Properties

Degree 50
Conductor $ 3^{25} \cdot 13^{25} \cdot 103^{25} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 25·3-s + 7·4-s + 7·5-s − 150·6-s + 17·7-s − 27·8-s + 325·9-s + 42·10-s + 21·11-s − 175·12-s + 25·13-s + 102·14-s − 175·15-s − 72·16-s + 14·17-s + 1.95e3·18-s + 12·19-s + 49·20-s − 425·21-s + 126·22-s + 41·23-s + 675·24-s − 23·25-s + 150·26-s − 2.92e3·27-s + 119·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 14.4·3-s + 7/2·4-s + 3.13·5-s − 61.2·6-s + 6.42·7-s − 9.54·8-s + 108.·9-s + 13.2·10-s + 6.33·11-s − 50.5·12-s + 6.93·13-s + 27.2·14-s − 45.1·15-s − 18·16-s + 3.39·17-s + 459.·18-s + 2.75·19-s + 10.9·20-s − 92.7·21-s + 26.8·22-s + 8.54·23-s + 137.·24-s − 4.59·25-s + 29.4·26-s − 562.·27-s + 22.4·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{25} \cdot 13^{25} \cdot 103^{25}\right)^{s/2} \, \Gamma_{\C}(s)^{25} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{25} \cdot 13^{25} \cdot 103^{25}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{25} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(50\)
\( N \)  =  \(3^{25} \cdot 13^{25} \cdot 103^{25}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4017} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(50,\ 3^{25} \cdot 13^{25} \cdot 103^{25} ,\ ( \ : [1/2]^{25} ),\ 1 )$
$L(1)$  $\approx$  $12870.75742$
$L(\frac12)$  $\approx$  $12870.75742$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \(F_p\) is a polynomial of degree 50. If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 49.
$p$$F_p$
bad3 \( ( 1 + T )^{25} \)
13 \( ( 1 - T )^{25} \)
103 \( ( 1 + T )^{25} \)
good2 \( 1 - 3 p T + 29 T^{2} - 105 T^{3} + 337 T^{4} - 947 T^{5} + 2457 T^{6} - 5875 T^{7} + 3315 p^{2} T^{8} - 28249 T^{9} + 57571 T^{10} - 112267 T^{11} + 13205 p^{4} T^{12} - 191909 p T^{13} + 21161 p^{5} T^{14} - 1160405 T^{15} + 970239 p T^{16} - 3166739 T^{17} + 1265579 p^{2} T^{18} - 3963303 p T^{19} + 12195155 T^{20} - 18429969 T^{21} + 27430469 T^{22} - 20091801 p T^{23} + 58060461 T^{24} - 20664065 p^{2} T^{25} + 58060461 p T^{26} - 20091801 p^{3} T^{27} + 27430469 p^{3} T^{28} - 18429969 p^{4} T^{29} + 12195155 p^{5} T^{30} - 3963303 p^{7} T^{31} + 1265579 p^{9} T^{32} - 3166739 p^{8} T^{33} + 970239 p^{10} T^{34} - 1160405 p^{10} T^{35} + 21161 p^{16} T^{36} - 191909 p^{13} T^{37} + 13205 p^{17} T^{38} - 112267 p^{14} T^{39} + 57571 p^{15} T^{40} - 28249 p^{16} T^{41} + 3315 p^{19} T^{42} - 5875 p^{18} T^{43} + 2457 p^{19} T^{44} - 947 p^{20} T^{45} + 337 p^{21} T^{46} - 105 p^{22} T^{47} + 29 p^{23} T^{48} - 3 p^{25} T^{49} + p^{25} T^{50} \)
