Properties

Degree 56
Conductor $ 2^{112} \cdot 503^{28} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 28

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 12·5-s − 31·9-s + 14·11-s − 31·13-s + 24·15-s − 9·17-s + 8·19-s + 4·23-s + 13·25-s + 68·27-s − 47·29-s + 5·31-s − 28·33-s − 67·37-s + 62·39-s − 28·41-s − 15·43-s + 372·45-s + 10·47-s − 88·49-s + 18·51-s − 58·53-s − 168·55-s − 16·57-s + 32·59-s − 55·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 5.36·5-s − 10.3·9-s + 4.22·11-s − 8.59·13-s + 6.19·15-s − 2.18·17-s + 1.83·19-s + 0.834·23-s + 13/5·25-s + 13.0·27-s − 8.72·29-s + 0.898·31-s − 4.87·33-s − 11.0·37-s + 9.92·39-s − 4.37·41-s − 2.28·43-s + 55.4·45-s + 1.45·47-s − 12.5·49-s + 2.52·51-s − 7.96·53-s − 22.6·55-s − 2.11·57-s + 4.16·59-s − 7.04·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{112} \cdot 503^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{112} \cdot 503^{28}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(56\)
\( N \)  =  \(2^{112} \cdot 503^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  28
Selberg data  =  $(56,\ 2^{112} \cdot 503^{28} ,\ ( \ : [1/2]^{28} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \(F_p\) is a polynomial of degree 56. If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 55.
$p$$F_p$
bad2 \( 1 \)
503 \( ( 1 + T )^{28} \)
good3 \( 1 + 2 T + 35 T^{2} + 64 T^{3} + 625 T^{4} + 1067 T^{5} + 7607 T^{6} + 12307 T^{7} + 7894 p^{2} T^{8} + 110128 T^{9} + 543200 T^{10} + 813250 T^{11} + 3540791 T^{12} + 5148910 T^{13} + 20224315 T^{14} + 28667257 T^{15} + 34396606 p T^{16} + 142831298 T^{17} + 476802785 T^{18} + 214890403 p T^{19} + 2014777436 T^{20} + 2658679579 T^{21} + 7840110031 T^{22} + 10079008553 T^{23} + 28231900405 T^{24} + 35267164864 T^{25} + 94383142742 T^{26} + 38064198188 p T^{27} + 293497015607 T^{28} + 38064198188 p^{2} T^{29} + 94383142742 p^{2} T^{30} + 35267164864 p^{3} T^{31} + 28231900405 p^{4} T^{32} + 10079008553 p^{5} T^{33} + 7840110031 p^{6} T^{34} + 2658679579 p^{7} T^{35} + 2014777436 p^{8} T^{36} + 214890403 p^{10} T^{37} + 476802785 p^{10} T^{38} + 142831298 p^{11} T^{39} + 34396606 p^{13} T^{40} + 28667257 p^{13} T^{41} + 20224315 p^{14} T^{42} + 5148910 p^{15} T^{43} + 3540791 p^{16} T^{44} + 813250 p^{17} T^{45} + 543200 p^{18} T^{46} + 110128 p^{19} T^{47} + 7894 p^{22} T^{48} + 12307 p^{21} T^{49} + 7607 p^{22} T^{50} + 1067 p^{23} T^{51} + 625 p^{24} T^{52} + 64 p^{25} T^{53} + 35 p^{26} T^{54} + 2 p^{27} T^{55} + p^{28} T^{56} \)
5 \( 1 + 12 T + 131 T^{2} + 199 p T^{3} + 6864 T^{4} + 40018 T^{5} + 215834 T^{6} + 1048341 T^{7} + 4781292 T^{8} + 20231362 T^{9} + 81272483 T^{10} + 308057929 T^{11} + 1117663323 T^{12} + 773534168 p T^{13} + 12890467427 T^{14} + 41283899974 T^{15} + 127949641144 T^{16} + 3064823391 p^{3} T^{17} + 1114093173204 T^{18} + 628449932707 p T^{19} + 8631363321752 T^{20} + 922382735724 p^{2} T^{21} + 60117621203543 T^{22} + 30546392183988 p T^{23} + 379157032738347 T^{24} + 918394403666308 T^{25} + 435095246296906 p T^{26} + 5031535329592807 T^{27} + 11384848558654458 T^{28} + 5031535329592807 p T^{29} + 435095246296906 p^{3} T^{30} + 918394403666308 p^{3} T^{31} + 379157032738347 p^{4} T^{32} + 30546392183988 p^{6} T^{33} + 60117621203543 p^{6} T^{34} + 922382735724 p^{9} T^{35} + 8631363321752 p^{8} T^{36} + 628449932707 p^{10} T^{37} + 1114093173204 p^{10} T^{38} + 3064823391 p^{14} T^{39} + 127949641144 p^{12} T^{40} + 41283899974 p^{13} T^{41} + 12890467427 p^{14} T^{42} + 773534168 p^{16} T^{43} + 1117663323 p^{16} T^{44} + 308057929 p^{17} T^{45} + 81272483 p^{18} T^{46} + 20231362 p^{19} T^{47} + 4781292 p^{20} T^{48} + 1048341 p^{21} T^{49} + 215834 p^{22} T^{50} + 40018 p^{23} T^{51} + 6864 p^{24} T^{52} + 199 p^{26} T^{53} + 131 p^{26} T^{54} + 12 p^{27} T^{55} + p^{28} T^{56} \)
7 \( 1 + 88 T^{2} + 3 p T^{3} + 3943 T^{4} + 1662 T^{5} + 120109 T^{6} + 9479 p T^{7} + 399688 p T^{8} + 1781281 T^{9} + 7584314 p T^{10} + 36176558 T^{11} + 121874328 p T^{12} + 592952562 T^{13} + 11914092077 T^{14} + 8171055117 T^{15} + 147294265177 T^{16} + 97399946121 T^{17} + 1634509420822 T^{18} + 146584650442 p T^{19} + 16454889604046 T^{20} + 9721824238391 T^{21} + 151531271020897 T^{22} + 84085164175925 T^{23} + 1284511578134090 T^{24} + 672302745934766 T^{25} + 10067935072291184 T^{26} + 5016577601596944 T^{27} + 73160446327105451 T^{28} + 5016577601596944 p T^{29} + 10067935072291184 p^{2} T^{30} + 672302745934766 p^{3} T^{31} + 1284511578134090 p^{4} T^{32} + 84085164175925 p^{5} T^{33} + 151531271020897 p^{6} T^{34} + 9721824238391 p^{7} T^{35} + 16454889604046 p^{8} T^{36} + 146584650442 p^{10} T^{37} + 1634509420822 p^{10} T^{38} + 97399946121 p^{11} T^{39} + 147294265177 p^{12} T^{40} + 8171055117 p^{13} T^{41} + 11914092077 p^{14} T^{42} + 592952562 p^{15} T^{43} + 121874328 p^{17} T^{44} + 36176558 p^{17} T^{45} + 7584314 p^{19} T^{46} + 1781281 p^{19} T^{47} + 399688 p^{21} T^{48} + 9479 p^{22} T^{49} + 120109 p^{22} T^{50} + 1662 p^{23} T^{51} + 3943 p^{24} T^{52} + 3 p^{26} T^{53} + 88 p^{26} T^{54} + p^{28} T^{56} \)
