Properties

Degree 50
Conductor $ 2^{25} \cdot 23^{25} \cdot 131^{25} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 25

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 25·2-s − 4·3-s + 325·4-s − 3·5-s + 100·6-s − 11·7-s − 2.92e3·8-s − 20·9-s + 75·10-s − 12·11-s − 1.30e3·12-s − 6·13-s + 275·14-s + 12·15-s + 2.04e4·16-s + 8·17-s + 500·18-s − 23·19-s − 975·20-s + 44·21-s + 300·22-s + 25·23-s + 1.17e4·24-s − 56·25-s + 150·26-s + 101·27-s − 3.57e3·28-s + ⋯
L(s)  = 1  − 17.6·2-s − 2.30·3-s + 162.5·4-s − 1.34·5-s + 40.8·6-s − 4.15·7-s − 1.03e3·8-s − 6.66·9-s + 23.7·10-s − 3.61·11-s − 375.·12-s − 1.66·13-s + 73.4·14-s + 3.09·15-s + 5.11e3·16-s + 1.94·17-s + 117.·18-s − 5.27·19-s − 218.·20-s + 9.60·21-s + 63.9·22-s + 5.21·23-s + 2.38e3·24-s − 11.1·25-s + 29.4·26-s + 19.4·27-s − 675.·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{25} \cdot 23^{25} \cdot 131^{25}\right)^{s/2} \, \Gamma_{\C}(s)^{25} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{25} \cdot 23^{25} \cdot 131^{25}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{25} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(50\)
\( N \)  =  \(2^{25} \cdot 23^{25} \cdot 131^{25}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6026} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  25
Selberg data  =  $(50,\ 2^{25} \cdot 23^{25} \cdot 131^{25} ,\ ( \ : [1/2]^{25} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \(F_p\) is a polynomial of degree 50. If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 49.
$p$$F_p$
bad2 \( ( 1 + T )^{25} \)
23 \( ( 1 - T )^{25} \)
131 \( ( 1 - T )^{25} \)
good3 \( 1 + 4 T + 4 p^{2} T^{2} + 41 p T^{3} + 649 T^{4} + 1960 T^{5} + 7868 T^{6} + 7169 p T^{7} + 72422 T^{8} + 182188 T^{9} + 540496 T^{10} + 1266298 T^{11} + 378421 p^{2} T^{12} + 2498648 p T^{13} + 18615235 T^{14} + 38740043 T^{15} + 29980817 p T^{16} + 177853955 T^{17} + 389437024 T^{18} + 734383985 T^{19} + 1526109965 T^{20} + 2751518750 T^{21} + 5450544704 T^{22} + 3136774108 p T^{23} + 5940558878 p T^{24} + 9826727194 p T^{25} + 5940558878 p^{2} T^{26} + 3136774108 p^{3} T^{27} + 5450544704 p^{3} T^{28} + 2751518750 p^{4} T^{29} + 1526109965 p^{5} T^{30} + 734383985 p^{6} T^{31} + 389437024 p^{7} T^{32} + 177853955 p^{8} T^{33} + 29980817 p^{10} T^{34} + 38740043 p^{10} T^{35} + 18615235 p^{11} T^{36} + 2498648 p^{13} T^{37} + 378421 p^{15} T^{38} + 1266298 p^{14} T^{39} + 540496 p^{15} T^{40} + 182188 p^{16} T^{41} + 72422 p^{17} T^{42} + 7169 p^{19} T^{43} + 7868 p^{19} T^{44} + 1960 p^{20} T^{45} + 649 p^{21} T^{46} + 41 p^{23} T^{47} + 4 p^{25} T^{48} + 4 p^{24} T^{49} + p^{25} T^{50} \)
5 \( 1 + 3 T + 13 p T^{2} + 186 T^{3} + 2166 T^{4} + 1178 p T^{5} + 9804 p T^{6} + 25323 p T^{7} + 843357 T^{8} + 2070783 T^{9} + 11715109 T^{10} + 5475281 p T^{11} + 136341377 T^{12} + 303543922 T^{13} + 1362357737 T^{14} + 2891927262 T^{15} + 11888001196 T^{16} + 24070546476 T^{17} + 91685486001 T^{18} + 177071640968 T^{19} + 126062904728 p T^{20} + 1160520673032 T^{21} + 777029548501 p T^{22} + 1362417848596 p T^{23} + 4309884491498 p T^{24} + 35919398024172 T^{25} + 4309884491498 p^{2} T^{26} + 1362417848596 p^{3} T^{27} + 777029548501 p^{4} T^{28} + 1160520673032 p^{4} T^{29} + 126062904728 p^{6} T^{30} + 177071640968 p^{6} T^{31} + 91685486001 p^{7} T^{32} + 24070546476 p^{8} T^{33} + 11888001196 p^{9} T^{34} + 2891927262 p^{10} T^{35} + 1362357737 p^{11} T^{36} + 303543922 p^{12} T^{37} + 136341377 p^{13} T^{38} + 5475281 p^{15} T^{39} + 11715109 p^{15} T^{40} + 2070783 p^{16} T^{41} + 843357 p^{17} T^{42} + 25323 p^{19} T^{43} + 9804 p^{20} T^{44} + 1178 p^{21} T^{45} + 2166 p^{21} T^{46} + 186 p^{22} T^{47} + 13 p^{24} T^{48} + 3 p^{24} T^{49} + p^{25} T^{50} \)