5 \( 1 - 7 T + 72 T^{2} - 392 T^{3} + 2464 T^{4} - 11268 T^{5} + 54828 T^{6} - 219403 T^{7} + 901671 T^{8} - 3240291 T^{9} + 11755642 T^{10} - 38627232 T^{11} + 127041957 T^{12} - 386790877 T^{13} + 1173826118 T^{14} - 3345416416 T^{15} + 9486985656 T^{16} - 25514532248 T^{17} + 68247631747 T^{18} - 34862963446 p T^{19} + 88585285678 p T^{20} - 1079734530299 T^{21} + 523970652843 p T^{22} - 6116868103759 T^{23} + 14221796548591 T^{24} - 31865004751812 T^{25} + 14221796548591 p T^{26} - 6116868103759 p^{2} T^{27} + 523970652843 p^{4} T^{28} - 1079734530299 p^{4} T^{29} + 88585285678 p^{6} T^{30} - 34862963446 p^{7} T^{31} + 68247631747 p^{7} T^{32} - 25514532248 p^{8} T^{33} + 9486985656 p^{9} T^{34} - 3345416416 p^{10} T^{35} + 1173826118 p^{11} T^{36} - 386790877 p^{12} T^{37} + 127041957 p^{13} T^{38} - 38627232 p^{14} T^{39} + 11755642 p^{15} T^{40} - 3240291 p^{16} T^{41} + 901671 p^{17} T^{42} - 219403 p^{18} T^{43} + 54828 p^{19} T^{44} - 11268 p^{20} T^{45} + 2464 p^{21} T^{46} - 392 p^{22} T^{47} + 72 p^{23} T^{48} - 7 p^{24} T^{49} + p^{25} T^{50} \)
7 \( 1 - 17 T + 234 T^{2} - 2321 T^{3} + 20102 T^{4} - 148888 T^{5} + 1000753 T^{6} - 6082876 T^{7} + 34307874 T^{8} - 179654060 T^{9} + 885229830 T^{10} - 4110618998 T^{11} + 18129276593 T^{12} - 76057915279 T^{13} + 305020281249 T^{14} - 1170859030030 T^{15} + 4315953490874 T^{16} - 15293062862639 T^{17} + 52204697215604 T^{18} - 171810717842810 T^{19} + 545964821160407 T^{20} - 1675968677919891 T^{21} + 4974903440646642 T^{22} - 14283121865396329 T^{23} + 39686770697892093 T^{24} - 106719349305035820 T^{25} + 39686770697892093 p T^{26} - 14283121865396329 p^{2} T^{27} + 4974903440646642 p^{3} T^{28} - 1675968677919891 p^{4} T^{29} + 545964821160407 p^{5} T^{30} - 171810717842810 p^{6} T^{31} + 52204697215604 p^{7} T^{32} - 15293062862639 p^{8} T^{33} + 4315953490874 p^{9} T^{34} - 1170859030030 p^{10} T^{35} + 305020281249 p^{11} T^{36} - 76057915279 p^{12} T^{37} + 18129276593 p^{13} T^{38} - 4110618998 p^{14} T^{39} + 885229830 p^{15} T^{40} - 179654060 p^{16} T^{41} + 34307874 p^{17} T^{42} - 6082876 p^{18} T^{43} + 1000753 p^{19} T^{44} - 148888 p^{20} T^{45} + 20102 p^{21} T^{46} - 2321 p^{22} T^{47} + 234 p^{23} T^{48} - 17 p^{24} T^{49} + p^{25} T^{50} \)
11 \( 1 - 21 T + 340 T^{2} - 4049 T^{3} + 41579 T^{4} - 367728 T^{5} + 2936977 T^{6} - 21271691 T^{7} + 142630215 T^{8} - 80831192 p T^{9} + 473729580 p T^{10} - 28814604167 T^{11} + 151369664131 T^{12} - 757618232274 T^{13} + 3629830675677 T^{14} - 16685465625138 T^{15} + 73836620837458 T^{16} - 28645736408854 p T^{17} + 1300038891379357 T^{18} - 5192082588985555 T^{19} + 20108892269886719 T^{20} - 75589394461955940 T^{21} + 276120819499365509 T^{22} - 89144316324343156 p T^{23} + 3388118565048160145 T^{24} - 11390110643277548030 T^{25} + 3388118565048160145 p T^{26} - 89144316324343156 p^{3} T^{27} + 276120819499365509 p^{3} T^{28} - 75589394461955940 p^{4} T^{29} + 20108892269886719 p^{5} T^{30} - 5192082588985555 p^{6} T^{31} + 1300038891379357 p^{7} T^{32} - 28645736408854 p^{9} T^{33} + 73836620837458 p^{9} T^{34} - 16685465625138 p^{10} T^{35} + 3629830675677 p^{11} T^{36} - 757618232274 p^{12} T^{37} + 151369664131 p^{13} T^{38} - 28814604167 p^{14} T^{39} + 473729580 p^{16} T^{40} - 80831192 p^{17} T^{41} + 142630215 p^{17} T^{42} - 21271691 p^{18} T^{43} + 2936977 p^{19} T^{44} - 367728 p^{20} T^{45} + 41579 p^{21} T^{46} - 4049 p^{22} T^{47} + 340 p^{23} T^{48} - 21 p^{24} T^{49} + p^{25} T^{50} \)