11 \( 1 - 14 T + 255 T^{2} - 2745 T^{3} + 2751 p T^{4} - 266278 T^{5} + 2265798 T^{6} - 16992179 T^{7} + 121670117 T^{8} - 800772160 T^{9} + 5032116390 T^{10} - 29684168147 T^{11} + 167766333813 T^{12} - 900909381912 T^{13} + 4653496518533 T^{14} - 23017856970158 T^{15} + 109912290440433 T^{16} - 505335270195828 T^{17} + 2249822104137249 T^{18} - 9682646067809708 T^{19} + 40454654262029859 T^{20} - 163868216199639079 T^{21} + 645661412900267771 T^{22} - 2471660352451851345 T^{23} + 9216896876111189959 T^{24} - 3040095282181524278 p T^{25} + \)\(11\!\cdots\!27\)\( T^{26} - \)\(40\!\cdots\!92\)\( T^{27} + \)\(13\!\cdots\!73\)\( T^{28} - \)\(40\!\cdots\!92\)\( p T^{29} + \)\(11\!\cdots\!27\)\( p^{2} T^{30} - 3040095282181524278 p^{4} T^{31} + 9216896876111189959 p^{4} T^{32} - 2471660352451851345 p^{5} T^{33} + 645661412900267771 p^{6} T^{34} - 163868216199639079 p^{7} T^{35} + 40454654262029859 p^{8} T^{36} - 9682646067809708 p^{9} T^{37} + 2249822104137249 p^{10} T^{38} - 505335270195828 p^{11} T^{39} + 109912290440433 p^{12} T^{40} - 23017856970158 p^{13} T^{41} + 4653496518533 p^{14} T^{42} - 900909381912 p^{15} T^{43} + 167766333813 p^{16} T^{44} - 29684168147 p^{17} T^{45} + 5032116390 p^{18} T^{46} - 800772160 p^{19} T^{47} + 121670117 p^{20} T^{48} - 16992179 p^{21} T^{49} + 2265798 p^{22} T^{50} - 266278 p^{23} T^{51} + 2751 p^{25} T^{52} - 2745 p^{25} T^{53} + 255 p^{26} T^{54} - 14 p^{27} T^{55} + p^{28} T^{56} \)
13 \( 1 + 31 T + 660 T^{2} + 10503 T^{3} + 140111 T^{4} + 1612891 T^{5} + 16574573 T^{6} + 154303657 T^{7} + 1321495411 T^{8} + 62145223 p^{2} T^{9} + 78122191807 T^{10} + 546962420984 T^{11} + 3623917360456 T^{12} + 22809878075702 T^{13} + 136890488910593 T^{14} + 785479947155492 T^{15} + 4320465380577609 T^{16} + 22826659659763810 T^{17} + 116061235006019375 T^{18} + 568743967526434320 T^{19} + 2689861382399404133 T^{20} + 12291460620874985897 T^{21} + 321428243321194010 p^{2} T^{22} + \)\(23\!\cdots\!15\)\( T^{23} + \)\(96\!\cdots\!16\)\( T^{24} + \)\(38\!\cdots\!49\)\( T^{25} + \)\(15\!\cdots\!91\)\( T^{26} + \)\(56\!\cdots\!51\)\( T^{27} + \)\(20\!\cdots\!95\)\( T^{28} + \)\(56\!\cdots\!51\)\( p T^{29} + \)\(15\!\cdots\!91\)\( p^{2} T^{30} + \)\(38\!\cdots\!49\)\( p^{3} T^{31} + \)\(96\!\cdots\!16\)\( p^{4} T^{32} + \)\(23\!\cdots\!15\)\( p^{5} T^{33} + 321428243321194010 p^{8} T^{34} + 12291460620874985897 p^{7} T^{35} + 2689861382399404133 p^{8} T^{36} + 568743967526434320 p^{9} T^{37} + 116061235006019375 p^{10} T^{38} + 22826659659763810 p^{11} T^{39} + 4320465380577609 p^{12} T^{40} + 785479947155492 p^{13} T^{41} + 136890488910593 p^{14} T^{42} + 22809878075702 p^{15} T^{43} + 3623917360456 p^{16} T^{44} + 546962420984 p^{17} T^{45} + 78122191807 p^{18} T^{46} + 62145223 p^{21} T^{47} + 1321495411 p^{20} T^{48} + 154303657 p^{21} T^{49} + 16574573 p^{22} T^{50} + 1612891 p^{23} T^{51} + 140111 p^{24} T^{52} + 10503 p^{25} T^{53} + 660 p^{26} T^{54} + 31 p^{27} T^{55} + p^{28} T^{56} \)
17 \( 1 + 9 T + 250 T^{2} + 1783 T^{3} + 28622 T^{4} + 166782 T^{5} + 2035156 T^{6} + 9728333 T^{7} + 101823454 T^{8} + 389314389 T^{9} + 13250255 p^{2} T^{10} + 10821000036 T^{11} + 112663495250 T^{12} + 185017834471 T^{13} + 2674996940469 T^{14} + 219642106847 T^{15} + 53526479648370 T^{16} - 108961694569977 T^{17} + 990002308362366 T^{18} - 4395495900239885 T^{19} + 19711640666759310 T^{20} - 111860114946300184 T^{21} + 454877042466759364 T^{22} - 2188296392041045785 T^{23} + 10896760172683487119 T^{24} - 36307538850657650184 T^{25} + 13942247962040946284 p T^{26} - \)\(57\!\cdots\!27\)\( T^{27} + \)\(43\!\cdots\!80\)\( T^{28} - \)\(57\!\cdots\!27\)\( p T^{29} + 13942247962040946284 p^{3} T^{30} - 36307538850657650184 p^{3} T^{31} + 10896760172683487119 p^{4} T^{32} - 2188296392041045785 p^{5} T^{33} + 454877042466759364 p^{6} T^{34} - 111860114946300184 p^{7} T^{35} + 19711640666759310 p^{8} T^{36} - 4395495900239885 p^{9} T^{37} + 990002308362366 p^{10} T^{38} - 108961694569977 p^{11} T^{39} + 53526479648370 p^{12} T^{40} + 219642106847 p^{13} T^{41} + 2674996940469 p^{14} T^{42} + 185017834471 p^{15} T^{43} + 112663495250 p^{16} T^{44} + 10821000036 p^{17} T^{45} + 13250255 p^{20} T^{46} + 389314389 p^{19} T^{47} + 101823454 p^{20} T^{48} + 9728333 p^{21} T^{49} + 2035156 p^{22} T^{50} + 166782 p^{23} T^{51} + 28622 p^{24} T^{52} + 1783 p^{25} T^{53} + 250 p^{26} T^{54} + 9 p^{27} T^{55} + p^{28} T^{56} \)
