Properties

Degree 56
Conductor $ 2^{28} \cdot 3^{56} \cdot 5^{28} \cdot 7^{28} $
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  − 2·3-s + 7·4-s − 28·5-s − 4·7-s + 5·9-s − 14·12-s + 56·15-s + 21·16-s − 6·17-s − 6·19-s − 196·20-s + 8·21-s + 406·25-s + 2·27-s − 28·28-s − 12·31-s + 112·35-s + 35·36-s + 4·37-s + 18·41-s + 28·43-s − 140·45-s + 30·47-s − 42·48-s + 49-s + 12·51-s − 42·53-s + ⋯
L(s)  = 1  − 1.15·3-s + 7/2·4-s − 12.5·5-s − 1.51·7-s + 5/3·9-s − 4.04·12-s + 14.4·15-s + 21/4·16-s − 1.45·17-s − 1.37·19-s − 43.8·20-s + 1.74·21-s + 81.1·25-s + 0.384·27-s − 5.29·28-s − 2.15·31-s + 18.9·35-s + 35/6·36-s + 0.657·37-s + 2.81·41-s + 4.26·43-s − 20.8·45-s + 4.37·47-s − 6.06·48-s + 1/7·49-s + 1.68·51-s − 5.76·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{56} \cdot 5^{28} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{56} \cdot 5^{28} \cdot 7^{28}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{28} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(56\)
\( N \)  =  \(2^{28} \cdot 3^{56} \cdot 5^{28} \cdot 7^{28}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{630} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((56,\ 2^{28} \cdot 3^{56} \cdot 5^{28} \cdot 7^{28} ,\ ( \ : [1/2]^{28} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.101764\)
\(L(\frac12)\)  \(\approx\)  \(0.101764\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 56. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 55.
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{7} \)
3 \( 1 + 2 T - T^{2} - 14 T^{3} - 8 T^{4} + 8 p T^{5} + 14 p T^{6} - 4 p^{2} T^{7} - 13 p T^{8} + 2 p^{3} T^{9} + 17 p^{2} T^{10} + 74 p^{2} T^{11} + 127 p^{2} T^{12} - 56 p^{3} T^{13} - 302 p^{3} T^{14} - 56 p^{4} T^{15} + 127 p^{4} T^{16} + 74 p^{5} T^{17} + 17 p^{6} T^{18} + 2 p^{8} T^{19} - 13 p^{7} T^{20} - 4 p^{9} T^{21} + 14 p^{9} T^{22} + 8 p^{10} T^{23} - 8 p^{10} T^{24} - 14 p^{11} T^{25} - p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
5 \( ( 1 + T )^{28} \)
7 \( 1 + 4 T + 15 T^{2} - 20 T^{3} - 83 T^{4} - 480 T^{5} + 762 T^{6} + 792 T^{7} + 9690 T^{8} - 28752 T^{9} - 4278 T^{10} - 32976 p T^{11} + 500531 T^{12} - 578140 T^{13} + 4798425 T^{14} - 578140 p T^{15} + 500531 p^{2} T^{16} - 32976 p^{4} T^{17} - 4278 p^{4} T^{18} - 28752 p^{5} T^{19} + 9690 p^{6} T^{20} + 792 p^{7} T^{21} + 762 p^{8} T^{22} - 480 p^{9} T^{23} - 83 p^{10} T^{24} - 20 p^{11} T^{25} + 15 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \)
good11 \( 1 - 146 T^{2} + 10707 T^{4} - 528598 T^{6} + 19822535 T^{8} - 602979252 T^{10} + 15497516684 T^{12} - 345863239582 T^{14} + 621110297667 p T^{16} - 121126189453744 T^{18} + 1946721082636661 T^{20} - 28570482275825892 T^{22} + 384860978278850096 T^{24} - 4774578808865700478 T^{26} + 54657509597834027646 T^{28} - 4774578808865700478 p^{2} T^{30} + 384860978278850096 p^{4} T^{32} - 28570482275825892 p^{6} T^{34} + 1946721082636661 p^{8} T^{36} - 121126189453744 p^{10} T^{38} + 621110297667 p^{13} T^{40} - 345863239582 p^{14} T^{42} + 15497516684 p^{16} T^{44} - 602979252 p^{18} T^{46} + 19822535 p^{20} T^{48} - 528598 p^{22} T^{50} + 10707 p^{24} T^{52} - 146 p^{26} T^{54} + p^{28} T^{56} \)
13 \( 1 + 62 T^{2} + 1785 T^{4} + 816 T^{5} + 35514 T^{6} + 56898 T^{7} + 599418 T^{8} + 1847532 T^{9} + 9062991 T^{10} + 41356296 T^{11} + 132388313 T^{12} + 793167990 T^{13} + 2060169148 T^{14} + 13652454990 T^{15} + 33307311411 T^{16} + 15977149206 p T^{17} + 555687909389 T^{18} + 2916163155276 T^{19} + 56038055410 p^{2} T^{20} + 39540157480272 T^{21} + 152406218956887 T^{22} + 521846288616894 T^{23} + 2268353673552987 T^{24} + 526417740418266 p T^{25} + 32152049439722133 T^{26} + 90283322204514678 T^{27} + 432369083532394875 T^{28} + 90283322204514678 p T^{29} + 32152049439722133 p^{2} T^{30} + 526417740418266 p^{4} T^{31} + 2268353673552987 p^{4} T^{32} + 521846288616894 p^{5} T^{33} + 152406218956887 p^{6} T^{34} + 39540157480272 p^{7} T^{35} + 56038055410 p^{10} T^{36} + 2916163155276 p^{9} T^{37} + 555687909389 p^{10} T^{38} + 15977149206 p^{12} T^{39} + 33307311411 p^{12} T^{40} + 13652454990 p^{13} T^{41} + 2060169148 p^{14} T^{42} + 793167990 p^{15} T^{43} + 132388313 p^{16} T^{44} + 41356296 p^{17} T^{45} + 9062991 p^{18} T^{46} + 1847532 p^{19} T^{47} + 599418 p^{20} T^{48} + 56898 p^{21} T^{49} + 35514 p^{22} T^{50} + 816 p^{23} T^{51} + 1785 p^{24} T^{52} + 62 p^{26} T^{54} + p^{28} T^{56} \)