7 \( 1 + 11 T + 3 p^{2} T^{2} + 1213 T^{3} + 10048 T^{4} + 67413 T^{5} + 436101 T^{6} + 356730 p T^{7} + 13678656 T^{8} + 68930831 T^{9} + 332510037 T^{10} + 1505035835 T^{11} + 6538474231 T^{12} + 3851169914 p T^{13} + 106987028058 T^{14} + 405814847906 T^{15} + 1485328965583 T^{16} + 5220180570874 T^{17} + 17737741711083 T^{18} + 58053101996257 T^{19} + 183966468917081 T^{20} + 562670472393327 T^{21} + 1667990232304007 T^{22} + 4778349897301485 T^{23} + 13275226384917897 T^{24} + 35662286060270112 T^{25} + 13275226384917897 p T^{26} + 4778349897301485 p^{2} T^{27} + 1667990232304007 p^{3} T^{28} + 562670472393327 p^{4} T^{29} + 183966468917081 p^{5} T^{30} + 58053101996257 p^{6} T^{31} + 17737741711083 p^{7} T^{32} + 5220180570874 p^{8} T^{33} + 1485328965583 p^{9} T^{34} + 405814847906 p^{10} T^{35} + 106987028058 p^{11} T^{36} + 3851169914 p^{13} T^{37} + 6538474231 p^{13} T^{38} + 1505035835 p^{14} T^{39} + 332510037 p^{15} T^{40} + 68930831 p^{16} T^{41} + 13678656 p^{17} T^{42} + 356730 p^{19} T^{43} + 436101 p^{19} T^{44} + 67413 p^{20} T^{45} + 10048 p^{21} T^{46} + 1213 p^{22} T^{47} + 3 p^{25} T^{48} + 11 p^{24} T^{49} + p^{25} T^{50} \)
11 \( 1 + 12 T + 224 T^{2} + 188 p T^{3} + 23069 T^{4} + 177158 T^{5} + 1506517 T^{6} + 10039318 T^{7} + 71176645 T^{8} + 422449857 T^{9} + 2611717376 T^{10} + 14047311554 T^{11} + 77745743381 T^{12} + 383555594780 T^{13} + 1932723096536 T^{14} + 8822408406588 T^{15} + 40941147925796 T^{16} + 174018424268358 T^{17} + 749690542736200 T^{18} + 2980702022108623 T^{19} + 11988769891226730 T^{20} + 44728400173469654 T^{21} + 168616835001240477 T^{22} + 591466985532752297 T^{23} + 2095041464592827116 T^{24} + 6915385250688367610 T^{25} + 2095041464592827116 p T^{26} + 591466985532752297 p^{2} T^{27} + 168616835001240477 p^{3} T^{28} + 44728400173469654 p^{4} T^{29} + 11988769891226730 p^{5} T^{30} + 2980702022108623 p^{6} T^{31} + 749690542736200 p^{7} T^{32} + 174018424268358 p^{8} T^{33} + 40941147925796 p^{9} T^{34} + 8822408406588 p^{10} T^{35} + 1932723096536 p^{11} T^{36} + 383555594780 p^{12} T^{37} + 77745743381 p^{13} T^{38} + 14047311554 p^{14} T^{39} + 2611717376 p^{15} T^{40} + 422449857 p^{16} T^{41} + 71176645 p^{17} T^{42} + 10039318 p^{18} T^{43} + 1506517 p^{19} T^{44} + 177158 p^{20} T^{45} + 23069 p^{21} T^{46} + 188 p^{23} T^{47} + 224 p^{23} T^{48} + 12 p^{24} T^{49} + p^{25} T^{50} \)
13 \( 1 + 6 T + 15 p T^{2} + 1143 T^{3} + 18901 T^{4} + 105421 T^{5} + 1202106 T^{6} + 6272715 T^{7} + 55937140 T^{8} + 270290947 T^{9} + 2016912127 T^{10} + 8968076126 T^{11} + 58388082674 T^{12} + 18294892821 p T^{13} + 1391239316546 T^{14} + 5171318970284 T^{15} + 27839861043840 T^{16} + 94114971193129 T^{17} + 477453232363876 T^{18} + 1466614385504607 T^{19} + 7201592769490499 T^{20} + 20240855174753202 T^{21} + 99096119384310202 T^{22} + 260898525707701941 T^{23} + 1301926706590398517 T^{24} + 3346701769668275004 T^{25} + 1301926706590398517 p T^{26} + 260898525707701941 p^{2} T^{27} + 99096119384310202 p^{3} T^{28} + 20240855174753202 p^{4} T^{29} + 7201592769490499 p^{5} T^{30} + 1466614385504607 p^{6} T^{31} + 477453232363876 p^{7} T^{32} + 94114971193129 p^{8} T^{33} + 27839861043840 p^{9} T^{34} + 5171318970284 p^{10} T^{35} + 1391239316546 p^{11} T^{36} + 18294892821 p^{13} T^{37} + 58388082674 p^{13} T^{38} + 8968076126 p^{14} T^{39} + 2016912127 p^{15} T^{40} + 270290947 p^{16} T^{41} + 55937140 p^{17} T^{42} + 6272715 p^{18} T^{43} + 1202106 p^{19} T^{44} + 105421 p^{20} T^{45} + 18901 p^{21} T^{46} + 1143 p^{22} T^{47} + 15 p^{24} T^{48} + 6 p^{24} T^{49} + p^{25} T^{50} \)
17 \( 1 - 8 T + 254 T^{2} - 1761 T^{3} + 30656 T^{4} - 11096 p T^{5} + 2365586 T^{6} - 13124465 T^{7} + 132018728 T^{8} - 667395347 T^{9} + 5704868733 T^{10} - 26459963694 T^{11} + 199514873037 T^{12} - 852800228274 T^{13} + 5833859774108 T^{14} - 23050119965532 T^{15} + 146548997673265 T^{16} - 536773502303019 T^{17} + 3244439107650812 T^{18} - 11067162092562277 T^{19} + 64962704558909663 T^{20} - 208218004494195338 T^{21} + 1207410978703358152 T^{22} - 3692055082893607157 T^{23} + 21313320819549866289 T^{24} - 63416324504934393776 T^{25} + 21313320819549866289 p T^{26} - 3692055082893607157 p^{2} T^{27} + 1207410978703358152 p^{3} T^{28} - 208218004494195338 p^{4} T^{29} + 64962704558909663 p^{5} T^{30} - 11067162092562277 p^{6} T^{31} + 3244439107650812 p^{7} T^{32} - 536773502303019 p^{8} T^{33} + 146548997673265 p^{9} T^{34} - 23050119965532 p^{10} T^{35} + 5833859774108 p^{11} T^{36} - 852800228274 p^{12} T^{37} + 199514873037 p^{13} T^{38} - 26459963694 p^{14} T^{39} + 5704868733 p^{15} T^{40} - 667395347 p^{16} T^{41} + 132018728 p^{17} T^{42} - 13124465 p^{18} T^{43} + 2365586 p^{19} T^{44} - 11096 p^{21} T^{45} + 30656 p^{21} T^{46} - 1761 p^{22} T^{47} + 254 p^{23} T^{48} - 8 p^{24} T^{49} + p^{25} T^{50} \)
