10011746505763742 p^{14} T^{41} + 890198037778932 p^{15} T^{42} + 74062952942258 p^{16} T^{43} + 6099602096846 p^{17} T^{44} + 461690332476 p^{18} T^{45} + 34705009110 p^{19} T^{46} + 2331109100 p^{20} T^{47} + 156416274 p^{21} T^{48} + 8935104 p^{22} T^{49} + 516533 p^{23} T^{50} + 23067 p^{24} T^{51} + 1080 p^{25} T^{52} + 30 p^{26} T^{53} + p^{27} T^{54} \)
61 \( 1 + 6 T + 766 T^{2} + 5305 T^{3} + 289615 T^{4} + 2227702 T^{5} + 72533653 T^{6} + 598761189 T^{7} + 13580925360 T^{8} + 116774217796 T^{9} + 2027002095768 T^{10} + 17711049821541 T^{11} + 250411445466627 T^{12} + 2180961480966837 T^{13} + 26214758703216842 T^{14} + 224377651591467606 T^{15} + 2363222422113854717 T^{16} + 19687311574361619258 T^{17} + \)\(18\!\cdots\!40\)\( T^{18} + \)\(14\!\cdots\!91\)\( T^{19} + \)\(12\!\cdots\!31\)\( T^{20} + \)\(10\!\cdots\!52\)\( T^{21} + \)\(81\!\cdots\!33\)\( T^{22} + \)\(62\!\cdots\!86\)\( T^{23} + \)\(48\!\cdots\!78\)\( T^{24} + \)\(36\!\cdots\!70\)\( T^{25} + \)\(28\!\cdots\!28\)\( T^{26} + \)\(22\!\cdots\!26\)\( T^{27} + \)\(28\!\cdots\!28\)\( p T^{28} + \)\(36\!\cdots\!70\)\( p^{2} T^{29} + \)\(48\!\cdots\!78\)\( p^{3} T^{30} + \)\(62\!\cdots\!86\)\( p^{4} T^{31} + \)\(81\!\cdots\!33\)\( p^{5} T^{32} + \)\(10\!\cdots\!52\)\( p^{6} T^{33} + \)\(12\!\cdots\!31\)\( p^{7} T^{34} + \)\(14\!\cdots\!91\)\( p^{8} T^{35} + \)\(18\!\cdots\!40\)\( p^{9} T^{36} + 19687311574361619258 p^{10} T^{37} + 2363222422113854717 p^{11} T^{38} + 224377651591467606 p^{12} T^{39} + 26214758703216842 p^{13} T^{40} + 2180961480966837 p^{14} T^{41} + 250411445466627 p^{15} T^{42} + 17711049821541 p^{16} T^{43} + 2027002095768 p^{17} T^{44} + 116774217796 p^{18} T^{45} + 13580925360 p^{19} T^{46} + 598761189 p^{20} T^{47} + 72533653 p^{21} T^{48} + 2227702 p^{22} T^{49} + 289615 p^{23} T^{50} + 5305 p^{24} T^{51} + 766 p^{25} T^{52} + 6 p^{26} T^{53} + p^{27} T^{54} \)
67 \( 1 - 13 T + 1073 T^{2} - 11212 T^{3} + 547364 T^{4} - 4664529 T^{5} + 180361827 T^{6} - 1255672343 T^{7} + 43782047747 T^{8} - 246939244539 T^{9} + 8432887734411 T^{10} - 37875694208054 T^{11} + 1351123998407398 T^{12} - 4710656529941836 T^{13} + 185888278775632339 T^{14} - 486274777687919040 T^{15} + 22446994522830817895 T^{16} - 42199182570815186272 T^{17} + \)\(24\!\cdots\!51\)\( T^{18} - \)\(30\!\cdots\!13\)\( T^{19} + \)\(23\!\cdots\!37\)\( T^{20} - \)\(19\!\cdots\!84\)\( T^{21} + \)\(20\!\cdots\!26\)\( T^{22} - \)\(10\!\cdots\!22\)\( T^{23} + \)\(16\!\cdots\!03\)\( T^{24} - \)\(50\!\cdots\!83\)\( T^{25} + \)\(12\!\cdots\!60\)\( T^{26} - \)\(28\!\cdots\!04\)\( T^{27} + \)\(12\!\cdots\!60\)\( p T^{28} - \)\(50\!\cdots\!83\)\( p^{2} T^{29} + \)\(16\!