17 \( 1 - 14 T + 305 T^{2} - 3470 T^{3} + 2651 p T^{4} - 433724 T^{5} + 4324570 T^{6} - 36287136 T^{7} + 304256524 T^{8} - 2276191194 T^{9} + 16772809053 T^{10} - 113721812793 T^{11} + 755097463566 T^{12} - 4695913614860 T^{13} + 28537897111709 T^{14} - 164227700627114 T^{15} + 922974358952711 T^{16} - 4946847297533977 T^{17} + 25893723994347641 T^{18} - 129860398820996482 T^{19} + 636191347408142916 T^{20} - 2995200532068771020 T^{21} + 13778526426791207444 T^{22} - 61023578896486044557 T^{23} + \)\(26\!\cdots\!53\)\( T^{24} - \)\(11\!\cdots\!18\)\( T^{25} + \)\(26\!\cdots\!53\)\( p T^{26} - 61023578896486044557 p^{2} T^{27} + 13778526426791207444 p^{3} T^{28} - 2995200532068771020 p^{4} T^{29} + 636191347408142916 p^{5} T^{30} - 129860398820996482 p^{6} T^{31} + 25893723994347641 p^{7} T^{32} - 4946847297533977 p^{8} T^{33} + 922974358952711 p^{9} T^{34} - 164227700627114 p^{10} T^{35} + 28537897111709 p^{11} T^{36} - 4695913614860 p^{12} T^{37} + 755097463566 p^{13} T^{38} - 113721812793 p^{14} T^{39} + 16772809053 p^{15} T^{40} - 2276191194 p^{16} T^{41} + 304256524 p^{17} T^{42} - 36287136 p^{18} T^{43} + 4324570 p^{19} T^{44} - 433724 p^{20} T^{45} + 2651 p^{22} T^{46} - 3470 p^{22} T^{47} + 305 p^{23} T^{48} - 14 p^{24} T^{49} + p^{25} T^{50} \)
19 \( 1 - 12 T + 310 T^{2} - 3050 T^{3} + 45102 T^{4} - 381493 T^{5} + 4196957 T^{6} - 31500365 T^{7} + 285046376 T^{8} - 102170845 p T^{9} + 15217912957 T^{10} - 95566201888 T^{11} + 669437661968 T^{12} - 3921983482406 T^{13} + 25049373371910 T^{14} - 138057754843580 T^{15} + 815202336310217 T^{16} - 4251465254728846 T^{17} + 23441330493302804 T^{18} - 116138504090109336 T^{19} + 602339120122916694 T^{20} - 149569172125675423 p T^{21} + 13938880278497175369 T^{22} - 62690427615860752997 T^{23} + \)\(29\!\cdots\!77\)\( T^{24} - \)\(12\!\cdots\!18\)\( T^{25} + \)\(29\!\cdots\!77\)\( p T^{26} - 62690427615860752997 p^{2} T^{27} + 13938880278497175369 p^{3} T^{28} - 149569172125675423 p^{5} T^{29} + 602339120122916694 p^{5} T^{30} - 116138504090109336 p^{6} T^{31} + 23441330493302804 p^{7} T^{32} - 4251465254728846 p^{8} T^{33} + 815202336310217 p^{9} T^{34} - 138057754843580 p^{10} T^{35} + 25049373371910 p^{11} T^{36} - 3921983482406 p^{12} T^{37} + 669437661968 p^{13} T^{38} - 95566201888 p^{14} T^{39} + 15217912957 p^{15} T^{40} - 102170845 p^{17} T^{41} + 285046376 p^{17} T^{42} - 31500365 p^{18} T^{43} + 4196957 p^{19} T^{44} - 381493 p^{20} T^{45} + 45102 p^{21} T^{46} - 3050 p^{22} T^{47} + 310 p^{23} T^{48} - 12 p^{24} T^{49} + p^{25} T^{50} \)