19 \( 1 + 23 T + 492 T^{2} + 7129 T^{3} + 95499 T^{4} + 1065329 T^{5} + 11134220 T^{6} + 5468196 p T^{7} + 918272263 T^{8} + 7486208390 T^{9} + 58265982150 T^{10} + 425888019553 T^{11} + 2988835282520 T^{12} + 19915548482618 T^{13} + 127937768048400 T^{14} + 785904253287251 T^{15} + 4668374863461711 T^{16} + 26643579839519210 T^{17} + 147365878601316231 T^{18} + 785656379985798806 T^{19} + 4065574800967877032 T^{20} + 20322262360042211113 T^{21} + 98702351225203576283 T^{22} + \)\(46\!\cdots\!85\)\( T^{23} + \)\(21\!\cdots\!98\)\( T^{24} + \)\(93\!\cdots\!86\)\( T^{25} + \)\(21\!\cdots\!98\)\( p T^{26} + \)\(46\!\cdots\!85\)\( p^{2} T^{27} + 98702351225203576283 p^{3} T^{28} + 20322262360042211113 p^{4} T^{29} + 4065574800967877032 p^{5} T^{30} + 785656379985798806 p^{6} T^{31} + 147365878601316231 p^{7} T^{32} + 26643579839519210 p^{8} T^{33} + 4668374863461711 p^{9} T^{34} + 785904253287251 p^{10} T^{35} + 127937768048400 p^{11} T^{36} + 19915548482618 p^{12} T^{37} + 2988835282520 p^{13} T^{38} + 425888019553 p^{14} T^{39} + 58265982150 p^{15} T^{40} + 7486208390 p^{16} T^{41} + 918272263 p^{17} T^{42} + 5468196 p^{19} T^{43} + 11134220 p^{19} T^{44} + 1065329 p^{20} T^{45} + 95499 p^{21} T^{46} + 7129 p^{22} T^{47} + 492 p^{23} T^{48} + 23 p^{24} T^{49} + p^{25} T^{50} \)
29 \( 1 + 7 T + 378 T^{2} + 2260 T^{3} + 69986 T^{4} + 365784 T^{5} + 8567840 T^{6} + 39912593 T^{7} + 787135149 T^{8} + 3322298978 T^{9} + 58188494702 T^{10} + 225527831901 T^{11} + 3613619587943 T^{12} + 12997197590838 T^{13} + 193965698798852 T^{14} + 652654124927508 T^{15} + 9177996398864113 T^{16} + 29070080156328633 T^{17} + 388236555618679487 T^{18} + 1163176381134780698 T^{19} + 14828160508011904061 T^{20} + 42190725100437640110 T^{21} + \)\(51\!\cdots\!58\)\( T^{22} + \)\(13\!\cdots\!32\)\( T^{23} + \)\(16\!\cdots\!54\)\( T^{24} + \)\(14\!\cdots\!60\)\( p T^{25} + \)\(16\!\cdots\!54\)\( p T^{26} + \)\(13\!\cdots\!32\)\( p^{2} T^{27} + \)\(51\!\cdots\!58\)\( p^{3} T^{28} + 42190725100437640110 p^{4} T^{29} + 14828160508011904061 p^{5} T^{30} + 1163176381134780698 p^{6} T^{31} + 388236555618679487 p^{7} T^{32} + 29070080156328633 p^{8} T^{33} + 9177996398864113 p^{9} T^{34} + 652654124927508 p^{10} T^{35} + 193965698798852 p^{11} T^{36} + 12997197590838 p^{12} T^{37} + 3613619587943 p^{13} T^{38} + 225527831901 p^{14} T^{39} + 58188494702 p^{15} T^{40} + 3322298978 p^{16} T^{41} + 787135149 p^{17} T^{42} + 39912593 p^{18} T^{43} + 8567840 p^{19} T^{44} + 365784 p^{20} T^{45} + 69986 p^{21} T^{46} + 2260 p^{22} T^{47} + 378 p^{23} T^{48} + 7 p^{24} T^{49} + p^{25} T^{50} \)
31 \( 1 + 7 T + 426 T^{2} + 2922 T^{3} + 91033 T^{4} + 611217 T^{5} + 13010180 T^{6} + 85362684 T^{7} + 1398590763 T^{8} + 8949119470 T^{9} + 120557118615 T^{10} + 750608457060 T^{11} + 8671060012051 T^{12} + 52405872476548 T^{13} + 534414971469295 T^{14} + 3127285061706857 T^{15} + 28748820961766251 T^{16} + 162449603424638697 T^{17} + 1367494445569306561 T^{18} + 7440125482601251801 T^{19} + 58040108210511317129 T^{20} + \)\(30\!\cdots\!15\)\( T^{21} + \)\(22\!\cdots\!90\)\( T^{22} + \)\(11\!\cdots\!40\)\( T^{23} + \)\(75\!\cdots\!51\)\( T^{24} + \)\(36\!\cdots\!80\)\( T^{25} + \)\(75\!\cdots\!51\)\( p T^{26} + \)\(11\!\cdots\!40\)\( p^{2} T^{27} + \)\(22\!\cdots\!90\)\( p^{3} T^{28} + \)\(30\!\cdots\!15\)\( p^{4} T^{29} + 58040108210511317129 p^{5} T^{30} + 7440125482601251801 p^{6} T^{31} + 1367494445569306561 p^{7} T^{32} + 162449603424638697 p^{8} T^{33} + 28748820961766251 p^{9} T^{34} + 3127285061706857 p^{10} T^{35} + 534414971469295 p^{11} T^{36} + 52405872476548 p^{12} T^{37} + 8671060012051 p^{13} T^{38} + 750608457060 p^{14} T^{39} + 120557118615 p^{15} T^{40} + 8949119470 p^{16} T^{41} + 1398590763 p^{17} T^{42} + 85362684 p^{18} T^{43} + 13010180 p^{19} T^{44} + 611217 p^{20} T^{45} + 91033 p^{21} T^{46} + 2922 p^{22} T^{47} + 426 p^{23} T^{48} + 7 p^{24} T^{49} + p^{25} T^{50} \)