\cdots\!03\)\( p^{3} T^{30} - \)\(10\!\cdots\!22\)\( p^{4} T^{31} + \)\(20\!\cdots\!26\)\( p^{5} T^{32} - \)\(19\!\cdots\!84\)\( p^{6} T^{33} + \)\(23\!\cdots\!37\)\( p^{7} T^{34} - \)\(30\!\cdots\!13\)\( p^{8} T^{35} + \)\(24\!\cdots\!51\)\( p^{9} T^{36} - 42199182570815186272 p^{10} T^{37} + 22446994522830817895 p^{11} T^{38} - 486274777687919040 p^{12} T^{39} + 185888278775632339 p^{13} T^{40} - 4710656529941836 p^{14} T^{41} + 1351123998407398 p^{15} T^{42} - 37875694208054 p^{16} T^{43} + 8432887734411 p^{17} T^{44} - 246939244539 p^{18} T^{45} + 43782047747 p^{19} T^{46} - 1255672343 p^{20} T^{47} + 180361827 p^{21} T^{48} - 4664529 p^{22} T^{49} + 547364 p^{23} T^{50} - 11212 p^{24} T^{51} + 1073 p^{25} T^{52} - 13 p^{26} T^{53} + p^{27} T^{54} \)
71 \( 1 + 59 T + 2640 T^{2} + 85456 T^{3} + 2373541 T^{4} + 56147872 T^{5} + 1196806441 T^{6} + 22966050344 T^{7} + 407685282229 T^{8} + 6697682506989 T^{9} + 103475613821330 T^{10} + 21186098733856 p T^{11} + 20795640564028388 T^{12} + 273524130332594896 T^{13} + 3449256026147885573 T^{14} + 41707472991815152914 T^{15} + \)\(48\!\cdots\!34\)\( T^{16} + \)\(54\!\cdots\!52\)\( T^{17} + \)\(59\!\cdots\!21\)\( T^{18} + \)\(62\!\cdots\!67\)\( T^{19} + \)\(64\!\cdots\!73\)\( T^{20} + \)\(64\!\cdots\!13\)\( T^{21} + \)\(62\!\cdots\!51\)\( T^{22} + \)\(58\!\cdots\!20\)\( T^{23} + \)\(54\!\cdots\!82\)\( T^{24} + \)\(48\!\cdots\!45\)\( T^{25} + \)\(42\!\cdots\!94\)\( T^{26} + \)\(36\!\cdots\!40\)\( T^{27} + \)\(42\!\cdots\!94\)\( p T^{28} + \)\(48\!\cdots\!45\)\( p^{2} T^{29} + \)\(54\!\cdots\!82\)\( p^{3} T^{30} + \)\(58\!\cdots\!20\)\( p^{4} T^{31} + \)\(62\!\cdots\!51\)\( p^{5} T^{32} + \)\(64\!\cdots\!13\)\( p^{6} T^{33} + \)\(64\!\cdots\!73\)\( p^{7} T^{34} + \)\(62\!\cdots\!67\)\( p^{8} T^{35} + \)\(59\!\cdots\!21\)\( p^{9} T^{36} + \)\(54\!\cdots\!52\)\( p^{10} T^{37} + \)\(48\!\cdots\!34\)\( p^{11} T^{38} + 41707472991815152914 p^{12} T^{39} + 3449256026147885573 p^{13} T^{40} + 273524130332594896 p^{14} T^{41} + 20795640564028388 p^{15} T^{42} + 21186098733856 p^{17} T^{43} + 103475613821330 p^{17} T^{44} + 6697682506989 p^{18} T^{45} + 407685282229 p^{19} T^{46} + 22966050344 p^{20} T^{47} + 1196806441 p^{21} T^{48} + 56147872 p^{22} T^{49} + 2373541 p^{23} T^{50} + 85456 p^{24} T^{51} + 2640 p^{25} T^{52} + 59 p^{26} T^{53} + p^{27} T^{54} \)