23 \( 1 - 41 T + 1153 T^{2} - 24361 T^{3} + 430931 T^{4} - 6583733 T^{5} + 89644617 T^{6} - 1104648939 T^{7} + 12493918942 T^{8} - 130850991838 T^{9} + 1278961930345 T^{10} - 11731527696600 T^{11} + 101481214946512 T^{12} - 830930580699258 T^{13} + 6461052358604229 T^{14} - 47832044342337036 T^{15} + 337899105222489276 T^{16} - 2281883271253272594 T^{17} + 14754325384391061150 T^{18} - 91456199711641375877 T^{19} + \)\(54\!\cdots\!99\)\( T^{20} - \)\(31\!\cdots\!67\)\( T^{21} + \)\(17\!\cdots\!94\)\( T^{22} - \)\(90\!\cdots\!35\)\( T^{23} + \)\(45\!\cdots\!51\)\( T^{24} - \)\(22\!\cdots\!14\)\( T^{25} + \)\(45\!\cdots\!51\)\( p T^{26} - \)\(90\!\cdots\!35\)\( p^{2} T^{27} + \)\(17\!\cdots\!94\)\( p^{3} T^{28} - \)\(31\!\cdots\!67\)\( p^{4} T^{29} + \)\(54\!\cdots\!99\)\( p^{5} T^{30} - 91456199711641375877 p^{6} T^{31} + 14754325384391061150 p^{7} T^{32} - 2281883271253272594 p^{8} T^{33} + 337899105222489276 p^{9} T^{34} - 47832044342337036 p^{10} T^{35} + 6461052358604229 p^{11} T^{36} - 830930580699258 p^{12} T^{37} + 101481214946512 p^{13} T^{38} - 11731527696600 p^{14} T^{39} + 1278961930345 p^{15} T^{40} - 130850991838 p^{16} T^{41} + 12493918942 p^{17} T^{42} - 1104648939 p^{18} T^{43} + 89644617 p^{19} T^{44} - 6583733 p^{20} T^{45} + 430931 p^{21} T^{46} - 24361 p^{22} T^{47} + 1153 p^{23} T^{48} - 41 p^{24} T^{49} + p^{25} T^{50} \)
29 \( 1 - 22 T + 656 T^{2} - 10300 T^{3} + 181875 T^{4} - 2259413 T^{5} + 29960790 T^{6} - 310441300 T^{7} + 3376001029 T^{8} - 1037842900 p T^{9} + 280916211513 T^{10} - 2197822561055 T^{11} + 18113421412442 T^{12} - 126005707401429 T^{13} + 936198011349249 T^{14} - 5842313493136403 T^{15} + 39843880650944655 T^{16} - 224529081468421807 T^{17} + 1432897947695388074 T^{18} - 7345034412040939105 T^{19} + 44964066613705879528 T^{20} - \)\(21\!\cdots\!86\)\( T^{21} + \)\(12\!\cdots\!12\)\( T^{22} - \)\(58\!\cdots\!17\)\( T^{23} + \)\(12\!\cdots\!16\)\( p T^{24} - \)\(16\!\cdots\!98\)\( T^{25} + \)\(12\!\cdots\!16\)\( p^{2} T^{26} - \)\(58\!\cdots\!17\)\( p^{2} T^{27} + \)\(12\!\cdots\!12\)\( p^{3} T^{28} - \)\(21\!\cdots\!86\)\( p^{4} T^{29} + 44964066613705879528 p^{5} T^{30} - 7345034412040939105 p^{6} T^{31} + 1432897947695388074 p^{7} T^{32} - 224529081468421807 p^{8} T^{33} + 39843880650944655 p^{9} T^{34} - 5842313493136403 p^{10} T^{35} + 936198011349249 p^{11} T^{36} - 126005707401429 p^{12} T^{37} + 18113421412442 p^{13} T^{38} - 2197822561055 p^{14} T^{39} + 280916211513 p^{15} T^{40} - 1037842900 p^{17} T^{41} + 3376001029 p^{17} T^{42} - 310441300 p^{18} T^{43} + 29960790 p^{19} T^{44} - 2259413 p^{20} T^{45} + 181875 p^{21} T^{46} - 10300 p^{22} T^{47} + 656 p^{23} T^{48} - 22 p^{24} T^{49} + p^{25} T^{50} \)