37 \( 1 + 7 T + 556 T^{2} + 3918 T^{3} + 154851 T^{4} + 1081483 T^{5} + 28690675 T^{6} + 196456575 T^{7} + 3965669931 T^{8} + 26427692834 T^{9} + 435100680859 T^{10} + 2807878149542 T^{11} + 39396377987690 T^{12} + 245356279444231 T^{13} + 3023275911396863 T^{14} + 18125209438152690 T^{15} + 200445179797511741 T^{16} + 1154435783357719457 T^{17} + 11646257061361572358 T^{18} + 64306506977144074378 T^{19} + \)\(59\!\cdots\!90\)\( T^{20} + \)\(31\!\cdots\!70\)\( T^{21} + \)\(27\!\cdots\!47\)\( T^{22} + \)\(37\!\cdots\!17\)\( p T^{23} + \)\(11\!\cdots\!02\)\( T^{24} + \)\(14\!\cdots\!72\)\( p T^{25} + \)\(11\!\cdots\!02\)\( p T^{26} + \)\(37\!\cdots\!17\)\( p^{3} T^{27} + \)\(27\!\cdots\!47\)\( p^{3} T^{28} + \)\(31\!\cdots\!70\)\( p^{4} T^{29} + \)\(59\!\cdots\!90\)\( p^{5} T^{30} + 64306506977144074378 p^{6} T^{31} + 11646257061361572358 p^{7} T^{32} + 1154435783357719457 p^{8} T^{33} + 200445179797511741 p^{9} T^{34} + 18125209438152690 p^{10} T^{35} + 3023275911396863 p^{11} T^{36} + 245356279444231 p^{12} T^{37} + 39396377987690 p^{13} T^{38} + 2807878149542 p^{14} T^{39} + 435100680859 p^{15} T^{40} + 26427692834 p^{16} T^{41} + 3965669931 p^{17} T^{42} + 196456575 p^{18} T^{43} + 28690675 p^{19} T^{44} + 1081483 p^{20} T^{45} + 154851 p^{21} T^{46} + 3918 p^{22} T^{47} + 556 p^{23} T^{48} + 7 p^{24} T^{49} + p^{25} T^{50} \)
41 \( 1 + 10 T + 525 T^{2} + 5148 T^{3} + 142031 T^{4} + 1345666 T^{5} + 26200765 T^{6} + 237739769 T^{7} + 3682899427 T^{8} + 31857112279 T^{9} + 418397756551 T^{10} + 3443749158903 T^{11} + 39829812691969 T^{12} + 311852204477639 T^{13} + 3254950282171436 T^{14} + 24252524080986034 T^{15} + 232261503676572640 T^{16} + 1647785283392262423 T^{17} + 14649203757338095020 T^{18} + 98994754544791672770 T^{19} + \)\(82\!\cdots\!38\)\( T^{20} + \)\(53\!\cdots\!39\)\( T^{21} + \)\(41\!\cdots\!76\)\( T^{22} + \)\(25\!\cdots\!10\)\( T^{23} + \)\(18\!\cdots\!57\)\( T^{24} + \)\(11\!\cdots\!44\)\( T^{25} + \)\(18\!\cdots\!57\)\( p T^{26} + \)\(25\!\cdots\!10\)\( p^{2} T^{27} + \)\(41\!\cdots\!76\)\( p^{3} T^{28} + \)\(53\!\cdots\!39\)\( p^{4} T^{29} + \)\(82\!\cdots\!38\)\( p^{5} T^{30} + 98994754544791672770 p^{6} T^{31} + 14649203757338095020 p^{7} T^{32} + 1647785283392262423 p^{8} T^{33} + 232261503676572640 p^{9} T^{34} + 24252524080986034 p^{10} T^{35} + 3254950282171436 p^{11} T^{36} + 311852204477639 p^{12} T^{37} + 39829812691969 p^{13} T^{38} + 3443749158903 p^{14} T^{39} + 418397756551 p^{15} T^{40} + 31857112279 p^{16} T^{41} + 3682899427 p^{17} T^{42} + 237739769 p^{18} T^{43} + 26200765 p^{19} T^{44} + 1345666 p^{20} T^{45} + 142031 p^{21} T^{46} + 5148 p^{22} T^{47} + 525 p^{23} T^{48} + 10 p^{24} T^{49} + p^{25} T^{50} \)
43 \( 1 + 26 T + 867 T^{2} + 17083 T^{3} + 347567 T^{4} + 5617829 T^{5} + 88651191 T^{6} + 1229693584 T^{7} + 16400886083 T^{8} + 200929131059 T^{9} + 2361304461897 T^{10} + 26053475175233 T^{11} + 276209897807587 T^{12} + 2782644663873452 T^{13} + 27005878299638577 T^{14} + 250876451610762478 T^{15} + 2250599006209122951 T^{16} + 19416220205729568873 T^{17} + \)\(16\!\cdots\!70\)\( T^{18} + \)\(13\!\cdots\!08\)\( T^{19} + \)\(10\!\cdots\!89\)\( T^{20} + \)\(76\!\cdots\!35\)\( T^{21} + \)\(56\!\cdots\!97\)\( T^{22} + \)\(39\!\cdots\!50\)\( T^{23} + \)\(27\!\cdots\!93\)\( T^{24} + \)\(18\!\cdots\!64\)\( T^{25} + \)\(27\!\cdots\!93\)\( p T^{26} + \)\(39\!\cdots\!50\)\( p^{2} T^{27} + \)\(56\!\cdots\!97\)\( p^{3} T^{28} + \)\(76\!\cdots\!35\)\( p^{4} T^{29} + \)\(10\!\cdots\!89\)\( p^{5} T^{30} + \)\(13\!\cdots\!08\)\( p^{6} T^{31} + \)\(16\!\cdots\!70\)\( p^{7} T^{32} + 19416220205729568873 p^{8} T^{33} + 2250599006209122951 p^{9} T^{34} + 250876451610762478 p^{10} T^{35} + 27005878299638577 p^{11} T^{36} + 2782644663873452 p^{12} T^{37} + 276209897807587 p^{13} T^{38} + 26053475175233 p^{14} T^{39} + 2361304461897 p^{15} T^{40} + 200929131059 p^{16} T^{41} + 16400886083 p^{17} T^{42} + 1229693584 p^{18} T^{43} + 88651191 p^{19} T^{44} + 5617829 p^{20} T^{45} + 347567 p^{21} T^{46} + 17083 p^{22} T^{47} + 867 p^{23} T^{48} + 26 p^{24} T^{49} + p^{25} T^{50} \)