73 \( 1 + 30 T + 1452 T^{2} + 35914 T^{3} + 1028688 T^{4} + 21665211 T^{5} + 474525951 T^{6} + 8737480816 T^{7} + 160584231349 T^{8} + 2638404866364 T^{9} + 42563100125473 T^{10} + 633729891108453 T^{11} + 9204940721110336 T^{12} + 125657149590841624 T^{13} + 1669621432008254155 T^{14} + 21080930278548471859 T^{15} + \)\(25\!\cdots\!70\)\( T^{16} + \)\(30\!\cdots\!88\)\( T^{17} + \)\(34\!\cdots\!40\)\( T^{18} + \)\(38\!\cdots\!87\)\( T^{19} + \)\(41\!\cdots\!30\)\( T^{20} + \)\(42\!\cdots\!59\)\( T^{21} + \)\(42\!\cdots\!16\)\( T^{22} + \)\(41\!\cdots\!13\)\( T^{23} + \)\(39\!\cdots\!79\)\( T^{24} + \)\(36\!\cdots\!48\)\( T^{25} + \)\(32\!\cdots\!73\)\( T^{26} + \)\(27\!\cdots\!50\)\( T^{27} + \)\(32\!\cdots\!73\)\( p T^{28} + \)\(36\!\cdots\!48\)\( p^{2} T^{29} + \)\(39\!\cdots\!79\)\( p^{3} T^{30} + \)\(41\!\cdots\!13\)\( p^{4} T^{31} + \)\(42\!\cdots\!16\)\( p^{5} T^{32} + \)\(42\!\cdots\!59\)\( p^{6} T^{33} + \)\(41\!\cdots\!30\)\( p^{7} T^{34} + \)\(38\!\cdots\!87\)\( p^{8} T^{35} + \)\(34\!\cdots\!40\)\( p^{9} T^{36} + \)\(30\!\cdots\!88\)\( p^{10} T^{37} + \)\(25\!\cdots\!70\)\( p^{11} T^{38} + 21080930278548471859 p^{12} T^{39} + 1669621432008254155 p^{13} T^{40} + 125657149590841624 p^{14} T^{41} + 9204940721110336 p^{15} T^{42} + 633729891108453 p^{16} T^{43} + 42563100125473 p^{17} T^{44} + 2638404866364 p^{18} T^{45} + 160584231349 p^{19} T^{46} + 8737480816 p^{20} T^{47} + 474525951 p^{21} T^{48} + 21665211 p^{22} T^{49} + 1028688 p^{23} T^{50} + 35914 p^{24} T^{51} + 1452 p^{25} T^{52} + 30 p^{26} T^{53} + p^{27} T^{54} \)
83 \( 1 - 4 T + 1145 T^{2} - 5299 T^{3} + 673057 T^{4} - 3495659 T^{5} + 268980248 T^{6} - 1534085472 T^{7} + 81801460924 T^{8} - 503553711919 T^{9} + 20110033117496 T^{10} - 131662087241702 T^{11} + 4148151744869807 T^{12} - 28504206192648402 T^{13} + 736055831576330280 T^{14} - 5242718910567890825 T^{15} + \)\(11\!\cdots\!62\)\( T^{16} - \)\(83\!\cdots\!71\)\( T^{17} + \)\(15\!\cdots\!73\)\( T^{18} - \)\(11\!\cdots\!16\)\( T^{19} + \)\(19\!\cdots\!58\)\( T^{20} - \)\(14\!\cdots\!38\)\( T^{21} + \)\(21\!\cdots\!40\)\( T^{22} - \)\(15\!\cdots\!28\)\( T^{23} + \)\(21\!\cdots\!53\)\( T^{24} - \)\(15\!\cdots\!23\)\( T^{25} + \)\(19\!\cdots\!72\)\( T^{26} - \)\(13\!\cdots\!56\)\( T^{27} + \)\(19\!\cdots\!72\)\( p T^{28} - \)\(15\!\cdots\!23\)\( p^{2} T^{29} + \)\(21\!\cdots\!53\)\( p^{3} T^{30} - \)\(15\!\cdots\!28\)\( p^{4} T^{31} + \)\(21\!\cdots\!40\)\( p^{5} T^{32} - \)\(14\!\cdots\!38\)\( p^{6} T^{33} + \)\(19\!\cdots\!58\)\( p^{7} T^{34} - \)\(11\!\cdots\!16\)\( p^{8} T^{35} + \)\(15\!\cdots\!73\)\( p^{9} T^{36} - \)\(83\!\cdots\!71\)\( p^{10} T^{37} + \)\(11\!\cdots\!62\)\( p^{11} T^{38} - 5242718910567890825 p^{12} T^{39} + 736055831576330280 p^{13} T^{40} - 28504206192648402 p^{14} T^{41} + 4148151744869807 p^{15} T^{42} - 131662087241702 p^{16} T^{43} + 20110033117496 p^{17} T^{44} - 503553711919 p^{18} T^{45} + 81801460924 p^{19} T^{46} - 1534085472 p^{20} T^{47} + 268980248 p^{21} T^{48} - 3495659 p^{22} T^{49} + 673057 p^{23} T^{50} - 5299 p^{24} T^{51} + 1145 p^{25} T^{52} - 4 p^{26} T^{53} + p^{27} T^{54} \)