31 \( 1 - 14 T + 470 T^{2} - 5975 T^{3} + 111225 T^{4} - 1274741 T^{5} + 563980 p T^{6} - 181162763 T^{7} + 65859794 p T^{8} - 19265526371 T^{9} + 188404699324 T^{10} - 1632417143863 T^{11} + 14294253294284 T^{12} - 114616108101197 T^{13} + 916645113784616 T^{14} - 6849112784395200 T^{15} + 50701174443126814 T^{16} - 355097140368252419 T^{17} + 2456017945668855817 T^{18} - 16200727215232313055 T^{19} + \)\(10\!\cdots\!57\)\( T^{20} - \)\(65\!\cdots\!22\)\( T^{21} + \)\(40\!\cdots\!38\)\( T^{22} - \)\(23\!\cdots\!04\)\( T^{23} + \)\(13\!\cdots\!42\)\( T^{24} - \)\(77\!\cdots\!64\)\( T^{25} + \)\(13\!\cdots\!42\)\( p T^{26} - \)\(23\!\cdots\!04\)\( p^{2} T^{27} + \)\(40\!\cdots\!38\)\( p^{3} T^{28} - \)\(65\!\cdots\!22\)\( p^{4} T^{29} + \)\(10\!\cdots\!57\)\( p^{5} T^{30} - 16200727215232313055 p^{6} T^{31} + 2456017945668855817 p^{7} T^{32} - 355097140368252419 p^{8} T^{33} + 50701174443126814 p^{9} T^{34} - 6849112784395200 p^{10} T^{35} + 916645113784616 p^{11} T^{36} - 114616108101197 p^{12} T^{37} + 14294253294284 p^{13} T^{38} - 1632417143863 p^{14} T^{39} + 188404699324 p^{15} T^{40} - 19265526371 p^{16} T^{41} + 65859794 p^{18} T^{42} - 181162763 p^{18} T^{43} + 563980 p^{20} T^{44} - 1274741 p^{20} T^{45} + 111225 p^{21} T^{46} - 5975 p^{22} T^{47} + 470 p^{23} T^{48} - 14 p^{24} T^{49} + p^{25} T^{50} \)
37 \( 1 + 6 T + 442 T^{2} + 2733 T^{3} + 102070 T^{4} + 637249 T^{5} + 16201101 T^{6} + 100687180 T^{7} + 1969697795 T^{8} + 12063718254 T^{9} + 194291088462 T^{10} + 1164206277174 T^{11} + 16110996946926 T^{12} + 93943821558654 T^{13} + 1150390358759125 T^{14} + 6501364080608448 T^{15} + 71966372735936543 T^{16} + 392941222962472873 T^{17} + 3995522428107922807 T^{18} + 21021101624097774734 T^{19} + \)\(19\!\cdots\!83\)\( T^{20} + \)\(10\!\cdots\!06\)\( T^{21} + \)\(89\!\cdots\!85\)\( T^{22} + \)\(43\!\cdots\!39\)\( T^{23} + \)\(36\!\cdots\!64\)\( T^{24} + \)\(16\!\cdots\!24\)\( T^{25} + \)\(36\!\cdots\!64\)\( p T^{26} + \)\(43\!\cdots\!39\)\( p^{2} T^{27} + \)\(89\!\cdots\!85\)\( p^{3} T^{28} + \)\(10\!\cdots\!06\)\( p^{4} T^{29} + \)\(19\!\cdots\!83\)\( p^{5} T^{30} + 21021101624097774734 p^{6} T^{31} + 3995522428107922807 p^{7} T^{32} + 392941222962472873 p^{8} T^{33} + 71966372735936543 p^{9} T^{34} + 6501364080608448 p^{10} T^{35} + 1150390358759125 p^{11} T^{36} + 93943821558654 p^{12} T^{37} + 16110996946926 p^{13} T^{38} + 1164206277174 p^{14} T^{39} + 194291088462 p^{15} T^{40} + 12063718254 p^{16} T^{41} + 1969697795 p^{17} T^{42} + 100687180 p^{18} T^{43} + 16201101 p^{19} T^{44} + 637249 p^{20} T^{45} + 102070 p^{21} T^{46} + 2733 p^{22} T^{47} + 442 p^{23} T^{48} + 6 p^{24} T^{49} + p^{25} T^{50} \)