19 \( 1 - 8 T + 299 T^{2} - 2323 T^{3} + 45002 T^{4} - 337967 T^{5} + 4549348 T^{6} - 32850950 T^{7} + 347161056 T^{8} - 2399422962 T^{9} + 21284620245 T^{10} - 140370591702 T^{11} + 1089197728220 T^{12} - 6843320346570 T^{13} + 47714128202988 T^{14} - 285480119045105 T^{15} + 1821404983104956 T^{16} - 10380502042775395 T^{17} + 61375521524940195 T^{18} - 333365229277524317 T^{19} + 1843105661345973338 T^{20} - 9545778308879129669 T^{21} + 49671880853751226506 T^{22} - \)\(24\!\cdots\!65\)\( T^{23} + \)\(12\!\cdots\!59\)\( T^{24} - \)\(56\!\cdots\!19\)\( T^{25} + \)\(26\!\cdots\!83\)\( T^{26} - \)\(11\!\cdots\!84\)\( T^{27} + \)\(52\!\cdots\!32\)\( T^{28} - \)\(11\!\cdots\!84\)\( p T^{29} + \)\(26\!\cdots\!83\)\( p^{2} T^{30} - \)\(56\!\cdots\!19\)\( p^{3} T^{31} + \)\(12\!\cdots\!59\)\( p^{4} T^{32} - \)\(24\!\cdots\!65\)\( p^{5} T^{33} + 49671880853751226506 p^{6} T^{34} - 9545778308879129669 p^{7} T^{35} + 1843105661345973338 p^{8} T^{36} - 333365229277524317 p^{9} T^{37} + 61375521524940195 p^{10} T^{38} - 10380502042775395 p^{11} T^{39} + 1821404983104956 p^{12} T^{40} - 285480119045105 p^{13} T^{41} + 47714128202988 p^{14} T^{42} - 6843320346570 p^{15} T^{43} + 1089197728220 p^{16} T^{44} - 140370591702 p^{17} T^{45} + 21284620245 p^{18} T^{46} - 2399422962 p^{19} T^{47} + 347161056 p^{20} T^{48} - 32850950 p^{21} T^{49} + 4549348 p^{22} T^{50} - 337967 p^{23} T^{51} + 45002 p^{24} T^{52} - 2323 p^{25} T^{53} + 299 p^{26} T^{54} - 8 p^{27} T^{55} + p^{28} T^{56} \)
23 \( 1 - 4 T + 258 T^{2} - 865 T^{3} + 34757 T^{4} - 96640 T^{5} + 3225191 T^{6} - 7210116 T^{7} + 230280505 T^{8} - 391239702 T^{9} + 13433858366 T^{10} - 15622196042 T^{11} + 665080667844 T^{12} - 417063334782 T^{13} + 28694616081563 T^{14} - 2820840175246 T^{15} + 1100510300084084 T^{16} + 450940581904809 T^{17} + 38103913050684945 T^{18} + 33418520816317831 T^{19} + 1205722368813170436 T^{20} + 1542090441078683443 T^{21} + 35204662771016675154 T^{22} + 55973907182027566803 T^{23} + 41537997089775006226 p T^{24} + \)\(17\!\cdots\!17\)\( T^{25} + \)\(24\!\cdots\!47\)\( T^{26} + \)\(45\!\cdots\!32\)\( T^{27} + \)\(57\!\cdots\!38\)\( T^{28} + \)\(45\!\cdots\!32\)\( p T^{29} + \)\(24\!\cdots\!47\)\( p^{2} T^{30} + \)\(17\!\cdots\!17\)\( p^{3} T^{31} + 41537997089775006226 p^{5} T^{32} + 55973907182027566803 p^{5} T^{33} + 35204662771016675154 p^{6} T^{34} + 1542090441078683443 p^{7} T^{35} + 1205722368813170436 p^{8} T^{36} + 33418520816317831 p^{9} T^{37} + 38103913050684945 p^{10} T^{38} + 450940581904809 p^{11} T^{39} + 1100510300084084 p^{12} T^{40} - 2820840175246 p^{13} T^{41} + 28694616081563 p^{14} T^{42} - 417063334782 p^{15} T^{43} + 665080667844 p^{16} T^{44} - 15622196042 p^{17} T^{45} + 13433858366 p^{18} T^{46} - 391239702 p^{19} T^{47} + 230280505 p^{20} T^{48} - 7210116 p^{21} T^{49} + 3225191 p^{22} T^{50} - 96640 p^{23} T^{51} + 34757 p^{24} T^{52} - 865 p^{25} T^{53} + 258 p^{26} T^{54} - 4 p^{27} T^{55} + p^{28} T^{56} \)
29 \( 1 + 47 T + 1432 T^{2} + 32107 T^{3} + 592706 T^{4} + 9337899 T^{5} + 130096744 T^{6} + 1630616477 T^{7} + 18704188455 T^{8} + 198343135082 T^{9} + 1964141394849 T^{10} + 18285239581279 T^{11} + 161104457146519 T^{12} + 1349704676898073 T^{13} + 10803460647146022 T^{14} + 82907007045720735 T^{15} + 612169611204770125 T^{16} + 4360777676227460104 T^{17} + 30052377896555458227 T^{18} + \)\(20\!\cdots\!33\)\( T^{19} + \)\(13\!\cdots\!90\)\( T^{20} + \)\(82\!\cdots\!97\)\( T^{21} + \)\(50\!\cdots\!01\)\( T^{22} + \)\(10\!\cdots\!00\)\( p T^{23} + \)\(17\!\cdots\!91\)\( T^{24} + \)\(10\!\cdots\!55\)\( T^{25} + \)\(58\!\cdots\!69\)\( T^{26} + \)\(32\!\cdots\!20\)\( T^{27} + \)\(17\!\cdots\!26\)\( T^{28} + \)\(32\!\cdots\!20\)\( p T^{29} + \)\(58\!\cdots\!69\)\( p^{2} T^{30} + \)\(10\!\cdots\!55\)\( p^{3} T^{31} + \)\(17\!\cdots\!91\)\( p^{4} T^{32} + \)\(10\!\cdots\!00\)\( p^{6} T^{33} + \)\(50\!\cdots\!01\)\( p^{6} T^{34} + \)\(82\!\cdots\!97\)\( p^{7} T^{35} + \)\(13\!\cdots\!90\)\( p^{8} T^{36} + \)\(20\!\cdots\!33\)\( p^{9} T^{37} + 30052377896555458227 p^{10} T^{38} + 4360777676227460104 p^{11} T^{39} + 612169611204770125 p^{12} T^{40} + 82907007045720735 p^{13} T^{41} + 10803460647146022 p^{14} T^{42} + 1349704676898073 p^{15} T^{43} + 161104457146519 p^{16} T^{44} + 18285239581279 p^{17} T^{45} + 1964141394849 p^{18} T^{46} + 198343135082 p^{19} T^{47} + 18704188455 p^{20} T^{48} + 1630616477 p^{21} T^{49} + 130096744 p^{22} T^{50} + 9337899 p^{23} T^{51} + 592706 p^{24} T^{52} + 32107 p^{25} T^{53} + 1432 p^{26} T^{54} + 47 p^{27} T^{55} + p^{28} T^{56} \)
31 \( 1 - 5 T + 373 T^{2} - 1650 T^{3} + 67999 T^{4} - 253984 T^{5} + 8105255 T^{6} - 24140352 T^{7} + 716013959 T^{8} - 1568395073 T^{9} + 50635276573 T^{10} - 71552269649 T^{11} + 3034867370738 T^{12} - 2132119202627 T^{13} + 161061677414049 T^{14} - 19321847695320 T^{15} + 7797306777288479 T^{16} + 2341000542723051 T^{17} + 349620173299242493 T^{18} + 200410033426461122 T^{19} + 14584471317268117653 T^{20} + 10902851809568381724 T^{21} + \)\(56\!\cdots\!11\)\( T^{22} + \)\(49\!\cdots\!74\)\( T^{23} + \)\(20\!\cdots\!57\)\( T^{24} + \)\(19\!\cdots\!65\)\( T^{25} + \)\(69\!\cdots\!98\)\( T^{26} + \)\(67\!\cdots\!86\)\( T^{27} + \)\(22\!\cdots\!56\)\( T^{28} + \)\(67\!\cdots\!86\)\( p T^{29} + \)\(69\!\cdots\!98\)\( p^{2} T^{30} + \)\(19\!\cdots\!65\)\( p^{3} T^{31} + \)\(20\!\cdots\!57\)\( p^{4} T^{32} + \)\(49\!\cdots\!74\)\( p^{5} T^{33} + \)\(56\!\cdots\!11\)\( p^{6} T^{34} + 10902851809568381724 p^{7} T^{35} + 14584471317268117653 p^{8} T^{36} + 200410033426461122 p^{9} T^{37} + 349620173299242493 p^{10} T^{38} + 2341000542723051 p^{11} T^{39} + 7797306777288479 p^{12} T^{40} - 19321847695320 p^{13} T^{41} + 161061677414049 p^{14} T^{42} - 2132119202627 p^{15} T^{43} + 3034867370738 p^{16} T^{44} - 71552269649 p^{17} T^{45} + 50635276573 p^{18} T^{46} - 1568395073 p^{19} T^{47} + 716013959 p^{20} T^{48} - 24140352 p^{21} T^{49} + 8105255 p^{22} T^{50} - 253984 p^{23} T^{51} + 67999 p^{24} T^{52} - 1650 p^{25} T^{53} + 373 p^{26} T^{54} - 5 p^{27} T^{55} + p^{28} T^{56} \)