89 \( 1 + 56 T + 2874 T^{2} + 101065 T^{3} + 3221794 T^{4} + 86101638 T^{5} + 2119831412 T^{6} + 46762541625 T^{7} + 964097651144 T^{8} + 18370948285987 T^{9} + 330766718728237 T^{10} + 5600630912925382 T^{11} + 90338800219936040 T^{12} + 1385268614580645899 T^{13} + 20359219202093892222 T^{14} + \)\(28\!\cdots\!64\)\( T^{15} + \)\(38\!\cdots\!02\)\( T^{16} + \)\(50\!\cdots\!80\)\( T^{17} + \)\(63\!\cdots\!77\)\( T^{18} + \)\(77\!\cdots\!72\)\( T^{19} + \)\(91\!\cdots\!82\)\( T^{20} + \)\(10\!\cdots\!03\)\( T^{21} + \)\(11\!\cdots\!89\)\( T^{22} + \)\(12\!\cdots\!94\)\( T^{23} + \)\(12\!\cdots\!89\)\( T^{24} + \)\(13\!\cdots\!73\)\( T^{25} + \)\(12\!\cdots\!59\)\( T^{26} + \)\(12\!\cdots\!62\)\( T^{27} + \)\(12\!\cdots\!59\)\( p T^{28} + \)\(13\!\cdots\!73\)\( p^{2} T^{29} + \)\(12\!\cdots\!89\)\( p^{3} T^{30} + \)\(12\!\cdots\!94\)\( p^{4} T^{31} + \)\(11\!\cdots\!89\)\( p^{5} T^{32} + \)\(10\!\cdots\!03\)\( p^{6} T^{33} + \)\(91\!\cdots\!82\)\( p^{7} T^{34} + \)\(77\!\cdots\!72\)\( p^{8} T^{35} + \)\(63\!\cdots\!77\)\( p^{9} T^{36} + \)\(50\!\cdots\!80\)\( p^{10} T^{37} + \)\(38\!\cdots\!02\)\( p^{11} T^{38} + \)\(28\!\cdots\!64\)\( p^{12} T^{39} + 20359219202093892222 p^{13} T^{40} + 1385268614580645899 p^{14} T^{41} + 90338800219936040 p^{15} T^{42} + 5600630912925382 p^{16} T^{43} + 330766718728237 p^{17} T^{44} + 18370948285987 p^{18} T^{45} + 964097651144 p^{19} T^{46} + 46762541625 p^{20} T^{47} + 2119831412 p^{21} T^{48} + 86101638 p^{22} T^{49} + 3221794 p^{23} T^{50} + 101065 p^{24} T^{51} + 2874 p^{25} T^{52} + 56 p^{26} T^{53} + p^{27} T^{54} \)
97 \( 1 + 30 T + 1565 T^{2} + 36079 T^{3} + 1120461 T^{4} + 21757305 T^{5} + 516925097 T^{6} + 8879294679 T^{7} + 177084934490 T^{8} + 2768076833777 T^{9} + 48531128413098 T^{10} + 702683764010450 T^{11} + 11126198489575895 T^{12} + 150993225445835027 T^{13} + 2197002077089120035 T^{14} + 28173979280287281963 T^{15} + \)\(38\!\cdots\!39\)\( T^{16} + \)\(46\!\cdots\!74\)\( T^{17} + \)\(58\!\cdots\!05\)\( T^{18} + \)\(68\!\cdots\!63\)\( T^{19} + \)\(82\!\cdots\!63\)\( T^{20} + \)\(91\!\cdots\!41\)\( T^{21} + \)\(10\!\cdots\!66\)\( T^{22} + \)\(11\!\cdots\!35\)\( T^{23} + \)\(12\!\cdots\!65\)\( T^{24} + \)\(12\!\cdots\!04\)\( T^{25} + \)\(12\!\cdots\!80\)\( T^{26} + \)\(12\!\cdots\!26\)\( T^{27} + \)\(12\!\cdots\!80\)\( p T^{28} + \)\(12\!\cdots\!04\)\( p^{2} T^{29} + \)\(12\!\cdots\!65\)\( p^{3} T^{30} + \)\(11\!\cdots\!35\)\( p^{4} T^{31} + \)\(10\!