47 \( 1 + 2 T + 462 T^{2} - 261 T^{3} + 106961 T^{4} - 320583 T^{5} + 17176429 T^{6} - 89115590 T^{7} + 2200925325 T^{8} - 15202835473 T^{9} + 241441501395 T^{10} - 1927459127007 T^{11} + 23435421162385 T^{12} - 197599872746931 T^{13} + 2037088317547527 T^{14} - 17177400527363072 T^{15} + 159033379216308383 T^{16} - 1303548054808662142 T^{17} + 11163420802340037497 T^{18} - 87901005477866862511 T^{19} + \)\(70\!\cdots\!17\)\( T^{20} - \)\(53\!\cdots\!34\)\( T^{21} + \)\(40\!\cdots\!59\)\( T^{22} - \)\(29\!\cdots\!63\)\( T^{23} + \)\(20\!\cdots\!01\)\( T^{24} - \)\(14\!\cdots\!54\)\( T^{25} + \)\(20\!\cdots\!01\)\( p T^{26} - \)\(29\!\cdots\!63\)\( p^{2} T^{27} + \)\(40\!\cdots\!59\)\( p^{3} T^{28} - \)\(53\!\cdots\!34\)\( p^{4} T^{29} + \)\(70\!\cdots\!17\)\( p^{5} T^{30} - 87901005477866862511 p^{6} T^{31} + 11163420802340037497 p^{7} T^{32} - 1303548054808662142 p^{8} T^{33} + 159033379216308383 p^{9} T^{34} - 17177400527363072 p^{10} T^{35} + 2037088317547527 p^{11} T^{36} - 197599872746931 p^{12} T^{37} + 23435421162385 p^{13} T^{38} - 1927459127007 p^{14} T^{39} + 241441501395 p^{15} T^{40} - 15202835473 p^{16} T^{41} + 2200925325 p^{17} T^{42} - 89115590 p^{18} T^{43} + 17176429 p^{19} T^{44} - 320583 p^{20} T^{45} + 106961 p^{21} T^{46} - 261 p^{22} T^{47} + 462 p^{23} T^{48} + 2 p^{24} T^{49} + p^{25} T^{50} \)
53 \( 1 - 47 T + 1832 T^{2} - 51143 T^{3} + 1251742 T^{4} - 26130534 T^{5} + 495059310 T^{6} - 8463440912 T^{7} + 134114896028 T^{8} - 1969077603870 T^{9} + 27165921903177 T^{10} - 352494642730998 T^{11} + 4338092744945484 T^{12} - 50691807587519428 T^{13} + 565520529830574072 T^{14} - 6028954062410643731 T^{15} + 61654261043045950965 T^{16} - \)\(60\!\cdots\!96\)\( T^{17} + \)\(57\!\cdots\!35\)\( T^{18} - \)\(52\!\cdots\!10\)\( T^{19} + \)\(45\!\cdots\!33\)\( T^{20} - \)\(38\!\cdots\!78\)\( T^{21} + \)\(31\!\cdots\!91\)\( T^{22} - \)\(25\!\cdots\!38\)\( T^{23} + \)\(19\!\cdots\!94\)\( T^{24} - \)\(14\!\cdots\!70\)\( T^{25} + \)\(19\!\cdots\!94\)\( p T^{26} - \)\(25\!\cdots\!38\)\( p^{2} T^{27} + \)\(31\!\cdots\!91\)\( p^{3} T^{28} - \)\(38\!\cdots\!78\)\( p^{4} T^{29} + \)\(45\!\cdots\!33\)\( p^{5} T^{30} - \)\(52\!\cdots\!10\)\( p^{6} T^{31} + \)\(57\!\cdots\!35\)\( p^{7} T^{32} - \)\(60\!\cdots\!96\)\( p^{8} T^{33} + 61654261043045950965 p^{9} T^{34} - 6028954062410643731 p^{10} T^{35} + 565520529830574072 p^{11} T^{36} - 50691807587519428 p^{12} T^{37} + 4338092744945484 p^{13} T^{38} - 352494642730998 p^{14} T^{39} + 27165921903177 p^{15} T^{40} - 1969077603870 p^{16} T^{41} + 134114896028 p^{17} T^{42} - 8463440912 p^{18} T^{43} + 495059310 p^{19} T^{44} - 26130534 p^{20} T^{45} + 1251742 p^{21} T^{46} - 51143 p^{22} T^{47} + 1832 p^{23} T^{48} - 47 p^{24} T^{49} + p^{25} T^{50} \)
59 \( 1 + 19 T + 897 T^{2} + 13784 T^{3} + 365172 T^{4} + 4693523 T^{5} + 91171238 T^{6} + 999157694 T^{7} + 15833498436 T^{8} + 149214848477 T^{9} + 2051117106194 T^{10} + 16575986233595 T^{11} + 207133396883003 T^{12} + 1407800092809722 T^{13} + 16794854805673068 T^{14} + 91051342444965268 T^{15} + 1115171014660403136 T^{16} + 4171935857408687142 T^{17} + 61401853743096124235 T^{18} + 84827938450284748331 T^{19} + \)\(28\!\cdots\!22\)\( T^{20} - \)\(66\!\cdots\!99\)\( T^{21} + \)\(11\!\cdots\!35\)\( T^{22} - \)\(94\!\cdots\!20\)\( T^{23} + \)\(46\!\cdots\!53\)\( T^{24} - \)\(68\!\cdots\!04\)\( T^{25} + \)\(46\!\cdots\!53\)\( p T^{26} - \)\(94\!\cdots\!20\)\( p^{2} T^{27} + \)\(11\!\cdots\!35\)\( p^{3} T^{28} - \)\(66\!\cdots\!99\)\( p^{4} T^{29} + \)\(28\!\cdots\!22\)\( p^{5} T^{30} + 84827938450284748331 p^{6} T^{31} + 61401853743096124235 p^{7} T^{32} + 4171935857408687142 p^{8} T^{33} + 1115171014660403136 p^{9} T^{34} + 91051342444965268 p^{10} T^{35} + 16794854805673068 p^{11} T^{36} + 1407800092809722 p^{12} T^{37} + 207133396883003 p^{13} T^{38} + 16575986233595 p^{14} T^{39} + 2051117106194 p^{15} T^{40} + 149214848477 p^{16} T^{41} + 15833498436 p^{17} T^{42} + 999157694 p^{18} T^{43} + 91171238 p^{19} T^{44} + 4693523 p^{20} T^{45} + 365172 p^{21} T^{46} + 13784 p^{22} T^{47} + 897 p^{23} T^{48} + 19 p^{24} T^{49} + p^{25} T^{50} \)