37 \( 1 + 67 T + 2763 T^{2} + 84265 T^{3} + 2096067 T^{4} + 44472295 T^{5} + 830032839 T^{6} + 13895854654 T^{7} + 211766280283 T^{8} + 2969320701858 T^{9} + 38636632537757 T^{10} + 469673493534282 T^{11} + 5363773403861265 T^{12} + 57812634847015732 T^{13} + 590438493452580982 T^{14} + 5733325204335342595 T^{15} + 53092691253652712237 T^{16} + \)\(47\!\cdots\!61\)\( T^{17} + \)\(39\!\cdots\!44\)\( T^{18} + \)\(32\!\cdots\!93\)\( T^{19} + \)\(25\!\cdots\!17\)\( T^{20} + \)\(19\!\cdots\!71\)\( T^{21} + \)\(14\!\cdots\!02\)\( T^{22} + \)\(10\!\cdots\!42\)\( T^{23} + \)\(69\!\cdots\!23\)\( T^{24} + \)\(46\!\cdots\!63\)\( T^{25} + \)\(30\!\cdots\!01\)\( T^{26} + \)\(19\!\cdots\!82\)\( T^{27} + \)\(11\!\cdots\!54\)\( T^{28} + \)\(19\!\cdots\!82\)\( p T^{29} + \)\(30\!\cdots\!01\)\( p^{2} T^{30} + \)\(46\!\cdots\!63\)\( p^{3} T^{31} + \)\(69\!\cdots\!23\)\( p^{4} T^{32} + \)\(10\!\cdots\!42\)\( p^{5} T^{33} + \)\(14\!\cdots\!02\)\( p^{6} T^{34} + \)\(19\!\cdots\!71\)\( p^{7} T^{35} + \)\(25\!\cdots\!17\)\( p^{8} T^{36} + \)\(32\!\cdots\!93\)\( p^{9} T^{37} + \)\(39\!\cdots\!44\)\( p^{10} T^{38} + \)\(47\!\cdots\!61\)\( p^{11} T^{39} + 53092691253652712237 p^{12} T^{40} + 5733325204335342595 p^{13} T^{41} + 590438493452580982 p^{14} T^{42} + 57812634847015732 p^{15} T^{43} + 5363773403861265 p^{16} T^{44} + 469673493534282 p^{17} T^{45} + 38636632537757 p^{18} T^{46} + 2969320701858 p^{19} T^{47} + 211766280283 p^{20} T^{48} + 13895854654 p^{21} T^{49} + 830032839 p^{22} T^{50} + 44472295 p^{23} T^{51} + 2096067 p^{24} T^{52} + 84265 p^{25} T^{53} + 2763 p^{26} T^{54} + 67 p^{27} T^{55} + p^{28} T^{56} \)
41 \( 1 + 28 T + 916 T^{2} + 18018 T^{3} + 362036 T^{4} + 5681729 T^{5} + 88378305 T^{6} + 1178707286 T^{7} + 15458722100 T^{8} + 181851535791 T^{9} + 2100718056467 T^{10} + 22330740406210 T^{11} + 233265446212670 T^{12} + 2278416948796973 T^{13} + 21896814752235920 T^{14} + 198877750927602543 T^{15} + 1779563682252876400 T^{16} + 15159448365070526084 T^{17} + \)\(12\!\cdots\!52\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{19} + \)\(81\!\cdots\!44\)\( T^{20} + \)\(61\!\cdots\!25\)\( T^{21} + \)\(46\!\cdots\!59\)\( T^{22} + \)\(82\!\cdots\!90\)\( p T^{23} + \)\(24\!\cdots\!19\)\( T^{24} + \)\(16\!\cdots\!18\)\( T^{25} + \)\(11\!\cdots\!49\)\( T^{26} + \)\(75\!\cdots\!29\)\( T^{27} + \)\(49\!\cdots\!68\)\( T^{28} + \)\(75\!\cdots\!29\)\( p T^{29} + \)\(11\!\cdots\!49\)\( p^{2} T^{30} + \)\(16\!\cdots\!18\)\( p^{3} T^{31} + \)\(24\!\cdots\!19\)\( p^{4} T^{32} + \)\(82\!\cdots\!90\)\( p^{6} T^{33} + \)\(46\!\cdots\!59\)\( p^{6} T^{34} + \)\(61\!\cdots\!25\)\( p^{7} T^{35} + \)\(81\!\cdots\!44\)\( p^{8} T^{36} + \)\(10\!\cdots\!92\)\( p^{9} T^{37} + \)\(12\!\cdots\!52\)\( p^{10} T^{38} + 15159448365070526084 p^{11} T^{39} + 1779563682252876400 p^{12} T^{40} + 198877750927602543 p^{13} T^{41} + 21896814752235920 p^{14} T^{42} + 2278416948796973 p^{15} T^{43} + 233265446212670 p^{16} T^{44} + 22330740406210 p^{17} T^{45} + 2100718056467 p^{18} T^{46} + 181851535791 p^{19} T^{47} + 15458722100 p^{20} T^{48} + 1178707286 p^{21} T^{49} + 88378305 p^{22} T^{50} + 5681729 p^{23} T^{51} + 362036 p^{24} T^{52} + 18018 p^{25} T^{53} + 916 p^{26} T^{54} + 28 p^{27} T^{55} + p^{28} T^{56} \)
43 \( 1 + 15 T + 585 T^{2} + 7573 T^{3} + 169980 T^{4} + 1947031 T^{5} + 32878180 T^{6} + 340204782 T^{7} + 4784703630 T^{8} + 45455567697 T^{9} + 560807285336 T^{10} + 4952797508684 T^{11} + 55278976208049 T^{12} + 10656303939083 p T^{13} + 4720019050705272 T^{14} + 36999011995478884 T^{15} + 356582776622644566 T^{16} + 2658718682854124603 T^{17} + 24206663555886452646 T^{18} + \)\(17\!\cdots\!01\)\( T^{19} + \)\(14\!\cdots\!07\)\( T^{20} + \)\(10\!\cdots\!68\)\( T^{21} + \)\(84\!\cdots\!23\)\( T^{22} + \)\(55\!\cdots\!33\)\( T^{23} + \)\(43\!\cdots\!32\)\( T^{24} + \)\(27\!\cdots\!53\)\( T^{25} + \)\(21\!\cdots\!87\)\( T^{26} + \)\(12\!\cdots\!73\)\( T^{27} + \)\(94\!\cdots\!79\)\( T^{28} + \)\(12\!\cdots\!73\)\( p T^{29} + \)\(21\!\cdots\!87\)\( p^{2} T^{30} + \)\(27\!\cdots\!53\)\( p^{3} T^{31} + \)\(43\!\cdots\!32\)\( p^{4} T^{32} + \)\(55\!\cdots\!33\)\( p^{5} T^{33} + \)\(84\!\cdots\!23\)\( p^{6} T^{34} + \)\(10\!\cdots\!68\)\( p^{7} T^{35} + \)\(14\!\cdots\!07\)\( p^{8} T^{36} + \)\(17\!\cdots\!01\)\( p^{9} T^{37} + 24206663555886452646 p^{10} T^{38} + 2658718682854124603 p^{11} T^{39} + 356582776622644566 p^{12} T^{40} + 36999011995478884 p^{13} T^{41} + 4720019050705272 p^{14} T^{42} + 10656303939083 p^{16} T^{43} + 55278976208049 p^{16} T^{44} + 4952797508684 p^{17} T^{45} + 560807285336 p^{18} T^{46} + 45455567697 p^{19} T^{47} + 4784703630 p^{20} T^{48} + 340204782 p^{21} T^{49} + 32878180 p^{22} T^{50} + 1947031 p^{23} T^{51} + 169980 p^{24} T^{52} + 7573 p^{25} T^{53} + 585 p^{26} T^{54} + 15 p^{27} T^{55} + p^{28} T^{56} \)