41 \( 1 - 33 T + 1006 T^{2} - 21262 T^{3} + 10008 p T^{4} - 6692288 T^{5} + 101230278 T^{6} - 1379624807 T^{7} + 17675384569 T^{8} - 5124649912 p T^{9} + 2371468711797 T^{10} - 25241237434298 T^{11} + 256953574117468 T^{12} - 2491649439682041 T^{13} + 23232297820307307 T^{14} - 207719125211141655 T^{15} + 1792856747558270632 T^{16} - 14906559188713848563 T^{17} + \)\(11\!\cdots\!17\)\( T^{18} - \)\(93\!\cdots\!09\)\( T^{19} + \)\(70\!\cdots\!70\)\( T^{20} - \)\(51\!\cdots\!37\)\( T^{21} + \)\(36\!\cdots\!03\)\( T^{22} - \)\(25\!\cdots\!77\)\( T^{23} + \)\(16\!\cdots\!04\)\( T^{24} - \)\(10\!\cdots\!64\)\( T^{25} + \)\(16\!\cdots\!04\)\( p T^{26} - \)\(25\!\cdots\!77\)\( p^{2} T^{27} + \)\(36\!\cdots\!03\)\( p^{3} T^{28} - \)\(51\!\cdots\!37\)\( p^{4} T^{29} + \)\(70\!\cdots\!70\)\( p^{5} T^{30} - \)\(93\!\cdots\!09\)\( p^{6} T^{31} + \)\(11\!\cdots\!17\)\( p^{7} T^{32} - 14906559188713848563 p^{8} T^{33} + 1792856747558270632 p^{9} T^{34} - 207719125211141655 p^{10} T^{35} + 23232297820307307 p^{11} T^{36} - 2491649439682041 p^{12} T^{37} + 256953574117468 p^{13} T^{38} - 25241237434298 p^{14} T^{39} + 2371468711797 p^{15} T^{40} - 5124649912 p^{17} T^{41} + 17675384569 p^{17} T^{42} - 1379624807 p^{18} T^{43} + 101230278 p^{19} T^{44} - 6692288 p^{20} T^{45} + 10008 p^{22} T^{46} - 21262 p^{22} T^{47} + 1006 p^{23} T^{48} - 33 p^{24} T^{49} + p^{25} T^{50} \)
43 \( 1 - 35 T + 1168 T^{2} - 26232 T^{3} + 542903 T^{4} - 9306845 T^{5} + 148198223 T^{6} - 2092398639 T^{7} + 27722359392 T^{8} - 336072774175 T^{9} + 3855501171841 T^{10} - 41185277405547 T^{11} + 419148726038672 T^{12} - 4016248447167268 T^{13} + 36871614422651770 T^{14} - 321249514758078721 T^{15} + 2695973896150276456 T^{16} - 21615255028544588929 T^{17} + \)\(16\!\cdots\!33\)\( T^{18} - \)\(12\!\cdots\!25\)\( T^{19} + \)\(91\!\cdots\!94\)\( T^{20} - \)\(15\!\cdots\!01\)\( p T^{21} + \)\(44\!\cdots\!38\)\( T^{22} - \)\(30\!\cdots\!92\)\( T^{23} + \)\(20\!\cdots\!47\)\( T^{24} - \)\(13\!\cdots\!98\)\( T^{25} + \)\(20\!\cdots\!47\)\( p T^{26} - \)\(30\!\cdots\!92\)\( p^{2} T^{27} + \)\(44\!\cdots\!38\)\( p^{3} T^{28} - \)\(15\!\cdots\!01\)\( p^{5} T^{29} + \)\(91\!\cdots\!94\)\( p^{5} T^{30} - \)\(12\!\cdots\!25\)\( p^{6} T^{31} + \)\(16\!\cdots\!33\)\( p^{7} T^{32} - 21615255028544588929 p^{8} T^{33} + 2695973896150276456 p^{9} T^{34} - 321249514758078721 p^{10} T^{35} + 36871614422651770 p^{11} T^{36} - 4016248447167268 p^{12} T^{37} + 419148726038672 p^{13} T^{38} - 41185277405547 p^{14} T^{39} + 3855501171841 p^{15} T^{40} - 336072774175 p^{16} T^{41} + 27722359392 p^{17} T^{42} - 2092398639 p^{18} T^{43} + 148198223 p^{19} T^{44} - 9306845 p^{20} T^{45} + 542903 p^{21} T^{46} - 26232 p^{22} T^{47} + 1168 p^{23} T^{48} - 35 p^{24} T^{49} + p^{25} T^{50} \)