\cdots\!66\)\( p^{5} T^{32} + \)\(91\!\cdots\!41\)\( p^{6} T^{33} + \)\(82\!\cdots\!63\)\( p^{7} T^{34} + \)\(68\!\cdots\!63\)\( p^{8} T^{35} + \)\(58\!\cdots\!05\)\( p^{9} T^{36} + \)\(46\!\cdots\!74\)\( p^{10} T^{37} + \)\(38\!\cdots\!39\)\( p^{11} T^{38} + 28173979280287281963 p^{12} T^{39} + 2197002077089120035 p^{13} T^{40} + 150993225445835027 p^{14} T^{41} + 11126198489575895 p^{15} T^{42} + 702683764010450 p^{16} T^{43} + 48531128413098 p^{17} T^{44} + 2768076833777 p^{18} T^{45} + 177084934490 p^{19} T^{46} + 8879294679 p^{20} T^{47} + 516925097 p^{21} T^{48} + 21757305 p^{22} T^{49} + 1120461 p^{23} T^{50} + 36079 p^{24} T^{51} + 1565 p^{25} T^{52} + 30 p^{26} T^{53} + p^{27} T^{54} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{54} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.66201084311837848829378398419, −1.65228235002582736535200043720, −1.60900310338533282701315999410, −1.59352188894863065929323614885, −1.59195134373130681617193127889, −1.57482760596646002229890479813, −1.48292504214974470030995127225, −1.44798197986885863401234133857, −1.43228508078177691804410637923, −1.42853497338071704016563511861, −1.42103566000731217464926442850, −1.35656518152428276287307997563, −1.26960611677553576538793974657, −1.24670349009693493441986806405, −1.24582298967227430404226807957, −1.21016927207050466288531287407, −1.17074265416223335376936033655, −1.10929933429005821256921550193, −1.08578522346039099856117117325, −1.07134899847730775915191384810, −1.02664138539665128640691545062, −0.956738257191548000861116598894, −0.937208967167987134861266365243, −0.877322514979659159441187585786, −0.64776199887710157436986602223, 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, 0.64776199887710157436986602223, 0.877322514979659159441187585786, 0.937208967167987134861266365243, 0.956738257191548000861116598894, 1.02664138539665128640691545062, 1.07134899847730775915191384810, 1.08578522346039099856117117325, 1.10929933429005821256921550193, 1.17074265416223335376936033655, 1.21016927207050466288531287407, 1.24582298967227430404226807957, 1.24670349009693493441986806405, 1.26960611677553576538793974657, 1.35656518152428276287307997563, 1.42103566000731217464926442850, 1.42853497338071704016563511861, 1.43228508078177691804410637923, 1.44798197986885863401234133857, 1.48292504214974470030995127225, 1.57482760596646002229890479813, 1.59195134373130681617193127889, 1.59352188894863065929323614885, 1.60900310338533282701315999410, 1.65228235002582736535200043720, 1.66201084311837848829378398419

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

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