17 \( 1 + 6 T - 97 T^{2} - 630 T^{3} + 5277 T^{4} + 36858 T^{5} - 186344 T^{6} - 1424706 T^{7} + 4559246 T^{8} + 39502578 T^{9} - 67221138 T^{10} - 766311672 T^{11} + 20225366 T^{12} + 8829696066 T^{13} + 33337532584 T^{14} + 18591684990 T^{15} - 1145783182632 T^{16} - 3374338338990 T^{17} + 24153219006664 T^{18} + 81247386203520 T^{19} - 334271156293354 T^{20} - 1058045461059420 T^{21} + 1988935568054286 T^{22} + 5916426589874418 T^{23} + 47562877849271798 T^{24} + 48546937323325950 T^{25} - 113442747286484761 p T^{26} - 703240226536052964 T^{27} + 40250052208883640585 T^{28} - 703240226536052964 p T^{29} - 113442747286484761 p^{3} T^{30} + 48546937323325950 p^{3} T^{31} + 47562877849271798 p^{4} T^{32} + 5916426589874418 p^{5} T^{33} + 1988935568054286 p^{6} T^{34} - 1058045461059420 p^{7} T^{35} - 334271156293354 p^{8} T^{36} + 81247386203520 p^{9} T^{37} + 24153219006664 p^{10} T^{38} - 3374338338990 p^{11} T^{39} - 1145783182632 p^{12} T^{40} + 18591684990 p^{13} T^{41} + 33337532584 p^{14} T^{42} + 8829696066 p^{15} T^{43} + 20225366 p^{16} T^{44} - 766311672 p^{17} T^{45} - 67221138 p^{18} T^{46} + 39502578 p^{19} T^{47} + 4559246 p^{20} T^{48} - 1424706 p^{21} T^{49} - 186344 p^{22} T^{50} + 36858 p^{23} T^{51} + 5277 p^{24} T^{52} - 630 p^{25} T^{53} - 97 p^{26} T^{54} + 6 p^{27} T^{55} + p^{28} T^{56} \)
19 \( 1 + 6 T + 116 T^{2} + 624 T^{3} + 5919 T^{4} + 29460 T^{5} + 180552 T^{6} + 821760 T^{7} + 3759060 T^{8} + 14131998 T^{9} + 55148643 T^{10} + 122532066 T^{11} + 22491389 p T^{12} - 929311536 T^{13} - 6799542656 T^{14} - 73261416642 T^{15} - 401743016721 T^{16} - 2378150107674 T^{17} - 10635972626101 T^{18} - 53197400701974 T^{19} - 184962505169534 T^{20} - 704185314010074 T^{21} - 1718896528858959 T^{22} - 1225738150477542 T^{23} + 839641895482965 p T^{24} + 175395169019143680 T^{25} + 1098216365855368383 T^{26} + 5043026458418277684 T^{27} + 26820817009067926959 T^{28} + 5043026458418277684 p T^{29} + 1098216365855368383 p^{2} T^{30} + 175395169019143680 p^{3} T^{31} + 839641895482965 p^{5} T^{32} - 1225738150477542 p^{5} T^{33} - 1718896528858959 p^{6} T^{34} - 704185314010074 p^{7} T^{35} - 184962505169534 p^{8} T^{36} - 53197400701974 p^{9} T^{37} - 10635972626101 p^{10} T^{38} - 2378150107674 p^{11} T^{39} - 401743016721 p^{12} T^{40} - 73261416642 p^{13} T^{41} - 6799542656 p^{14} T^{42} - 929311536 p^{15} T^{43} + 22491389 p^{17} T^{44} + 122532066 p^{17} T^{45} + 55148643 p^{18} T^{46} + 14131998 p^{19} T^{47} + 3759060 p^{20} T^{48} + 821760 p^{21} T^{49} + 180552 p^{22} T^{50} + 29460 p^{23} T^{51} + 5919 p^{24} T^{52} + 624 p^{25} T^{53} + 116 p^{26} T^{54} + 6 p^{27} T^{55} + p^{28} T^{56} \)
23 \( 1 - 260 T^{2} + 35634 T^{4} - 3370502 T^{6} + 245135125 T^{8} - 14530274316 T^{10} + 728149117423 T^{12} - 1376010530128 p T^{14} + 1216326199759362 T^{16} - 41983701453102986 T^{18} + 1318739469207686821 T^{20} - 38127060990265255632 T^{22} + \)\(10\!\cdots\!67\)\( T^{24} - \)\(25\!\cdots\!98\)\( T^{26} + \)\(61\!\cdots\!15\)\( T^{28} - \)\(25\!\cdots\!98\)\( p^{2} T^{30} + \)\(10\!\cdots\!67\)\( p^{4} T^{32} - 38127060990265255632 p^{6} T^{34} + 1318739469207686821 p^{8} T^{36} - 41983701453102986 p^{10} T^{38} + 1216326199759362 p^{12} T^{40} - 1376010530128 p^{15} T^{42} + 728149117423 p^{16} T^{44} - 14530274316 p^{18} T^{46} + 245135125 p^{20} T^{48} - 3370502 p^{22} T^{50} + 35634 p^{24} T^{52} - 260 p^{26} T^{54} + p^{28} T^{56} \)