47 \( 1 - 48 T + 1592 T^{2} - 38856 T^{3} + 794451 T^{4} - 13915396 T^{5} + 217646446 T^{6} - 3077058353 T^{7} + 40124486072 T^{8} - 486340364968 T^{9} + 5545740631753 T^{10} - 59807003951818 T^{11} + 614857158532707 T^{12} - 6047430964254580 T^{13} + 57214582129839338 T^{14} - 521866399676896028 T^{15} + 4606238002521722102 T^{16} - 39392930235720874337 T^{17} + \)\(32\!\cdots\!40\)\( T^{18} - \)\(26\!\cdots\!65\)\( T^{19} + \)\(20\!\cdots\!32\)\( T^{20} - \)\(15\!\cdots\!85\)\( T^{21} + \)\(11\!\cdots\!64\)\( T^{22} - \)\(86\!\cdots\!88\)\( T^{23} + \)\(61\!\cdots\!28\)\( T^{24} - \)\(42\!\cdots\!32\)\( T^{25} + \)\(61\!\cdots\!28\)\( p T^{26} - \)\(86\!\cdots\!88\)\( p^{2} T^{27} + \)\(11\!\cdots\!64\)\( p^{3} T^{28} - \)\(15\!\cdots\!85\)\( p^{4} T^{29} + \)\(20\!\cdots\!32\)\( p^{5} T^{30} - \)\(26\!\cdots\!65\)\( p^{6} T^{31} + \)\(32\!\cdots\!40\)\( p^{7} T^{32} - 39392930235720874337 p^{8} T^{33} + 4606238002521722102 p^{9} T^{34} - 521866399676896028 p^{10} T^{35} + 57214582129839338 p^{11} T^{36} - 6047430964254580 p^{12} T^{37} + 614857158532707 p^{13} T^{38} - 59807003951818 p^{14} T^{39} + 5545740631753 p^{15} T^{40} - 486340364968 p^{16} T^{41} + 40124486072 p^{17} T^{42} - 3077058353 p^{18} T^{43} + 217646446 p^{19} T^{44} - 13915396 p^{20} T^{45} + 794451 p^{21} T^{46} - 38856 p^{22} T^{47} + 1592 p^{23} T^{48} - 48 p^{24} T^{49} + p^{25} T^{50} \)
53 \( 1 - 18 T + 784 T^{2} - 13783 T^{3} + 319868 T^{4} - 5208208 T^{5} + 1676317 p T^{6} - 1304592698 T^{7} + 18561304660 T^{8} - 244705586597 T^{9} + 3074662933262 T^{10} - 36680865518185 T^{11} + 418181649762549 T^{12} - 4566665410218196 T^{13} + 47914588177878469 T^{14} - 483902434555434299 T^{15} + 4714778760736304660 T^{16} - 44374798652206976916 T^{17} + \)\(40\!\cdots\!87\)\( T^{18} - \)\(35\!\cdots\!36\)\( T^{19} + \)\(30\!\cdots\!81\)\( T^{20} - \)\(25\!\cdots\!40\)\( T^{21} + \)\(20\!\cdots\!24\)\( T^{22} - \)\(15\!\cdots\!03\)\( T^{23} + \)\(12\!\cdots\!62\)\( T^{24} - \)\(89\!\cdots\!26\)\( T^{25} + \)\(12\!\cdots\!62\)\( p T^{26} - \)\(15\!\cdots\!03\)\( p^{2} T^{27} + \)\(20\!\cdots\!24\)\( p^{3} T^{28} - \)\(25\!\cdots\!40\)\( p^{4} T^{29} + \)\(30\!\cdots\!81\)\( p^{5} T^{30} - \)\(35\!\cdots\!36\)\( p^{6} T^{31} + \)\(40\!\cdots\!87\)\( p^{7} T^{32} - 44374798652206976916 p^{8} T^{33} + 4714778760736304660 p^{9} T^{34} - 483902434555434299 p^{10} T^{35} + 47914588177878469 p^{11} T^{36} - 4566665410218196 p^{12} T^{37} + 418181649762549 p^{13} T^{38} - 36680865518185 p^{14} T^{39} + 3074662933262 p^{15} T^{40} - 244705586597 p^{16} T^{41} + 18561304660 p^{17} T^{42} - 1304592698 p^{18} T^{43} + 1676317 p^{20} T^{44} - 5208208 p^{20} T^{45} + 319868 p^{21} T^{46} - 13783 p^{22} T^{47} + 784 p^{23} T^{48} - 18 p^{24} T^{49} + p^{25} T^{50} \)