47 \( 1 - 10 T + 643 T^{2} - 5245 T^{3} + 197162 T^{4} - 1345529 T^{5} + 39268627 T^{6} - 228634519 T^{7} + 5807692847 T^{8} - 29349686180 T^{9} + 687797823330 T^{10} - 3066487778028 T^{11} + 1454892589267 p T^{12} - 273253219928829 T^{13} + 5890235780094114 T^{14} - 21427383545910275 T^{15} + 449697963454222929 T^{16} - 1511936884139342991 T^{17} + 30963522880282124708 T^{18} - 97597573635958590332 T^{19} + \)\(19\!\cdots\!60\)\( T^{20} - \)\(58\!\cdots\!56\)\( T^{21} + \)\(11\!\cdots\!24\)\( T^{22} - \)\(32\!\cdots\!11\)\( T^{23} + \)\(61\!\cdots\!70\)\( T^{24} - \)\(17\!\cdots\!67\)\( T^{25} + \)\(31\!\cdots\!16\)\( T^{26} - \)\(85\!\cdots\!39\)\( T^{27} + \)\(15\!\cdots\!77\)\( T^{28} - \)\(85\!\cdots\!39\)\( p T^{29} + \)\(31\!\cdots\!16\)\( p^{2} T^{30} - \)\(17\!\cdots\!67\)\( p^{3} T^{31} + \)\(61\!\cdots\!70\)\( p^{4} T^{32} - \)\(32\!\cdots\!11\)\( p^{5} T^{33} + \)\(11\!\cdots\!24\)\( p^{6} T^{34} - \)\(58\!\cdots\!56\)\( p^{7} T^{35} + \)\(19\!\cdots\!60\)\( p^{8} T^{36} - 97597573635958590332 p^{9} T^{37} + 30963522880282124708 p^{10} T^{38} - 1511936884139342991 p^{11} T^{39} + 449697963454222929 p^{12} T^{40} - 21427383545910275 p^{13} T^{41} + 5890235780094114 p^{14} T^{42} - 273253219928829 p^{15} T^{43} + 1454892589267 p^{17} T^{44} - 3066487778028 p^{17} T^{45} + 687797823330 p^{18} T^{46} - 29349686180 p^{19} T^{47} + 5807692847 p^{20} T^{48} - 228634519 p^{21} T^{49} + 39268627 p^{22} T^{50} - 1345529 p^{23} T^{51} + 197162 p^{24} T^{52} - 5245 p^{25} T^{53} + 643 p^{26} T^{54} - 10 p^{27} T^{55} + p^{28} T^{56} \)
53 \( 1 + 58 T + 2466 T^{2} + 75884 T^{3} + 1969083 T^{4} + 43293584 T^{5} + 846197080 T^{6} + 14784017384 T^{7} + 236147967548 T^{8} + 3464649848799 T^{9} + 47319370098775 T^{10} + 604021491830465 T^{11} + 7274212169093864 T^{12} + 82946204121678851 T^{13} + 902153973645038174 T^{14} + 9388950274352582740 T^{15} + 94054221838537192190 T^{16} + \)\(90\!\cdots\!29\)\( T^{17} + \)\(85\!\cdots\!37\)\( T^{18} + \)\(77\!\cdots\!56\)\( T^{19} + \)\(68\!\cdots\!77\)\( T^{20} + \)\(59\!\cdots\!48\)\( T^{21} + \)\(49\!\cdots\!49\)\( T^{22} + \)\(40\!\cdots\!33\)\( T^{23} + \)\(32\!\cdots\!01\)\( T^{24} + \)\(25\!\cdots\!65\)\( T^{25} + \)\(19\!\cdots\!91\)\( T^{26} + \)\(14\!\cdots\!92\)\( T^{27} + \)\(10\!\cdots\!96\)\( T^{28} + \)\(14\!\cdots\!92\)\( p T^{29} + \)\(19\!\cdots\!91\)\( p^{2} T^{30} + \)\(25\!\cdots\!65\)\( p^{3} T^{31} + \)\(32\!\cdots\!01\)\( p^{4} T^{32} + \)\(40\!\cdots\!33\)\( p^{5} T^{33} + \)\(49\!\cdots\!49\)\( p^{6} T^{34} + \)\(59\!\cdots\!48\)\( p^{7} T^{35} + \)\(68\!\cdots\!77\)\( p^{8} T^{36} + \)\(77\!\cdots\!56\)\( p^{9} T^{37} + \)\(85\!\cdots\!37\)\( p^{10} T^{38} + \)\(90\!\cdots\!29\)\( p^{11} T^{39} + 94054221838537192190 p^{12} T^{40} + 9388950274352582740 p^{13} T^{41} + 902153973645038174 p^{14} T^{42} + 82946204121678851 p^{15} T^{43} + 7274212169093864 p^{16} T^{44} + 604021491830465 p^{17} T^{45} + 47319370098775 p^{18} T^{46} + 3464649848799 p^{19} T^{47} + 236147967548 p^{20} T^{48} + 14784017384 p^{21} T^{49} + 846197080 p^{22} T^{50} + 43293584 p^{23} T^{51} + 1969083 p^{24} T^{52} + 75884 p^{25} T^{53} + 2466 p^{26} T^{54} + 58 p^{27} T^{55} + p^{28} T^{56} \)
59 \( 1 - 32 T + 1598 T^{2} - 39417 T^{3} + 1157039 T^{4} - 23595670 T^{5} + 521867393 T^{6} - 9165846276 T^{7} + 167507819531 T^{8} - 2601620416183 T^{9} + 41165633163904 T^{10} - 575766309925914 T^{11} + 8109141521791547 T^{12} - 103484836360745130 T^{13} + 1320814989378259194 T^{14} - 15528809918461721487 T^{15} + \)\(18\!\cdots\!88\)\( T^{16} - \)\(19\!\cdots\!92\)\( T^{17} + \)\(21\!\cdots\!58\)\( T^{18} - \)\(21\!\cdots\!20\)\( T^{19} + \)\(22\!\cdots\!50\)\( T^{20} - \)\(21\!\cdots\!44\)\( T^{21} + \)\(19\!\cdots\!06\)\( T^{22} - \)\(17\!\cdots\!27\)\( T^{23} + \)\(15\!\cdots\!11\)\( T^{24} - \)\(13\!\cdots\!75\)\( T^{25} + \)\(11\!\cdots\!55\)\( T^{26} - \)\(88\!\cdots\!13\)\( T^{27} + \)\(69\!\cdots\!18\)\( T^{28} - \)\(88\!\cdots\!13\)\( p T^{29} + \)\(11\!\cdots\!55\)\( p^{2} T^{30} - \)\(13\!\cdots\!75\)\( p^{3} T^{31} + \)\(15\!\cdots\!11\)\( p^{4} T^{32} - \)\(17\!\cdots\!27\)\( p^{5} T^{33} + \)\(19\!\cdots\!06\)\( p^{6} T^{34} - \)\(21\!\cdots\!44\)\( p^{7} T^{35} + \)\(22\!\cdots\!50\)\( p^{8} T^{36} - \)\(21\!\cdots\!20\)\( p^{9} T^{37} + \)\(21\!\cdots\!58\)\( p^{10} T^{38} - \)\(19\!\cdots\!92\)\( p^{11} T^{39} + \)\(18\!\cdots\!88\)\( p^{12} T^{40} - 15528809918461721487 p^{13} T^{41} + 1320814989378259194 p^{14} T^{42} - 103484836360745130 p^{15} T^{43} + 8109141521791547 p^{16} T^{44} - 575766309925914 p^{17} T^{45} + 41165633163904 p^{18} T^{46} - 2601620416183 p^{19} T^{47} + 167507819531 p^{20} T^{48} - 9165846276 p^{21} T^{49} + 521867393 p^{22} T^{50} - 23595670 p^{23} T^{51} + 1157039 p^{24} T^{52} - 39417 p^{25} T^{53} + 1598 p^{26} T^{54} - 32 p^{27} T^{55} + p^{28} T^{56} \)