29 \( 1 + 115 T^{2} + 4860 T^{4} + 408 T^{5} + 100453 T^{6} + 244830 T^{7} + 3189871 T^{8} + 24742530 T^{9} + 190913187 T^{10} + 1182780864 T^{11} + 7045266874 T^{12} + 38937847956 T^{13} + 207118702579 T^{14} + 1245775431372 T^{15} + 7635551157435 T^{16} + 45020705507136 T^{17} + 286429856796202 T^{18} + 1763377551940788 T^{19} + 9554530260534577 T^{20} + 62362858949556516 T^{21} + 293902722668034729 T^{22} + 1776508128266645088 T^{23} + 8593852963902863922 T^{24} + 52213937162623351092 T^{25} + \)\(26\!\cdots\!29\)\( T^{26} + \)\(18\!\cdots\!02\)\( T^{27} + \)\(83\!\cdots\!93\)\( T^{28} + \)\(18\!\cdots\!02\)\( p T^{29} + \)\(26\!\cdots\!29\)\( p^{2} T^{30} + 52213937162623351092 p^{3} T^{31} + 8593852963902863922 p^{4} T^{32} + 1776508128266645088 p^{5} T^{33} + 293902722668034729 p^{6} T^{34} + 62362858949556516 p^{7} T^{35} + 9554530260534577 p^{8} T^{36} + 1763377551940788 p^{9} T^{37} + 286429856796202 p^{10} T^{38} + 45020705507136 p^{11} T^{39} + 7635551157435 p^{12} T^{40} + 1245775431372 p^{13} T^{41} + 207118702579 p^{14} T^{42} + 38937847956 p^{15} T^{43} + 7045266874 p^{16} T^{44} + 1182780864 p^{17} T^{45} + 190913187 p^{18} T^{46} + 24742530 p^{19} T^{47} + 3189871 p^{20} T^{48} + 244830 p^{21} T^{49} + 100453 p^{22} T^{50} + 408 p^{23} T^{51} + 4860 p^{24} T^{52} + 115 p^{26} T^{54} + p^{28} T^{56} \)
31 \( 1 + 12 T + 221 T^{2} + 2076 T^{3} + 20388 T^{4} + 148938 T^{5} + 963693 T^{6} + 5114166 T^{7} + 20633694 T^{8} + 51987480 T^{9} - 897345 T^{10} - 865422270 T^{11} + 2476815536 T^{12} + 87311517228 T^{13} + 1006904921248 T^{14} + 7936635157002 T^{15} + 44391336849525 T^{16} + 202491003829902 T^{17} + 656013026619083 T^{18} + 1159611372502170 T^{19} + 4616809484191717 T^{20} + 60999563094757278 T^{21} + 797051508721993167 T^{22} + 7160799610858377012 T^{23} + 42108701006574207741 T^{24} + \)\(21\!\cdots\!44\)\( T^{25} + \)\(71\!\cdots\!89\)\( T^{26} + \)\(16\!\cdots\!10\)\( T^{27} + \)\(45\!\cdots\!09\)\( T^{28} + \)\(16\!\cdots\!10\)\( p T^{29} + \)\(71\!\cdots\!89\)\( p^{2} T^{30} + \)\(21\!\cdots\!44\)\( p^{3} T^{31} + 42108701006574207741 p^{4} T^{32} + 7160799610858377012 p^{5} T^{33} + 797051508721993167 p^{6} T^{34} + 60999563094757278 p^{7} T^{35} + 4616809484191717 p^{8} T^{36} + 1159611372502170 p^{9} T^{37} + 656013026619083 p^{10} T^{38} + 202491003829902 p^{11} T^{39} + 44391336849525 p^{12} T^{40} + 7936635157002 p^{13} T^{41} + 1006904921248 p^{14} T^{42} + 87311517228 p^{15} T^{43} + 2476815536 p^{16} T^{44} - 865422270 p^{17} T^{45} - 897345 p^{18} T^{46} + 51987480 p^{19} T^{47} + 20633694 p^{20} T^{48} + 5114166 p^{21} T^{49} + 963693 p^{22} T^{50} + 148938 p^{23} T^{51} + 20388 p^{24} T^{52} + 2076 p^{25} T^{53} + 221 p^{26} T^{54} + 12 p^{27} T^{55} + p^{28} T^{56} \)
37 \( 1 - 4 T - 302 T^{2} + 1112 T^{3} + 46798 T^{4} - 150034 T^{5} - 4939016 T^{6} + 12407810 T^{7} + 399537817 T^{8} - 623322072 T^{9} - 26467499025 T^{10} + 9331384374 T^{11} + 1492921715686 T^{12} + 1587818024948 T^{13} - 73015380458120 T^{14} - 192598790951248 T^{15} + 3089392946569390 T^{16} + 13612251784293794 T^{17} - 110054727140300543 T^{18} - 731975891912536390 T^{19} + 3054155677354167247 T^{20} + 31940043706835689104 T^{21} - 48518727637216490757 T^{22} - \)\(11\!\cdots\!90\)\( T^{23} - \)\(92\!\cdots\!34\)\( T^{24} + \)\(30\!\cdots\!36\)\( T^{25} + \)\(11\!\cdots\!27\)\( T^{26} - \)\(41\!\cdots\!16\)\( T^{27} - \)\(53\!\cdots\!85\)\( T^{28} - \)\(41\!\cdots\!16\)\( p T^{29} + \)\(11\!\cdots\!27\)\( p^{2} T^{30} + \)\(30\!\cdots\!36\)\( p^{3} T^{31} - \)\(92\!\cdots\!34\)\( p^{4} T^{32} - \)\(11\!\cdots\!90\)\( p^{5} T^{33} - 48518727637216490757 p^{6} T^{34} + 31940043706835689104 p^{7} T^{35} + 3054155677354167247 p^{8} T^{36} - 731975891912536390 p^{9} T^{37} - 110054727140300543 p^{10} T^{38} + 13612251784293794 p^{11} T^{39} + 3089392946569390 p^{12} T^{40} - 192598790951248 p^{13} T^{41} - 73015380458120 p^{14} T^{42} + 1587818024948 p^{15} T^{43} + 1492921715686 p^{16} T^{44} + 9331384374 p^{17} T^{45} - 26467499025 p^{18} T^{46} - 623322072 p^{19} T^{47} + 399537817 p^{20} T^{48} + 12407810 p^{21} T^{49} - 4939016 p^{22} T^{50} - 150034 p^{23} T^{51} + 46798 p^{24} T^{52} + 1112 p^{25} T^{53} - 302 p^{26} T^{54} - 4 p^{27} T^{55} + p^{28} T^{56} \)