61 \( 1 + 26 T + 1115 T^{2} + 23214 T^{3} + 583218 T^{4} + 10260087 T^{5} + 194945858 T^{6} + 2997904891 T^{7} + 47389824401 T^{8} + 651825592726 T^{9} + 8993768530802 T^{10} + 1843545641959 p T^{11} + 1392838244489169 T^{12} + 16021765513708799 T^{13} + 181344952326830890 T^{14} + 1936265191546329903 T^{15} + 332331438440492502 p T^{16} + \)\(20\!\cdots\!55\)\( T^{17} + \)\(19\!\cdots\!76\)\( T^{18} + \)\(18\!\cdots\!61\)\( T^{19} + \)\(16\!\cdots\!72\)\( T^{20} + \)\(14\!\cdots\!60\)\( T^{21} + \)\(12\!\cdots\!24\)\( T^{22} + \)\(10\!\cdots\!76\)\( T^{23} + \)\(88\!\cdots\!96\)\( T^{24} + \)\(69\!\cdots\!58\)\( T^{25} + \)\(88\!\cdots\!96\)\( p T^{26} + \)\(10\!\cdots\!76\)\( p^{2} T^{27} + \)\(12\!\cdots\!24\)\( p^{3} T^{28} + \)\(14\!\cdots\!60\)\( p^{4} T^{29} + \)\(16\!\cdots\!72\)\( p^{5} T^{30} + \)\(18\!\cdots\!61\)\( p^{6} T^{31} + \)\(19\!\cdots\!76\)\( p^{7} T^{32} + \)\(20\!\cdots\!55\)\( p^{8} T^{33} + 332331438440492502 p^{10} T^{34} + 1936265191546329903 p^{10} T^{35} + 181344952326830890 p^{11} T^{36} + 16021765513708799 p^{12} T^{37} + 1392838244489169 p^{13} T^{38} + 1843545641959 p^{15} T^{39} + 8993768530802 p^{15} T^{40} + 651825592726 p^{16} T^{41} + 47389824401 p^{17} T^{42} + 2997904891 p^{18} T^{43} + 194945858 p^{19} T^{44} + 10260087 p^{20} T^{45} + 583218 p^{21} T^{46} + 23214 p^{22} T^{47} + 1115 p^{23} T^{48} + 26 p^{24} T^{49} + p^{25} T^{50} \)
67 \( 1 + 34 T + 1528 T^{2} + 38750 T^{3} + 1054343 T^{4} + 21805320 T^{5} + 455308376 T^{6} + 8058083690 T^{7} + 140734665943 T^{8} + 2195209943003 T^{9} + 33471990927120 T^{10} + 469211028327570 T^{11} + 6404030588019331 T^{12} + 81775773685420411 T^{13} + 1015121011894412327 T^{14} + 11922572766653799592 T^{15} + \)\(13\!\cdots\!53\)\( T^{16} + \)\(14\!\cdots\!81\)\( T^{17} + \)\(15\!\cdots\!35\)\( T^{18} + \)\(15\!\cdots\!18\)\( T^{19} + \)\(15\!\cdots\!37\)\( T^{20} + \)\(14\!\cdots\!37\)\( T^{21} + \)\(13\!\cdots\!80\)\( T^{22} + \)\(12\!\cdots\!76\)\( T^{23} + \)\(10\!\cdots\!66\)\( T^{24} + \)\(85\!\cdots\!00\)\( T^{25} + \)\(10\!\cdots\!66\)\( p T^{26} + \)\(12\!\cdots\!76\)\( p^{2} T^{27} + \)\(13\!\cdots\!80\)\( p^{3} T^{28} + \)\(14\!\cdots\!37\)\( p^{4} T^{29} + \)\(15\!\cdots\!37\)\( p^{5} T^{30} + \)\(15\!\cdots\!18\)\( p^{6} T^{31} + \)\(15\!\cdots\!35\)\( p^{7} T^{32} + \)\(14\!\cdots\!81\)\( p^{8} T^{33} + \)\(13\!\cdots\!53\)\( p^{9} T^{34} + 11922572766653799592 p^{10} T^{35} + 1015121011894412327 p^{11} T^{36} + 81775773685420411 p^{12} T^{37} + 6404030588019331 p^{13} T^{38} + 469211028327570 p^{14} T^{39} + 33471990927120 p^{15} T^{40} + 2195209943003 p^{16} T^{41} + 140734665943 p^{17} T^{42} + 8058083690 p^{18} T^{43} + 455308376 p^{19} T^{44} + 21805320 p^{20} T^{45} + 1054343 p^{21} T^{46} + 38750 p^{22} T^{47} + 1528 p^{23} T^{48} + 34 p^{24} T^{49} + p^{25} T^{50} \)
71 \( 1 + 10 T + 889 T^{2} + 8091 T^{3} + 404906 T^{4} + 3413892 T^{5} + 125629649 T^{6} + 992335916 T^{7} + 29763332079 T^{8} + 221887245915 T^{9} + 5720182363531 T^{10} + 40445583790424 T^{11} + 925183247611535 T^{12} + 6224306039756433 T^{13} + 129002778720673700 T^{14} + 827429267753471568 T^{15} + 15765618184275263193 T^{16} + 96511237685042091779 T^{17} + \)\(17\!\cdots\!76\)\( T^{18} + \)\(99\!\cdots\!85\)\( T^{19} + \)\(16\!\cdots\!59\)\( T^{20} + \)\(92\!\cdots\!35\)\( T^{21} + \)\(14\!\cdots\!19\)\( T^{22} + \)\(76\!\cdots\!48\)\( T^{23} + \)\(11\!\cdots\!95\)\( T^{24} + \)\(57\!\cdots\!96\)\( T^{25} + \)\(11\!\cdots\!95\)\( p T^{26} + \)\(76\!\cdots\!48\)\( p^{2} T^{27} + \)\(14\!\cdots\!19\)\( p^{3} T^{28} + \)\(92\!\cdots\!35\)\( p^{4} T^{29} + \)\(16\!\cdots\!59\)\( p^{5} T^{30} + \)\(99\!\cdots\!85\)\( p^{6} T^{31} + \)\(17\!\cdots\!76\)\( p^{7} T^{32} + 96511237685042091779 p^{8} T^{33} + 15765618184275263193 p^{9} T^{34} + 827429267753471568 p^{10} T^{35} + 129002778720673700 p^{11} T^{36} + 6224306039756433 p^{12} T^{37} + 925183247611535 p^{13} T^{38} + 40445583790424 p^{14} T^{39} + 5720182363531 p^{15} T^{40} + 221887245915 p^{16} T^{41} + 29763332079 p^{17} T^{42} + 992335916 p^{18} T^{43} + 125629649 p^{19} T^{44} + 3413892 p^{20} T^{45} + 404906 p^{21} T^{46} + 8091 p^{22} T^{47} + 889 p^{23} T^{48} + 10 p^{24} T^{49} + p^{25} T^{50} \)