61 \( 1 + 55 T + 2204 T^{2} + 64571 T^{3} + 1601380 T^{4} + 33952394 T^{5} + 643751982 T^{6} + 11000589745 T^{7} + 172890668004 T^{8} + 2512861527533 T^{9} + 34149669156977 T^{10} + 435553186735964 T^{11} + 5247795953263833 T^{12} + 59871941819544349 T^{13} + 649416612847666472 T^{14} + 6705709304831925095 T^{15} + 66072774293502495400 T^{16} + \)\(62\!\cdots\!26\)\( T^{17} + \)\(55\!\cdots\!82\)\( T^{18} + \)\(48\!\cdots\!12\)\( T^{19} + \)\(39\!\cdots\!47\)\( T^{20} + \)\(30\!\cdots\!59\)\( T^{21} + \)\(23\!\cdots\!29\)\( T^{22} + \)\(16\!\cdots\!80\)\( T^{23} + \)\(11\!\cdots\!91\)\( T^{24} + \)\(78\!\cdots\!71\)\( T^{25} + \)\(53\!\cdots\!20\)\( T^{26} + \)\(37\!\cdots\!78\)\( T^{27} + \)\(27\!\cdots\!44\)\( T^{28} + \)\(37\!\cdots\!78\)\( p T^{29} + \)\(53\!\cdots\!20\)\( p^{2} T^{30} + \)\(78\!\cdots\!71\)\( p^{3} T^{31} + \)\(11\!\cdots\!91\)\( p^{4} T^{32} + \)\(16\!\cdots\!80\)\( p^{5} T^{33} + \)\(23\!\cdots\!29\)\( p^{6} T^{34} + \)\(30\!\cdots\!59\)\( p^{7} T^{35} + \)\(39\!\cdots\!47\)\( p^{8} T^{36} + \)\(48\!\cdots\!12\)\( p^{9} T^{37} + \)\(55\!\cdots\!82\)\( p^{10} T^{38} + \)\(62\!\cdots\!26\)\( p^{11} T^{39} + 66072774293502495400 p^{12} T^{40} + 6705709304831925095 p^{13} T^{41} + 649416612847666472 p^{14} T^{42} + 59871941819544349 p^{15} T^{43} + 5247795953263833 p^{16} T^{44} + 435553186735964 p^{17} T^{45} + 34149669156977 p^{18} T^{46} + 2512861527533 p^{19} T^{47} + 172890668004 p^{20} T^{48} + 11000589745 p^{21} T^{49} + 643751982 p^{22} T^{50} + 33952394 p^{23} T^{51} + 1601380 p^{24} T^{52} + 64571 p^{25} T^{53} + 2204 p^{26} T^{54} + 55 p^{27} T^{55} + p^{28} T^{56} \)
67 \( 1 + 22 T + 1426 T^{2} + 28245 T^{3} + 1003298 T^{4} + 268842 p T^{5} + 462974570 T^{6} + 7587205235 T^{7} + 157273945264 T^{8} + 2368321422885 T^{9} + 41869165261207 T^{10} + 582759580785927 T^{11} + 9081694758284659 T^{12} + 117424989978841876 T^{13} + 1647603128546167146 T^{14} + 19873391353589209033 T^{15} + \)\(25\!\cdots\!64\)\( T^{16} + \)\(28\!\cdots\!19\)\( T^{17} + \)\(33\!\cdots\!34\)\( T^{18} + \)\(36\!\cdots\!21\)\( T^{19} + \)\(39\!\cdots\!64\)\( T^{20} + \)\(39\!\cdots\!12\)\( T^{21} + \)\(40\!\cdots\!71\)\( T^{22} + \)\(38\!\cdots\!58\)\( T^{23} + \)\(36\!\cdots\!69\)\( T^{24} + \)\(32\!\cdots\!56\)\( T^{25} + \)\(29\!\cdots\!28\)\( T^{26} + \)\(24\!\cdots\!95\)\( T^{27} + \)\(20\!\cdots\!17\)\( T^{28} + \)\(24\!\cdots\!95\)\( p T^{29} + \)\(29\!\cdots\!28\)\( p^{2} T^{30} + \)\(32\!\cdots\!56\)\( p^{3} T^{31} + \)\(36\!\cdots\!69\)\( p^{4} T^{32} + \)\(38\!\cdots\!58\)\( p^{5} T^{33} + \)\(40\!\cdots\!71\)\( p^{6} T^{34} + \)\(39\!\cdots\!12\)\( p^{7} T^{35} + \)\(39\!\cdots\!64\)\( p^{8} T^{36} + \)\(36\!\cdots\!21\)\( p^{9} T^{37} + \)\(33\!\cdots\!34\)\( p^{10} T^{38} + \)\(28\!\cdots\!19\)\( p^{11} T^{39} + \)\(25\!\cdots\!64\)\( p^{12} T^{40} + 19873391353589209033 p^{13} T^{41} + 1647603128546167146 p^{14} T^{42} + 117424989978841876 p^{15} T^{43} + 9081694758284659 p^{16} T^{44} + 582759580785927 p^{17} T^{45} + 41869165261207 p^{18} T^{46} + 2368321422885 p^{19} T^{47} + 157273945264 p^{20} T^{48} + 7587205235 p^{21} T^{49} + 462974570 p^{22} T^{50} + 268842 p^{24} T^{51} + 1003298 p^{24} T^{52} + 28245 p^{25} T^{53} + 1426 p^{26} T^{54} + 22 p^{27} T^{55} + p^{28} T^{56} \)
71 \( 1 - 47 T + 2031 T^{2} - 60971 T^{3} + 1658690 T^{4} - 38256012 T^{5} + 812688089 T^{6} - 15602570631 T^{7} + 280024618923 T^{8} - 65853731456 p T^{9} + 73807924571885 T^{10} - 1101297251537054 T^{11} + 15666448305524151 T^{12} - 212730177321146113 T^{13} + 2771281749084561974 T^{14} - 34687468353887850832 T^{15} + \)\(41\!\cdots\!75\)\( T^{16} - \)\(48\!\cdots\!45\)\( T^{17} + \)\(54\!\cdots\!50\)\( T^{18} - \)\(59\!\cdots\!11\)\( T^{19} + \)\(63\!\cdots\!02\)\( T^{20} - \)\(65\!\cdots\!10\)\( T^{21} + \)\(65\!\cdots\!14\)\( T^{22} - \)\(63\!\cdots\!21\)\( T^{23} + \)\(59\!\cdots\!53\)\( T^{24} - \)\(55\!\cdots\!08\)\( T^{25} + \)\(49\!\cdots\!89\)\( T^{26} - \)\(43\!\cdots\!27\)\( T^{27} + \)\(37\!\cdots\!46\)\( T^{28} - \)\(43\!\cdots\!27\)\( p T^{29} + \)\(49\!\cdots\!89\)\( p^{2} T^{30} - \)\(55\!\cdots\!08\)\( p^{3} T^{31} + \)\(59\!\cdots\!53\)\( p^{4} T^{32} - \)\(63\!\cdots\!21\)\( p^{5} T^{33} + \)\(65\!\cdots\!14\)\( p^{6} T^{34} - \)\(65\!\cdots\!10\)\( p^{7} T^{35} + \)\(63\!\cdots\!02\)\( p^{8} T^{36} - \)\(59\!\cdots\!11\)\( p^{9} T^{37} + \)\(54\!\cdots\!50\)\( p^{10} T^{38} - \)\(48\!\cdots\!45\)\( p^{11} T^{39} + \)\(41\!\cdots\!75\)\( p^{12} T^{40} - 34687468353887850832 p^{13} T^{41} + 2771281749084561974 p^{14} T^{42} - 212730177321146113 p^{15} T^{43} + 15666448305524151 p^{16} T^{44} - 1101297251537054 p^{17} T^{45} + 73807924571885 p^{18} T^{46} - 65853731456 p^{20} T^{47} + 280024618923 p^{20} T^{48} - 15602570631 p^{21} T^{49} + 812688089 p^{22} T^{50} - 38256012 p^{23} T^{51} + 1658690 p^{24} T^{52} - 60971 p^{25} T^{53} + 2031 p^{26} T^{54} - 47 p^{27} T^{55} + p^{28} T^{56} \)