41 \( 1 - 18 T - 127 T^{2} + 3946 T^{3} + 4386 T^{4} - 446428 T^{5} + 285437 T^{6} + 35475234 T^{7} - 41749184 T^{8} - 2237187844 T^{9} + 2355769080 T^{10} + 119090744674 T^{11} - 58281265006 T^{12} - 5564533310424 T^{13} - 1070895940409 T^{14} + 231905776430448 T^{15} + 92741694059067 T^{16} - 8350028733036504 T^{17} + 5884011799799757 T^{18} + 233992443337315182 T^{19} - 1198841082977414292 T^{20} - 3677999312322320606 T^{21} + \)\(10\!\cdots\!98\)\( T^{22} - 57224654246037609988 T^{23} - \)\(63\!\cdots\!62\)\( T^{24} + \)\(56\!\cdots\!10\)\( T^{25} + \)\(32\!\cdots\!68\)\( T^{26} - \)\(11\!\cdots\!62\)\( T^{27} - \)\(13\!\cdots\!59\)\( T^{28} - \)\(11\!\cdots\!62\)\( p T^{29} + \)\(32\!\cdots\!68\)\( p^{2} T^{30} + \)\(56\!\cdots\!10\)\( p^{3} T^{31} - \)\(63\!\cdots\!62\)\( p^{4} T^{32} - 57224654246037609988 p^{5} T^{33} + \)\(10\!\cdots\!98\)\( p^{6} T^{34} - 3677999312322320606 p^{7} T^{35} - 1198841082977414292 p^{8} T^{36} + 233992443337315182 p^{9} T^{37} + 5884011799799757 p^{10} T^{38} - 8350028733036504 p^{11} T^{39} + 92741694059067 p^{12} T^{40} + 231905776430448 p^{13} T^{41} - 1070895940409 p^{14} T^{42} - 5564533310424 p^{15} T^{43} - 58281265006 p^{16} T^{44} + 119090744674 p^{17} T^{45} + 2355769080 p^{18} T^{46} - 2237187844 p^{19} T^{47} - 41749184 p^{20} T^{48} + 35475234 p^{21} T^{49} + 285437 p^{22} T^{50} - 446428 p^{23} T^{51} + 4386 p^{24} T^{52} + 3946 p^{25} T^{53} - 127 p^{26} T^{54} - 18 p^{27} T^{55} + p^{28} T^{56} \)
43 \( 1 - 28 T + 100 T^{2} + 4536 T^{3} - 1110 p T^{4} - 222246 T^{5} + 5172562 T^{6} - 1791880 T^{7} - 308099978 T^{8} + 535564700 T^{9} + 14499172843 T^{10} - 18943577968 T^{11} - 702886675883 T^{12} + 166071238640 T^{13} + 32956244426155 T^{14} + 22967913473182 T^{15} - 1275057295326325 T^{16} - 3141683916617480 T^{17} + 38115846797059854 T^{18} + 289536562331315880 T^{19} - 1033939042989136512 T^{20} - 18752815905470720858 T^{21} + 43051667193441183401 T^{22} + \)\(79\!\cdots\!70\)\( T^{23} - \)\(18\!\cdots\!51\)\( T^{24} - \)\(20\!\cdots\!20\)\( T^{25} + \)\(42\!\cdots\!34\)\( T^{26} + \)\(24\!\cdots\!32\)\( T^{27} - \)\(54\!\cdots\!30\)\( T^{28} + \)\(24\!\cdots\!32\)\( p T^{29} + \)\(42\!\cdots\!34\)\( p^{2} T^{30} - \)\(20\!\cdots\!20\)\( p^{3} T^{31} - \)\(18\!\cdots\!51\)\( p^{4} T^{32} + \)\(79\!\cdots\!70\)\( p^{5} T^{33} + 43051667193441183401 p^{6} T^{34} - 18752815905470720858 p^{7} T^{35} - 1033939042989136512 p^{8} T^{36} + 289536562331315880 p^{9} T^{37} + 38115846797059854 p^{10} T^{38} - 3141683916617480 p^{11} T^{39} - 1275057295326325 p^{12} T^{40} + 22967913473182 p^{13} T^{41} + 32956244426155 p^{14} T^{42} + 166071238640 p^{15} T^{43} - 702886675883 p^{16} T^{44} - 18943577968 p^{17} T^{45} + 14499172843 p^{18} T^{46} + 535564700 p^{19} T^{47} - 308099978 p^{20} T^{48} - 1791880 p^{21} T^{49} + 5172562 p^{22} T^{50} - 222246 p^{23} T^{51} - 1110 p^{25} T^{52} + 4536 p^{25} T^{53} + 100 p^{26} T^{54} - 28 p^{27} T^{55} + p^{28} T^{56} \)
47 \( 1 - 30 T + 56 T^{2} + 6200 T^{3} - 32700 T^{4} - 809402 T^{5} + 4546118 T^{6} + 93534288 T^{7} - 463402766 T^{8} - 9357359012 T^{9} + 38331720267 T^{10} + 799662243800 T^{11} - 2457927457711 T^{12} - 60508568220210 T^{13} + 120378549748039 T^{14} + 4094868422113128 T^{15} - 3967871548136811 T^{16} - 245932477017716928 T^{17} - 12167541373919760 T^{18} + 13113115232680035630 T^{19} + 15222582691868639748 T^{20} - \)\(61\!\cdots\!16\)\( T^{21} - \)\(15\!\cdots\!89\)\( T^{22} + \)\(25\!\cdots\!74\)\( T^{23} + \)\(10\!\cdots\!39\)\( T^{24} - \)\(79\!\cdots\!98\)\( T^{25} - \)\(61\!\cdots\!98\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(30\!\cdots\!06\)\( T^{28} + \)\(13\!\cdots\!44\)\( p T^{29} - \)\(61\!\cdots\!98\)\( p^{2} T^{30} - \)\(79\!\cdots\!98\)\( p^{3} T^{31} + \)\(10\!\cdots\!39\)\( p^{4} T^{32} + \)\(25\!\cdots\!74\)\( p^{5} T^{33} - \)\(15\!\cdots\!89\)\( p^{6} T^{34} - \)\(61\!\cdots\!16\)\( p^{7} T^{35} + 15222582691868639748 p^{8} T^{36} + 13113115232680035630 p^{9} T^{37} - 12167541373919760 p^{10} T^{38} - 245932477017716928 p^{11} T^{39} - 3967871548136811 p^{12} T^{40} + 4094868422113128 p^{13} T^{41} + 120378549748039 p^{14} T^{42} - 60508568220210 p^{15} T^{43} - 2457927457711 p^{16} T^{44} + 799662243800 p^{17} T^{45} + 38331720267 p^{18} T^{46} - 9357359012 p^{19} T^{47} - 463402766 p^{20} T^{48} + 93534288 p^{21} T^{49} + 4546118 p^{22} T^{50} - 809402 p^{23} T^{51} - 32700 p^{24} T^{52} + 6200 p^{25} T^{53} + 56 p^{26} T^{54} - 30 p^{27} T^{55} + p^{28} T^{56} \)