73 \( 1 + 22 T + 1390 T^{2} + 27230 T^{3} + 940309 T^{4} + 16536599 T^{5} + 412184666 T^{6} + 6560796724 T^{7} + 131599686936 T^{8} + 1910490660307 T^{9} + 32624897538305 T^{10} + 435000081078348 T^{11} + 6539453490428381 T^{12} + 80576860306603625 T^{13} + 1089775755585441642 T^{14} + 12475597572958932637 T^{15} + \)\(15\!\cdots\!29\)\( T^{16} + \)\(16\!\cdots\!30\)\( T^{17} + \)\(18\!\cdots\!56\)\( T^{18} + \)\(18\!\cdots\!99\)\( T^{19} + \)\(19\!\cdots\!57\)\( T^{20} + \)\(18\!\cdots\!96\)\( T^{21} + \)\(18\!\cdots\!77\)\( T^{22} + \)\(16\!\cdots\!10\)\( T^{23} + \)\(15\!\cdots\!03\)\( T^{24} + \)\(12\!\cdots\!34\)\( T^{25} + \)\(15\!\cdots\!03\)\( p T^{26} + \)\(16\!\cdots\!10\)\( p^{2} T^{27} + \)\(18\!\cdots\!77\)\( p^{3} T^{28} + \)\(18\!\cdots\!96\)\( p^{4} T^{29} + \)\(19\!\cdots\!57\)\( p^{5} T^{30} + \)\(18\!\cdots\!99\)\( p^{6} T^{31} + \)\(18\!\cdots\!56\)\( p^{7} T^{32} + \)\(16\!\cdots\!30\)\( p^{8} T^{33} + \)\(15\!\cdots\!29\)\( p^{9} T^{34} + 12475597572958932637 p^{10} T^{35} + 1089775755585441642 p^{11} T^{36} + 80576860306603625 p^{12} T^{37} + 6539453490428381 p^{13} T^{38} + 435000081078348 p^{14} T^{39} + 32624897538305 p^{15} T^{40} + 1910490660307 p^{16} T^{41} + 131599686936 p^{17} T^{42} + 6560796724 p^{18} T^{43} + 412184666 p^{19} T^{44} + 16536599 p^{20} T^{45} + 940309 p^{21} T^{46} + 27230 p^{22} T^{47} + 1390 p^{23} T^{48} + 22 p^{24} T^{49} + p^{25} T^{50} \)
79 \( 1 + 21 T + 1173 T^{2} + 22277 T^{3} + 686353 T^{4} + 11867351 T^{5} + 266358192 T^{6} + 4221236983 T^{7} + 76991769930 T^{8} + 1125489340888 T^{9} + 17658961579795 T^{10} + 239525230530569 T^{11} + 3344498111549243 T^{12} + 42322984651894803 T^{13} + 537542937722599210 T^{14} + 6377878317477111114 T^{15} + 74784782954210802978 T^{16} + \)\(83\!\cdots\!05\)\( T^{17} + \)\(91\!\cdots\!58\)\( T^{18} + \)\(96\!\cdots\!15\)\( T^{19} + \)\(99\!\cdots\!37\)\( T^{20} + \)\(99\!\cdots\!62\)\( T^{21} + \)\(96\!\cdots\!67\)\( T^{22} + \)\(91\!\cdots\!30\)\( T^{23} + \)\(84\!\cdots\!93\)\( T^{24} + \)\(76\!\cdots\!20\)\( T^{25} + \)\(84\!\cdots\!93\)\( p T^{26} + \)\(91\!\cdots\!30\)\( p^{2} T^{27} + \)\(96\!\cdots\!67\)\( p^{3} T^{28} + \)\(99\!\cdots\!62\)\( p^{4} T^{29} + \)\(99\!\cdots\!37\)\( p^{5} T^{30} + \)\(96\!\cdots\!15\)\( p^{6} T^{31} + \)\(91\!\cdots\!58\)\( p^{7} T^{32} + \)\(83\!\cdots\!05\)\( p^{8} T^{33} + 74784782954210802978 p^{9} T^{34} + 6377878317477111114 p^{10} T^{35} + 537542937722599210 p^{11} T^{36} + 42322984651894803 p^{12} T^{37} + 3344498111549243 p^{13} T^{38} + 239525230530569 p^{14} T^{39} + 17658961579795 p^{15} T^{40} + 1125489340888 p^{16} T^{41} + 76991769930 p^{17} T^{42} + 4221236983 p^{18} T^{43} + 266358192 p^{19} T^{44} + 11867351 p^{20} T^{45} + 686353 p^{21} T^{46} + 22277 p^{22} T^{47} + 1173 p^{23} T^{48} + 21 p^{24} T^{49} + p^{25} T^{50} \)
83 \( 1 + 16 T + 1083 T^{2} + 14649 T^{3} + 572918 T^{4} + 6815007 T^{5} + 200707145 T^{6} + 2155427852 T^{7} + 52863587683 T^{8} + 522482545350 T^{9} + 11222375559126 T^{10} + 103581849378937 T^{11} + 2004531997435500 T^{12} + 17468684580639949 T^{13} + 309868739891387621 T^{14} + 2569705433333093795 T^{15} + 42253564428803508226 T^{16} + \)\(33\!\cdots\!04\)\( T^{17} + \)\(51\!\cdots\!55\)\( T^{18} + \)\(39\!\cdots\!30\)\( T^{19} + \)\(56\!\cdots\!36\)\( T^{20} + \)\(41\!\cdots\!36\)\( T^{21} + \)\(56\!\cdots\!28\)\( T^{22} + \)\(39\!\cdots\!37\)\( T^{23} + \)\(51\!\cdots\!82\)\( T^{24} + \)\(34\!\cdots\!28\)\( T^{25} + \)\(51\!\cdots\!82\)\( p T^{26} + \)\(39\!\cdots\!37\)\( p^{2} T^{27} + \)\(56\!\cdots\!28\)\( p^{3} T^{28} + \)\(41\!\cdots\!36\)\( p^{4} T^{29} + \)\(56\!\cdots\!36\)\( p^{5} T^{30} + \)\(39\!\cdots\!30\)\( p^{6} T^{31} + \)\(51\!\cdots\!55\)\( p^{7} T^{32} + \)\(33\!\cdots\!04\)\( p^{8} T^{33} + 42253564428803508226 p^{9} T^{34} + 2569705433333093795 p^{10} T^{35} + 309868739891387621 p^{11} T^{36} + 17468684580639949 p^{12} T^{37} + 2004531997435500 p^{13} T^{38} + 103581849378937 p^{14} T^{39} + 11222375559126 p^{15} T^{40} + 522482545350 p^{16} T^{41} + 52863587683 p^{17} T^{42} + 2155427852 p^{18} T^{43} + 200707145 p^{19} T^{44} + 6815007 p^{20} T^{45} + 572918 p^{21} T^{46} + 14649 p^{22} T^{47} + 1083 p^{23} T^{48} + 16 p^{24} T^{49} + p^{25} T^{50} \)