73 \( 1 + 5 T + 1054 T^{2} + 5820 T^{3} + 548375 T^{4} + 3336426 T^{5} + 188880491 T^{6} + 1258402341 T^{7} + 48746055155 T^{8} + 351989994597 T^{9} + 10109022458523 T^{10} + 78028863236410 T^{11} + 1761765623439162 T^{12} + 14303698065172187 T^{13} + 265931213488003163 T^{14} + 2233074147703036602 T^{15} + 35492506757709960845 T^{16} + \)\(30\!\cdots\!43\)\( T^{17} + \)\(42\!\cdots\!44\)\( T^{18} + \)\(36\!\cdots\!90\)\( T^{19} + \)\(46\!\cdots\!72\)\( T^{20} + \)\(39\!\cdots\!79\)\( T^{21} + \)\(45\!\cdots\!58\)\( T^{22} + \)\(37\!\cdots\!95\)\( T^{23} + \)\(41\!\cdots\!58\)\( T^{24} + \)\(33\!\cdots\!15\)\( T^{25} + \)\(34\!\cdots\!79\)\( T^{26} + \)\(26\!\cdots\!98\)\( T^{27} + \)\(25\!\cdots\!16\)\( T^{28} + \)\(26\!\cdots\!98\)\( p T^{29} + \)\(34\!\cdots\!79\)\( p^{2} T^{30} + \)\(33\!\cdots\!15\)\( p^{3} T^{31} + \)\(41\!\cdots\!58\)\( p^{4} T^{32} + \)\(37\!\cdots\!95\)\( p^{5} T^{33} + \)\(45\!\cdots\!58\)\( p^{6} T^{34} + \)\(39\!\cdots\!79\)\( p^{7} T^{35} + \)\(46\!\cdots\!72\)\( p^{8} T^{36} + \)\(36\!\cdots\!90\)\( p^{9} T^{37} + \)\(42\!\cdots\!44\)\( p^{10} T^{38} + \)\(30\!\cdots\!43\)\( p^{11} T^{39} + 35492506757709960845 p^{12} T^{40} + 2233074147703036602 p^{13} T^{41} + 265931213488003163 p^{14} T^{42} + 14303698065172187 p^{15} T^{43} + 1761765623439162 p^{16} T^{44} + 78028863236410 p^{17} T^{45} + 10109022458523 p^{18} T^{46} + 351989994597 p^{19} T^{47} + 48746055155 p^{20} T^{48} + 1258402341 p^{21} T^{49} + 188880491 p^{22} T^{50} + 3336426 p^{23} T^{51} + 548375 p^{24} T^{52} + 5820 p^{25} T^{53} + 1054 p^{26} T^{54} + 5 p^{27} T^{55} + p^{28} T^{56} \)
79 \( 1 - 14 T + 1039 T^{2} - 13190 T^{3} + 541535 T^{4} - 6423276 T^{5} + 190994754 T^{6} - 2156689716 T^{7} + 51517193681 T^{8} - 559630597488 T^{9} + 11340873279667 T^{10} - 119138069366967 T^{11} + 2119146036748560 T^{12} - 21574946697824485 T^{13} + 344846879320116431 T^{14} - 3403809419545265048 T^{15} + 49733776514167681909 T^{16} - \)\(47\!\cdots\!52\)\( T^{17} + \)\(64\!\cdots\!28\)\( T^{18} - \)\(59\!\cdots\!29\)\( T^{19} + \)\(75\!\cdots\!52\)\( T^{20} - \)\(67\!\cdots\!18\)\( T^{21} + \)\(80\!\cdots\!21\)\( T^{22} - \)\(69\!\cdots\!05\)\( T^{23} + \)\(78\!\cdots\!70\)\( T^{24} - \)\(65\!\cdots\!07\)\( T^{25} + \)\(70\!\cdots\!44\)\( T^{26} - \)\(56\!\cdots\!41\)\( T^{27} + \)\(57\!\cdots\!32\)\( T^{28} - \)\(56\!\cdots\!41\)\( p T^{29} + \)\(70\!\cdots\!44\)\( p^{2} T^{30} - \)\(65\!\cdots\!07\)\( p^{3} T^{31} + \)\(78\!\cdots\!70\)\( p^{4} T^{32} - \)\(69\!\cdots\!05\)\( p^{5} T^{33} + \)\(80\!\cdots\!21\)\( p^{6} T^{34} - \)\(67\!\cdots\!18\)\( p^{7} T^{35} + \)\(75\!\cdots\!52\)\( p^{8} T^{36} - \)\(59\!\cdots\!29\)\( p^{9} T^{37} + \)\(64\!\cdots\!28\)\( p^{10} T^{38} - \)\(47\!\cdots\!52\)\( p^{11} T^{39} + 49733776514167681909 p^{12} T^{40} - 3403809419545265048 p^{13} T^{41} + 344846879320116431 p^{14} T^{42} - 21574946697824485 p^{15} T^{43} + 2119146036748560 p^{16} T^{44} - 119138069366967 p^{17} T^{45} + 11340873279667 p^{18} T^{46} - 559630597488 p^{19} T^{47} + 51517193681 p^{20} T^{48} - 2156689716 p^{21} T^{49} + 190994754 p^{22} T^{50} - 6423276 p^{23} T^{51} + 541535 p^{24} T^{52} - 13190 p^{25} T^{53} + 1039 p^{26} T^{54} - 14 p^{27} T^{55} + p^{28} T^{56} \)
83 \( 1 - 16 T + 1270 T^{2} - 17884 T^{3} + 783098 T^{4} - 9924777 T^{5} + 315773076 T^{6} - 3670442989 T^{7} + 94480978006 T^{8} - 1023825029733 T^{9} + 22527376316003 T^{10} - 230779735794302 T^{11} + 4480261541853138 T^{12} - 43889811104560766 T^{13} + 766668584773161044 T^{14} - 7245232216223691072 T^{15} + \)\(11\!\cdots\!14\)\( T^{16} - \)\(10\!\cdots\!42\)\( T^{17} + \)\(15\!\cdots\!65\)\( T^{18} - \)\(13\!\cdots\!81\)\( T^{19} + \)\(18\!\cdots\!09\)\( T^{20} - \)\(16\!\cdots\!14\)\( T^{21} + \)\(20\!\cdots\!07\)\( T^{22} - \)\(17\!\cdots\!01\)\( T^{23} + \)\(21\!\cdots\!91\)\( T^{24} - \)\(17\!\cdots\!72\)\( T^{25} + \)\(19\!\cdots\!63\)\( T^{26} - \)\(15\!\cdots\!23\)\( T^{27} + \)\(17\!\cdots\!11\)\( T^{28} - \)\(15\!\cdots\!23\)\( p T^{29} + \)\(19\!\cdots\!63\)\( p^{2} T^{30} - \)\(17\!\cdots\!72\)\( p^{3} T^{31} + \)\(21\!\cdots\!91\)\( p^{4} T^{32} - \)\(17\!\cdots\!01\)\( p^{5} T^{33} + \)\(20\!\cdots\!07\)\( p^{6} T^{34} - \)\(16\!\cdots\!14\)\( p^{7} T^{35} + \)\(18\!\cdots\!09\)\( p^{8} T^{36} - \)\(13\!\cdots\!81\)\( p^{9} T^{37} + \)\(15\!\cdots\!65\)\( p^{10} T^{38} - \)\(10\!\cdots\!42\)\( p^{11} T^{39} + \)\(11\!\cdots\!14\)\( p^{12} T^{40} - 7245232216223691072 p^{13} T^{41} + 766668584773161044 p^{14} T^{42} - 43889811104560766 p^{15} T^{43} + 4480261541853138 p^{16} T^{44} - 230779735794302 p^{17} T^{45} + 22527376316003 p^{18} T^{46} - 1023825029733 p^{19} T^{47} + 94480978006 p^{20} T^{48} - 3670442989 p^{21} T^{49} + 315773076 p^{22} T^{50} - 9924777 p^{23} T^{51} + 783098 p^{24} T^{52} - 17884 p^{25} T^{53} + 1270 p^{26} T^{54} - 16 p^{27} T^{55} + p^{28} T^{56} \)