53 \( 1 + 42 T + 1336 T^{2} + 31416 T^{3} + 632670 T^{4} + 10995474 T^{5} + 172221852 T^{6} + 2450682306 T^{7} + 32307368187 T^{8} + 397025277738 T^{9} + 4599376651089 T^{10} + 50467223286540 T^{11} + 528440783219468 T^{12} + 5300355117277080 T^{13} + 51207277881285734 T^{14} + 477981337040342028 T^{15} + 4329183690514443000 T^{16} + 38139763661023761654 T^{17} + \)\(32\!\cdots\!79\)\( T^{18} + 52018797729608330238 p T^{19} + \)\(22\!\cdots\!23\)\( T^{20} + \)\(18\!\cdots\!82\)\( T^{21} + \)\(14\!\cdots\!75\)\( T^{22} + \)\(11\!\cdots\!04\)\( T^{23} + \)\(88\!\cdots\!62\)\( T^{24} + \)\(67\!\cdots\!78\)\( T^{25} + \)\(50\!\cdots\!09\)\( T^{26} + \)\(37\!\cdots\!08\)\( T^{27} + \)\(27\!\cdots\!43\)\( T^{28} + \)\(37\!\cdots\!08\)\( p T^{29} + \)\(50\!\cdots\!09\)\( p^{2} T^{30} + \)\(67\!\cdots\!78\)\( p^{3} T^{31} + \)\(88\!\cdots\!62\)\( p^{4} T^{32} + \)\(11\!\cdots\!04\)\( p^{5} T^{33} + \)\(14\!\cdots\!75\)\( p^{6} T^{34} + \)\(18\!\cdots\!82\)\( p^{7} T^{35} + \)\(22\!\cdots\!23\)\( p^{8} T^{36} + 52018797729608330238 p^{10} T^{37} + \)\(32\!\cdots\!79\)\( p^{10} T^{38} + 38139763661023761654 p^{11} T^{39} + 4329183690514443000 p^{12} T^{40} + 477981337040342028 p^{13} T^{41} + 51207277881285734 p^{14} T^{42} + 5300355117277080 p^{15} T^{43} + 528440783219468 p^{16} T^{44} + 50467223286540 p^{17} T^{45} + 4599376651089 p^{18} T^{46} + 397025277738 p^{19} T^{47} + 32307368187 p^{20} T^{48} + 2450682306 p^{21} T^{49} + 172221852 p^{22} T^{50} + 10995474 p^{23} T^{51} + 632670 p^{24} T^{52} + 31416 p^{25} T^{53} + 1336 p^{26} T^{54} + 42 p^{27} T^{55} + p^{28} T^{56} \)
59 \( 1 + 24 T - 151 T^{2} - 6660 T^{3} + 20007 T^{4} + 1115010 T^{5} - 2976098 T^{6} - 131076516 T^{7} + 456205208 T^{8} + 11449338984 T^{9} - 59279555616 T^{10} - 774103844952 T^{11} + 6056563112564 T^{12} + 42453053701302 T^{13} - 503655611915678 T^{14} - 2137383417525786 T^{15} + 36463789409648022 T^{16} + 116485191425439162 T^{17} - 2478076612224875126 T^{18} - 6515841827265772260 T^{19} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(27\!\cdots\!96\)\( T^{21} - \)\(10\!\cdots\!68\)\( T^{22} - \)\(47\!\cdots\!24\)\( T^{23} + \)\(59\!\cdots\!28\)\( T^{24} - \)\(23\!\cdots\!82\)\( T^{25} - \)\(32\!\cdots\!73\)\( T^{26} + \)\(11\!\cdots\!74\)\( T^{27} + \)\(18\!\cdots\!09\)\( T^{28} + \)\(11\!\cdots\!74\)\( p T^{29} - \)\(32\!\cdots\!73\)\( p^{2} T^{30} - \)\(23\!\cdots\!82\)\( p^{3} T^{31} + \)\(59\!\cdots\!28\)\( p^{4} T^{32} - \)\(47\!\cdots\!24\)\( p^{5} T^{33} - \)\(10\!\cdots\!68\)\( p^{6} T^{34} + \)\(27\!\cdots\!96\)\( p^{7} T^{35} + \)\(16\!\cdots\!76\)\( p^{8} T^{36} - 6515841827265772260 p^{9} T^{37} - 2478076612224875126 p^{10} T^{38} + 116485191425439162 p^{11} T^{39} + 36463789409648022 p^{12} T^{40} - 2137383417525786 p^{13} T^{41} - 503655611915678 p^{14} T^{42} + 42453053701302 p^{15} T^{43} + 6056563112564 p^{16} T^{44} - 774103844952 p^{17} T^{45} - 59279555616 p^{18} T^{46} + 11449338984 p^{19} T^{47} + 456205208 p^{20} T^{48} - 131076516 p^{21} T^{49} - 2976098 p^{22} T^{50} + 1115010 p^{23} T^{51} + 20007 p^{24} T^{52} - 6660 p^{25} T^{53} - 151 p^{26} T^{54} + 24 p^{27} T^{55} + p^{28} T^{56} \)
61 \( 1 - 24 T + 608 T^{2} - 9984 T^{3} + 153048 T^{4} - 1918908 T^{5} + 21958008 T^{6} - 220568016 T^{7} + 2014147668 T^{8} - 16624883328 T^{9} + 126270001986 T^{10} - 899621097102 T^{11} + 6401563616444 T^{12} - 47645556745344 T^{13} + 406486967121445 T^{14} - 3607328811960768 T^{15} + 32800646157766560 T^{16} - 264966896882603196 T^{17} + 1884485208208523324 T^{18} - 9930447731131451040 T^{19} + 22293691459218036466 T^{20} + \)\(33\!\cdots\!68\)\( T^{21} - \)\(61\!\cdots\!18\)\( T^{22} + \)\(66\!\cdots\!20\)\( T^{23} - \)\(55\!\cdots\!46\)\( T^{24} + \)\(40\!\cdots\!34\)\( T^{25} - \)\(24\!\cdots\!88\)\( T^{26} + \)\(15\!\cdots\!28\)\( T^{27} - \)\(10\!\cdots\!76\)\( T^{28} + \)\(15\!\cdots\!28\)\( p T^{29} - \)\(24\!\cdots\!88\)\( p^{2} T^{30} + \)\(40\!\cdots\!34\)\( p^{3} T^{31} - \)\(55\!\cdots\!46\)\( p^{4} T^{32} + \)\(66\!\cdots\!20\)\( p^{5} T^{33} - \)\(61\!\cdots\!18\)\( p^{6} T^{34} + \)\(33\!\cdots\!68\)\( p^{7} T^{35} + 22293691459218036466 p^{8} T^{36} - 9930447731131451040 p^{9} T^{37} + 1884485208208523324 p^{10} T^{38} - 264966896882603196 p^{11} T^{39} + 32800646157766560 p^{12} T^{40} - 3607328811960768 p^{13} T^{41} + 406486967121445 p^{14} T^{42} - 47645556745344 p^{15} T^{43} + 6401563616444 p^{16} T^{44} - 899621097102 p^{17} T^{45} + 126270001986 p^{18} T^{46} - 16624883328 p^{19} T^{47} + 2014147668 p^{20} T^{48} - 220568016 p^{21} T^{49} + 21958008 p^{22} T^{50} - 1918908 p^{23} T^{51} + 153048 p^{24} T^{52} - 9984 p^{25} T^{53} + 608 p^{26} T^{54} - 24 p^{27} T^{55} + p^{28} T^{56} \)