89 \( 1 - 27 T + 1733 T^{2} - 40227 T^{3} + 1445085 T^{4} - 29459962 T^{5} + 775009134 T^{6} - 14104194369 T^{7} + 301181803983 T^{8} - 4955758178821 T^{9} + 90548963546410 T^{10} - 1360893209521850 T^{11} + 21952489659797350 T^{12} - 303863530982546051 T^{13} + 4416861108054863891 T^{14} - 56693303758188972767 T^{15} + \)\(75\!\cdots\!90\)\( T^{16} - \)\(90\!\cdots\!41\)\( T^{17} + \)\(11\!\cdots\!16\)\( T^{18} - \)\(12\!\cdots\!52\)\( T^{19} + \)\(14\!\cdots\!76\)\( T^{20} - \)\(14\!\cdots\!83\)\( T^{21} + \)\(16\!\cdots\!24\)\( T^{22} - \)\(15\!\cdots\!15\)\( T^{23} + \)\(16\!\cdots\!95\)\( T^{24} - \)\(15\!\cdots\!98\)\( T^{25} + \)\(16\!\cdots\!95\)\( p T^{26} - \)\(15\!\cdots\!15\)\( p^{2} T^{27} + \)\(16\!\cdots\!24\)\( p^{3} T^{28} - \)\(14\!\cdots\!83\)\( p^{4} T^{29} + \)\(14\!\cdots\!76\)\( p^{5} T^{30} - \)\(12\!\cdots\!52\)\( p^{6} T^{31} + \)\(11\!\cdots\!16\)\( p^{7} T^{32} - \)\(90\!\cdots\!41\)\( p^{8} T^{33} + \)\(75\!\cdots\!90\)\( p^{9} T^{34} - 56693303758188972767 p^{10} T^{35} + 4416861108054863891 p^{11} T^{36} - 303863530982546051 p^{12} T^{37} + 21952489659797350 p^{13} T^{38} - 1360893209521850 p^{14} T^{39} + 90548963546410 p^{15} T^{40} - 4955758178821 p^{16} T^{41} + 301181803983 p^{17} T^{42} - 14104194369 p^{18} T^{43} + 775009134 p^{19} T^{44} - 29459962 p^{20} T^{45} + 1445085 p^{21} T^{46} - 40227 p^{22} T^{47} + 1733 p^{23} T^{48} - 27 p^{24} T^{49} + p^{25} T^{50} \)
97 \( 1 - 4 T + 1214 T^{2} - 4447 T^{3} + 718706 T^{4} - 2286485 T^{5} + 276346199 T^{6} - 699796352 T^{7} + 77607777917 T^{8} - 132295351641 T^{9} + 17009876862831 T^{10} - 12029474499315 T^{11} + 3049442183193082 T^{12} + 1253055642752476 T^{13} + 465890314451303799 T^{14} + 712019082451021822 T^{15} + 63182816665108868505 T^{16} + \)\(15\!\cdots\!43\)\( T^{17} + \)\(78\!\cdots\!55\)\( T^{18} + \)\(23\!\cdots\!21\)\( T^{19} + \)\(93\!\cdots\!88\)\( T^{20} + \)\(27\!\cdots\!67\)\( T^{21} + \)\(10\!\cdots\!38\)\( T^{22} + \)\(28\!\cdots\!11\)\( T^{23} + \)\(10\!\cdots\!09\)\( T^{24} + \)\(28\!\cdots\!36\)\( p T^{25} + \)\(10\!\cdots\!09\)\( p T^{26} + \)\(28\!\cdots\!11\)\( p^{2} T^{27} + \)\(10\!\cdots\!38\)\( p^{3} T^{28} + \)\(27\!\cdots\!67\)\( p^{4} T^{29} + \)\(93\!\cdots\!88\)\( p^{5} T^{30} + \)\(23\!\cdots\!21\)\( p^{6} T^{31} + \)\(78\!\cdots\!55\)\( p^{7} T^{32} + \)\(15\!\cdots\!43\)\( p^{8} T^{33} + 63182816665108868505 p^{9} T^{34} + 712019082451021822 p^{10} T^{35} + 465890314451303799 p^{11} T^{36} + 1253055642752476 p^{12} T^{37} + 3049442183193082 p^{13} T^{38} - 12029474499315 p^{14} T^{39} + 17009876862831 p^{15} T^{40} - 132295351641 p^{16} T^{41} + 77607777917 p^{17} T^{42} - 699796352 p^{18} T^{43} + 276346199 p^{19} T^{44} - 2286485 p^{20} T^{45} + 718706 p^{21} T^{46} - 4447 p^{22} T^{47} + 1214 p^{23} T^{48} - 4 p^{24} T^{49} + p^{25} T^{50} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{50} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.77665827740375787441580608098, −1.69828034101949573932502851923, −1.63591772496228960282424887750, −1.53153741202928632495112382485, −1.50918940921191755528648737472, −1.50323182184173167699525152506, −1.46274412810410308467676941216, −1.37823128279622864792447388545, −1.37670636111020642304279110276, −1.25810323930692203126072734414, −1.20674295107536777458686810796, −1.17941779297711817050049152493, −1.14982863448888390230547221877, −1.13429550678216000632682495509, −1.11748954655509267604340453113, −1.11104974692636494669543908376, −1.09255648901912492308391020268, −1.04838368086083808633468380519, −1.02760729263460372720102465612, −1.01967887789016260037552235534, −0.992074706346339689910436579230, −0.969732710232950058690111136428, −0.795886524362455280404955689776, −0.78074516153620293350883922369, −0.50940447870168803271480564601, 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, 0.50940447870168803271480564601, 0.78074516153620293350883922369, 0.795886524362455280404955689776, 0.969732710232950058690111136428, 0.992074706346339689910436579230, 1.01967887789016260037552235534, 1.02760729263460372720102465612, 1.04838368086083808633468380519, 1.09255648901912492308391020268, 1.11104974692636494669543908376, 1.11748954655509267604340453113, 1.13429550678216000632682495509, 1.14982863448888390230547221877, 1.17941779297711817050049152493, 1.20674295107536777458686810796, 1.25810323930692203126072734414, 1.37670636111020642304279110276, 1.37823128279622864792447388545, 1.46274412810410308467676941216, 1.50323182184173167699525152506, 1.50918940921191755528648737472, 1.53153741202928632495112382485, 1.63591772496228960282424887750, 1.69828034101949573932502851923, 1.77665827740375787441580608098

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

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