89 \( 1 + 20 T + 1142 T^{2} + 20277 T^{3} + 653774 T^{4} + 10446346 T^{5} + 249612802 T^{6} + 3636813750 T^{7} + 71583784746 T^{8} + 962508417392 T^{9} + 16500690728831 T^{10} + 207022407921773 T^{11} + 3199214032998501 T^{12} + 37827223465955463 T^{13} + 539192892781833744 T^{14} + 6059254371626724813 T^{15} + 80928174865229435446 T^{16} + \)\(86\!\cdots\!43\)\( T^{17} + \)\(11\!\cdots\!57\)\( T^{18} + \)\(11\!\cdots\!16\)\( T^{19} + \)\(13\!\cdots\!70\)\( T^{20} + \)\(13\!\cdots\!93\)\( T^{21} + \)\(15\!\cdots\!19\)\( T^{22} + \)\(15\!\cdots\!47\)\( T^{23} + \)\(16\!\cdots\!59\)\( T^{24} + \)\(15\!\cdots\!83\)\( T^{25} + \)\(16\!\cdots\!73\)\( T^{26} + \)\(14\!\cdots\!44\)\( T^{27} + \)\(15\!\cdots\!26\)\( T^{28} + \)\(14\!\cdots\!44\)\( p T^{29} + \)\(16\!\cdots\!73\)\( p^{2} T^{30} + \)\(15\!\cdots\!83\)\( p^{3} T^{31} + \)\(16\!\cdots\!59\)\( p^{4} T^{32} + \)\(15\!\cdots\!47\)\( p^{5} T^{33} + \)\(15\!\cdots\!19\)\( p^{6} T^{34} + \)\(13\!\cdots\!93\)\( p^{7} T^{35} + \)\(13\!\cdots\!70\)\( p^{8} T^{36} + \)\(11\!\cdots\!16\)\( p^{9} T^{37} + \)\(11\!\cdots\!57\)\( p^{10} T^{38} + \)\(86\!\cdots\!43\)\( p^{11} T^{39} + 80928174865229435446 p^{12} T^{40} + 6059254371626724813 p^{13} T^{41} + 539192892781833744 p^{14} T^{42} + 37827223465955463 p^{15} T^{43} + 3199214032998501 p^{16} T^{44} + 207022407921773 p^{17} T^{45} + 16500690728831 p^{18} T^{46} + 962508417392 p^{19} T^{47} + 71583784746 p^{20} T^{48} + 3636813750 p^{21} T^{49} + 249612802 p^{22} T^{50} + 10446346 p^{23} T^{51} + 653774 p^{24} T^{52} + 20277 p^{25} T^{53} + 1142 p^{26} T^{54} + 20 p^{27} T^{55} + p^{28} T^{56} \)
97 \( 1 + 8 T + 1091 T^{2} + 7379 T^{3} + 611805 T^{4} + 3483031 T^{5} + 235264942 T^{6} + 1111588976 T^{7} + 69663403955 T^{8} + 266232049780 T^{9} + 16901137635670 T^{10} + 50097732154922 T^{11} + 3491496225842300 T^{12} + 7489866700189743 T^{13} + 630504183491778502 T^{14} + 862100284098764665 T^{15} + \)\(10\!\cdots\!49\)\( T^{16} + 65084365273644623077 T^{17} + \)\(14\!\cdots\!93\)\( T^{18} - \)\(20\!\cdots\!11\)\( T^{19} + \)\(19\!\cdots\!13\)\( T^{20} - \)\(11\!\cdots\!16\)\( T^{21} + \)\(24\!\cdots\!30\)\( T^{22} - \)\(24\!\cdots\!48\)\( T^{23} + \)\(27\!\cdots\!32\)\( T^{24} - \)\(35\!\cdots\!94\)\( T^{25} + \)\(29\!\cdots\!08\)\( T^{26} - \)\(41\!\cdots\!00\)\( T^{27} + \)\(29\!\cdots\!54\)\( T^{28} - \)\(41\!\cdots\!00\)\( p T^{29} + \)\(29\!\cdots\!08\)\( p^{2} T^{30} - \)\(35\!\cdots\!94\)\( p^{3} T^{31} + \)\(27\!\cdots\!32\)\( p^{4} T^{32} - \)\(24\!\cdots\!48\)\( p^{5} T^{33} + \)\(24\!\cdots\!30\)\( p^{6} T^{34} - \)\(11\!\cdots\!16\)\( p^{7} T^{35} + \)\(19\!\cdots\!13\)\( p^{8} T^{36} - \)\(20\!\cdots\!11\)\( p^{9} T^{37} + \)\(14\!\cdots\!93\)\( p^{10} T^{38} + 65084365273644623077 p^{11} T^{39} + \)\(10\!\cdots\!49\)\( p^{12} T^{40} + 862100284098764665 p^{13} T^{41} + 630504183491778502 p^{14} T^{42} + 7489866700189743 p^{15} T^{43} + 3491496225842300 p^{16} T^{44} + 50097732154922 p^{17} T^{45} + 16901137635670 p^{18} T^{46} + 266232049780 p^{19} T^{47} + 69663403955 p^{20} T^{48} + 1111588976 p^{21} T^{49} + 235264942 p^{22} T^{50} + 3483031 p^{23} T^{51} + 611805 p^{24} T^{52} + 7379 p^{25} T^{53} + 1091 p^{26} T^{54} + 8 p^{27} T^{55} + p^{28} T^{56} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.63770770193908761206466838298, −1.63554498916254410383738748312, −1.61903122415485114802024391472, −1.61824800684637868941273546609, −1.56947505273624099826573715630, −1.47463457272959050984728300566, −1.46436350369554941712900149375, −1.35105329859266142473500319013, −1.34035791832109570084665018600, −1.28609591188360719363498497339, −1.26719144578086705595364236449, −1.18489289159193189254183927522, −1.18369158068410716079231707868, −1.16508087988359572997944701976, −1.16455713802300826113658606586, −1.15155196753552591261035668284, −1.07220362467879091832477961900, −1.04159619961036427729963435093, −1.04124368194136027110486166907, −1.00622329637241671208116431983, −1.00057532283843358102770025283, −0.850138043771136430203071057853, −0.825386241406238619251554680177, −0.812381581751239376011451306375, −0.60147958183965052945995551813, 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, 0.60147958183965052945995551813, 0.812381581751239376011451306375, 0.825386241406238619251554680177, 0.850138043771136430203071057853, 1.00057532283843358102770025283, 1.00622329637241671208116431983, 1.04124368194136027110486166907, 1.04159619961036427729963435093, 1.07220362467879091832477961900, 1.15155196753552591261035668284, 1.16455713802300826113658606586, 1.16508087988359572997944701976, 1.18369158068410716079231707868, 1.18489289159193189254183927522, 1.26719144578086705595364236449, 1.28609591188360719363498497339, 1.34035791832109570084665018600, 1.35105329859266142473500319013, 1.46436350369554941712900149375, 1.47463457272959050984728300566, 1.56947505273624099826573715630, 1.61824800684637868941273546609, 1.61903122415485114802024391472, 1.63554498916254410383738748312, 1.63770770193908761206466838298

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

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