67 \( 1 + 40 T + 307 T^{2} - 7136 T^{3} - 91847 T^{4} + 944590 T^{5} + 13704516 T^{6} - 127268808 T^{7} - 1536737769 T^{8} + 16227988928 T^{9} + 138846728024 T^{10} - 1779944551276 T^{11} - 9543081027150 T^{12} + 168719151784854 T^{13} + 429782773984731 T^{14} - 13984092686465918 T^{15} - 1090349039703053 T^{16} + 15238365431135554 p T^{17} - 2277847032421371066 T^{18} - 65959440039823715502 T^{19} + \)\(32\!\cdots\!97\)\( T^{20} + \)\(38\!\cdots\!28\)\( T^{21} - \)\(32\!\cdots\!10\)\( T^{22} - \)\(20\!\cdots\!18\)\( T^{23} + \)\(27\!\cdots\!72\)\( T^{24} + \)\(85\!\cdots\!88\)\( T^{25} - \)\(20\!\cdots\!53\)\( T^{26} - \)\(19\!\cdots\!64\)\( T^{27} + \)\(14\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!64\)\( p T^{29} - \)\(20\!\cdots\!53\)\( p^{2} T^{30} + \)\(85\!\cdots\!88\)\( p^{3} T^{31} + \)\(27\!\cdots\!72\)\( p^{4} T^{32} - \)\(20\!\cdots\!18\)\( p^{5} T^{33} - \)\(32\!\cdots\!10\)\( p^{6} T^{34} + \)\(38\!\cdots\!28\)\( p^{7} T^{35} + \)\(32\!\cdots\!97\)\( p^{8} T^{36} - 65959440039823715502 p^{9} T^{37} - 2277847032421371066 p^{10} T^{38} + 15238365431135554 p^{12} T^{39} - 1090349039703053 p^{12} T^{40} - 13984092686465918 p^{13} T^{41} + 429782773984731 p^{14} T^{42} + 168719151784854 p^{15} T^{43} - 9543081027150 p^{16} T^{44} - 1779944551276 p^{17} T^{45} + 138846728024 p^{18} T^{46} + 16227988928 p^{19} T^{47} - 1536737769 p^{20} T^{48} - 127268808 p^{21} T^{49} + 13704516 p^{22} T^{50} + 944590 p^{23} T^{51} - 91847 p^{24} T^{52} - 7136 p^{25} T^{53} + 307 p^{26} T^{54} + 40 p^{27} T^{55} + p^{28} T^{56} \)
71 \( 1 - 746 T^{2} + 293199 T^{4} - 80065146 T^{6} + 16991154681 T^{8} - 2978367955560 T^{10} + 448118639239358 T^{12} - 59399308356015598 T^{14} + 7064428682205812304 T^{16} - \)\(76\!\cdots\!86\)\( T^{18} + \)\(75\!\cdots\!38\)\( T^{20} - \)\(69\!\cdots\!40\)\( T^{22} + \)\(58\!\cdots\!03\)\( T^{24} - \)\(46\!\cdots\!92\)\( T^{26} + \)\(34\!\cdots\!44\)\( T^{28} - \)\(46\!\cdots\!92\)\( p^{2} T^{30} + \)\(58\!\cdots\!03\)\( p^{4} T^{32} - \)\(69\!\cdots\!40\)\( p^{6} T^{34} + \)\(75\!\cdots\!38\)\( p^{8} T^{36} - \)\(76\!\cdots\!86\)\( p^{10} T^{38} + 7064428682205812304 p^{12} T^{40} - 59399308356015598 p^{14} T^{42} + 448118639239358 p^{16} T^{44} - 2978367955560 p^{18} T^{46} + 16991154681 p^{20} T^{48} - 80065146 p^{22} T^{50} + 293199 p^{24} T^{52} - 746 p^{26} T^{54} + p^{28} T^{56} \)
73 \( 1 - 6 T + 332 T^{2} - 1920 T^{3} + 51237 T^{4} - 306318 T^{5} + 4529912 T^{6} - 25490076 T^{7} + 186789670 T^{8} - 304362900 T^{9} - 8214433419 T^{10} + 205883092314 T^{11} - 2117732454869 T^{12} + 28962680672154 T^{13} - 183927507035344 T^{14} + 1985470200505932 T^{15} - 6374331176973465 T^{16} + 38523888468736386 T^{17} + 524014905986659175 T^{18} - 7366946689365176688 T^{19} + 94052998955089014712 T^{20} - \)\(91\!\cdots\!68\)\( T^{21} + \)\(61\!\cdots\!65\)\( T^{22} - \)\(43\!\cdots\!42\)\( T^{23} + \)\(75\!\cdots\!63\)\( T^{24} + \)\(12\!\cdots\!48\)\( T^{25} - \)\(24\!\cdots\!73\)\( T^{26} + \)\(37\!\cdots\!16\)\( T^{27} - \)\(27\!\cdots\!77\)\( T^{28} + \)\(37\!\cdots\!16\)\( p T^{29} - \)\(24\!\cdots\!73\)\( p^{2} T^{30} + \)\(12\!\cdots\!48\)\( p^{3} T^{31} + \)\(75\!\cdots\!63\)\( p^{4} T^{32} - \)\(43\!\cdots\!42\)\( p^{5} T^{33} + \)\(61\!\cdots\!65\)\( p^{6} T^{34} - \)\(91\!\cdots\!68\)\( p^{7} T^{35} + 94052998955089014712 p^{8} T^{36} - 7366946689365176688 p^{9} T^{37} + 524014905986659175 p^{10} T^{38} + 38523888468736386 p^{11} T^{39} - 6374331176973465 p^{12} T^{40} + 1985470200505932 p^{13} T^{41} - 183927507035344 p^{14} T^{42} + 28962680672154 p^{15} T^{43} - 2117732454869 p^{16} T^{44} + 205883092314 p^{17} T^{45} - 8214433419 p^{18} T^{46} - 304362900 p^{19} T^{47} + 186789670 p^{20} T^{48} - 25490076 p^{21} T^{49} + 4529912 p^{22} T^{50} - 306318 p^{23} T^{51} + 51237 p^{24} T^{52} - 1920 p^{25} T^{53} + 332 p^{26} T^{54} - 6 p^{27} T^{55} + p^{28} T^{56} \)
79 \( 1 - 2 T - 749 T^{2} - 434 T^{3} + 299869 T^{4} + 819796 T^{5} - 81065156 T^{6} - 390615932 T^{7} + 16236563839 T^{8} + 113993397918 T^{9} - 2501999072505 T^{10} - 24173099162640 T^{11} + 298367199924703 T^{12} + 3993010401149020 T^{13} - 26458651503046229 T^{14} - 532836183078076922 T^{15} + 1438898467594141798 T^{16} + 58590074891348303074 T^{17} + 22059680972858848912 T^{18} - \)\(53\!\cdots\!58\)\( T^{19} - \)\(17\!\cdots\!98\)\( T^{20} + \)\(40\!\cdots\!00\)\( T^{21} + \)\(25\!\cdots\!93\)\( T^{22} - \)\(25\!\cdots\!16\)\( T^{23} - \)\(27\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!78\)\( T^{25} + \)\(24\!\cdots\!64\)\( T^{26} - \)\(33\!\cdots\!44\)\( T^{27} - \)\(20\!\cdots\!79\)\( T^{28} - \)\(33\!\cdots\!44\)\( p T^{29} + \)\(24\!\cdots\!64\)\( p^{2} T^{30} + \)\(12\!\cdots\!78\)\( p^{3} T^{31} - \)\(27\!\cdots\!12\)\( p^{4} T^{32} - \)\(25\!\cdots\!16\)\( p^{5} T^{33} + \)\(25\!\cdots\!93\)\( p^{6} T^{34} + \)\(40\!\cdots\!00\)\( p^{7} T^{35} - \)\(17\!\cdots\!98\)\( p^{8} T^{36} - \)\(53\!\cdots\!58\)\( p^{9} T^{37} + 22059680972858848912 p^{10} T^{38} + 58590074891348303074 p^{11} T^{39} + 1438898467594141798 p^{12} T^{40} - 532836183078076922 p^{13} T^{41} - 26458651503046229 p^{14} T^{42} + 3993010401149020 p^{15} T^{43} + 298367199924703 p^{16} T^{44} - 24173099162640 p^{17} T^{45} - 2501999072505 p^{18} T^{46} + 113993397918 p^{19} T^{47} + 16236563839 p^{20} T^{48} - 390615932 p^{21} T^{49} - 81065156 p^{22} T^{50} + 819796 p^{23} T^{51} + 299869 p^{24} T^{52} - 434 p^{25} T^{53} - 749 p^{26} T^{54} - 2 p^{27} T^{55} + p^{28} T^{56} \)
83 \( 1 + 18 T - 316 T^{2} - 7844 T^{3} + 33315 T^{4} + 1645196 T^{5} + 2031128 T^{6} - 217711800 T^{7} - 1224339872 T^{8} + 20253139646 T^{9} + 236423232255 T^{10} - 1303904621918 T^{11} - 33125826839353 T^{12} + 15295292722488 T^{13} + 3752333254535440 T^{14} + 13523900440960122 T^{15} - 333203617472561613 T^{16} - 2831640435805867014 T^{17} + 20616462278128583859 T^{18} + \)\(36\!\cdots\!22\)\( T^{19} - \)\(43\!\cdots\!02\)\( T^{20} - \)\(36\!\cdots\!62\)\( T^{21} - \)\(10\!\cdots\!35\)\( T^{22} + \)\(29\!\cdots\!30\)\( T^{23} + \)\(20\!\cdots\!11\)\( T^{24} - \)\(17\!\cdots\!04\)\( T^{25} - \)\(25\!\cdots\!21\)\( T^{26} + \)\(55\!\cdots\!00\)\( T^{27} + \)\(23\!\cdots\!07\)\( T^{28} + \)\(55\!\cdots\!00\)\( p T^{29} - \)\(25\!\cdots\!21\)\( p^{2} T^{30} - \)\(17\!\cdots\!04\)\( p^{3} T^{31} + \)\(20\!\cdots\!11\)\( p^{4} T^{32} + \)\(29\!\cdots\!30\)\( p^{5} T^{33} - \)\(10\!\cdots\!35\)\( p^{6} T^{34} - \)\(36\!\cdots\!62\)\( p^{7} T^{35} - \)\(43\!\cdots\!02\)\( p^{8} T^{36} + \)\(36\!\cdots\!22\)\( p^{9} T^{37} + 20616462278128583859 p^{10} T^{38} - 2831640435805867014 p^{11} T^{39} - 333203617472561613 p^{12} T^{40} + 13523900440960122 p^{13} T^{41} + 3752333254535440 p^{14} T^{42} + 15295292722488 p^{15} T^{43} - 33125826839353 p^{16} T^{44} - 1303904621918 p^{17} T^{45} + 236423232255 p^{18} T^{46} + 20253139646 p^{19} T^{47} - 1224339872 p^{20} T^{48} - 217711800 p^{21} T^{49} + 2031128 p^{22} T^{50} + 1645196 p^{23} T^{51} + 33315 p^{24} T^{52} - 7844 p^{25} T^{53} - 316 p^{26} T^{54} + 18 p^{27} T^{55} + p^{28} T^{56} \)
89 \( 1 - 6 T - 493 T^{2} + 5446 T^{3} + 112041 T^{4} - 1967290 T^{5} - 9631354 T^{6} + 409038702 T^{7} - 1467211604 T^{8} - 49009932946 T^{9} + 618547462659 T^{10} + 1780567785508 T^{11} - 99768438315166 T^{12} + 546973552088172 T^{13} + 8288952885907960 T^{14} - 126188638080293868 T^{15} + 34773596268721128 T^{16} + 152705904578912388 p T^{17} - \)\(11\!\cdots\!79\)\( T^{18} - \)\(57\!\cdots\!20\)\( T^{19} + \)\(16\!\cdots\!26\)\( T^{20} - \)\(67\!\cdots\!10\)\( T^{21} - \)\(12\!\cdots\!83\)\( T^{22} + \)\(15\!\cdots\!26\)\( T^{23} - \)\(10\!\cdots\!32\)\( T^{24} - \)\(14\!\cdots\!38\)\( T^{25} + \)\(11\!\cdots\!75\)\( T^{26} + \)\(57\!\cdots\!80\)\( T^{27} - \)\(14\!\cdots\!02\)\( T^{28} + \)\(57\!\cdots\!80\)\( p T^{29} + \)\(11\!\cdots\!75\)\( p^{2} T^{30} - \)\(14\!\cdots\!38\)\( p^{3} T^{31} - \)\(10\!\cdots\!32\)\( p^{4} T^{32} + \)\(15\!\cdots\!26\)\( p^{5} T^{33} - \)\(12\!\cdots\!83\)\( p^{6} T^{34} - \)\(67\!\cdots\!10\)\( p^{7} T^{35} + \)\(16\!\cdots\!26\)\( p^{8} T^{36} - \)\(57\!\cdots\!20\)\( p^{9} T^{37} - \)\(11\!\cdots\!79\)\( p^{10} T^{38} + 152705904578912388 p^{12} T^{39} + 34773596268721128 p^{12} T^{40} - 126188638080293868 p^{13} T^{41} + 8288952885907960 p^{14} T^{42} + 546973552088172 p^{15} T^{43} - 99768438315166 p^{16} T^{44} + 1780567785508 p^{17} T^{45} + 618547462659 p^{18} T^{46} - 49009932946 p^{19} T^{47} - 1467211604 p^{20} T^{48} + 409038702 p^{21} T^{49} - 9631354 p^{22} T^{50} - 1967290 p^{23} T^{51} + 112041 p^{24} T^{52} + 5446 p^{25} T^{53} - 493 p^{26} T^{54} - 6 p^{27} T^{55} + p^{28} T^{56} \)
97 \( 1 + 72 T + 3248 T^{2} + 109440 T^{3} + 3047319 T^{4} + 73171614 T^{5} + 1562797976 T^{6} + 30218825706 T^{7} + 536326152007 T^{8} + 8813515583682 T^{9} + 135028313607384 T^{10} + 1937114070530040 T^{11} + 26105081682692116 T^{12} + 330941962417734348 T^{13} + 3947300416589560790 T^{14} + 44209130386111763052 T^{15} + \)\(46\!\cdots\!78\)\( T^{16} + \)\(44\!\cdots\!62\)\( T^{17} + \)\(39\!\cdots\!66\)\( T^{18} + \)\(30\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!06\)\( T^{20} + \)\(26\!\cdots\!46\)\( T^{21} - \)\(13\!\cdots\!32\)\( T^{22} - \)\(29\!\cdots\!54\)\( T^{23} - \)\(43\!\cdots\!29\)\( T^{24} - \)\(54\!\cdots\!18\)\( T^{25} - \)\(63\!\cdots\!14\)\( T^{26} - \)\(67\!\cdots\!06\)\( T^{27} - \)\(68\!\cdots\!67\)\( T^{28} - \)\(67\!\cdots\!06\)\( p T^{29} - \)\(63\!\cdots\!14\)\( p^{2} T^{30} - \)\(54\!\cdots\!18\)\( p^{3} T^{31} - \)\(43\!\cdots\!29\)\( p^{4} T^{32} - \)\(29\!\cdots\!54\)\( p^{5} T^{33} - \)\(13\!\cdots\!32\)\( p^{6} T^{34} + \)\(26\!\cdots\!46\)\( p^{7} T^{35} + \)\(17\!\cdots\!06\)\( p^{8} T^{36} + \)\(30\!\cdots\!78\)\( p^{9} T^{37} + \)\(39\!\cdots\!66\)\( p^{10} T^{38} + \)\(44\!\cdots\!62\)\( p^{11} T^{39} + \)\(46\!\cdots\!78\)\( p^{12} T^{40} + 44209130386111763052 p^{13} T^{41} + 3947300416589560790 p^{14} T^{42} + 330941962417734348 p^{15} T^{43} + 26105081682692116 p^{16} T^{44} + 1937114070530040 p^{17} T^{45} + 135028313607384 p^{18} T^{46} + 8813515583682 p^{19} T^{47} + 536326152007 p^{20} T^{48} + 30218825706 p^{21} T^{49} + 1562797976 p^{22} T^{50} + 73171614 p^{23} T^{51} + 3047319 p^{24} T^{52} + 109440 p^{25} T^{53} + 3248 p^{26} T^{54} + 72 p^{27} T^{55} + p^{28} T^{56} \)
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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

−1.87395014966513309745371255815, −1.82516018646147424949308923425, −1.81243144575432624489488522833, −1.80316904556372218440248277197, −1.73751610796011111804761214891, −1.48884495043448573995217388845, −1.42460873702357336916485098648, −1.42062411935293469990971564488, −1.38958634286749851105565489562, −1.29283470923669305625254494568, −1.28932493440538180621631624753, −1.22687392638480392922718558731, −1.15422805191874139373068366268, −1.07234254807158215411778366456, −1.04410207723864341761084264740, −0.840684990308132201785092990972, −0.71846857834881256163283289561, −0.71239264162939872530613925872, −0.59348831546055381236240889898, −0.46559417303241220102925705007, −0.42596909628755233020467051758, −0.33046425434303248010718232347, −0.31915890134172177719732803373, −0.15160374395279728179089100916, −0.13581194752278408675600148424, 0.13581194752278408675600148424, 0.15160374395279728179089100916, 0.31915890134172177719732803373, 0.33046425434303248010718232347, 0.42596909628755233020467051758, 0.46559417303241220102925705007, 0.59348831546055381236240889898, 0.71239264162939872530613925872, 0.71846857834881256163283289561, 0.840684990308132201785092990972, 1.04410207723864341761084264740, 1.07234254807158215411778366456, 1.15422805191874139373068366268, 1.22687392638480392922718558731, 1.28932493440538180621631624753, 1.29283470923669305625254494568, 1.38958634286749851105565489562, 1.42062411935293469990971564488, 1.42460873702357336916485098648, 1.48884495043448573995217388845, 1.73751610796011111804761214891, 1.80316904556372218440248277197, 1.81243144575432624489488522833, 1.82516018646147424949308923425, 1.87395014966513309745371255815

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

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