59 \( 1 - 46 T + 1679 T^{2} - 44064 T^{3} + 1007322 T^{4} - 19621723 T^{5} + 347151251 T^{6} - 5539091567 T^{7} + 82172338661 T^{8} - 1130116320606 T^{9} + 14665467151684 T^{10} - 179302871463591 T^{11} + 2089508945551145 T^{12} - 23186023071059102 T^{13} + 247053685509672096 T^{14} - 2525700657472571197 T^{15} + 24938088433429271547 T^{16} - \)\(23\!\cdots\!30\)\( T^{17} + \)\(21\!\cdots\!95\)\( T^{18} - \)\(19\!\cdots\!69\)\( T^{19} + \)\(17\!\cdots\!06\)\( T^{20} - \)\(14\!\cdots\!27\)\( T^{21} + \)\(12\!\cdots\!22\)\( T^{22} - \)\(98\!\cdots\!04\)\( T^{23} + \)\(78\!\cdots\!01\)\( T^{24} - \)\(60\!\cdots\!24\)\( T^{25} + \)\(78\!\cdots\!01\)\( p T^{26} - \)\(98\!\cdots\!04\)\( p^{2} T^{27} + \)\(12\!\cdots\!22\)\( p^{3} T^{28} - \)\(14\!\cdots\!27\)\( p^{4} T^{29} + \)\(17\!\cdots\!06\)\( p^{5} T^{30} - \)\(19\!\cdots\!69\)\( p^{6} T^{31} + \)\(21\!\cdots\!95\)\( p^{7} T^{32} - \)\(23\!\cdots\!30\)\( p^{8} T^{33} + 24938088433429271547 p^{9} T^{34} - 2525700657472571197 p^{10} T^{35} + 247053685509672096 p^{11} T^{36} - 23186023071059102 p^{12} T^{37} + 2089508945551145 p^{13} T^{38} - 179302871463591 p^{14} T^{39} + 14665467151684 p^{15} T^{40} - 1130116320606 p^{16} T^{41} + 82172338661 p^{17} T^{42} - 5539091567 p^{18} T^{43} + 347151251 p^{19} T^{44} - 19621723 p^{20} T^{45} + 1007322 p^{21} T^{46} - 44064 p^{22} T^{47} + 1679 p^{23} T^{48} - 46 p^{24} T^{49} + p^{25} T^{50} \)
61 \( 1 + 19 T + 874 T^{2} + 13652 T^{3} + 369935 T^{4} + 5026284 T^{5} + 103482727 T^{6} + 1261113997 T^{7} + 21685312753 T^{8} + 241468201451 T^{9} + 3637455357034 T^{10} + 37465174332428 T^{11} + 508553413154016 T^{12} + 4887339329711174 T^{13} + 60882936829362133 T^{14} + 549443367597700925 T^{15} + 6361013248955529690 T^{16} + 54170228067545659163 T^{17} + \)\(58\!\cdots\!06\)\( T^{18} + \)\(47\!\cdots\!49\)\( T^{19} + \)\(48\!\cdots\!18\)\( T^{20} + \)\(37\!\cdots\!00\)\( T^{21} + \)\(36\!\cdots\!05\)\( T^{22} + \)\(26\!\cdots\!17\)\( T^{23} + \)\(24\!\cdots\!08\)\( T^{24} + \)\(16\!\cdots\!66\)\( T^{25} + \)\(24\!\cdots\!08\)\( p T^{26} + \)\(26\!\cdots\!17\)\( p^{2} T^{27} + \)\(36\!\cdots\!05\)\( p^{3} T^{28} + \)\(37\!\cdots\!00\)\( p^{4} T^{29} + \)\(48\!\cdots\!18\)\( p^{5} T^{30} + \)\(47\!\cdots\!49\)\( p^{6} T^{31} + \)\(58\!\cdots\!06\)\( p^{7} T^{32} + 54170228067545659163 p^{8} T^{33} + 6361013248955529690 p^{9} T^{34} + 549443367597700925 p^{10} T^{35} + 60882936829362133 p^{11} T^{36} + 4887339329711174 p^{12} T^{37} + 508553413154016 p^{13} T^{38} + 37465174332428 p^{14} T^{39} + 3637455357034 p^{15} T^{40} + 241468201451 p^{16} T^{41} + 21685312753 p^{17} T^{42} + 1261113997 p^{18} T^{43} + 103482727 p^{19} T^{44} + 5026284 p^{20} T^{45} + 369935 p^{21} T^{46} + 13652 p^{22} T^{47} + 874 p^{23} T^{48} + 19 p^{24} T^{49} + p^{25} T^{50} \)
67 \( 1 - 16 T + 947 T^{2} - 12219 T^{3} + 427655 T^{4} - 4671988 T^{5} + 126171703 T^{6} - 1198818882 T^{7} + 27698657384 T^{8} - 232475561272 T^{9} + 4854489879083 T^{10} - 36343469865942 T^{11} + 709658568161000 T^{12} - 4772209295069194 T^{13} + 89135724733892179 T^{14} - 541678108764113347 T^{15} + 9824055413571677714 T^{16} - 54306715984986083932 T^{17} + \)\(96\!\cdots\!64\)\( T^{18} - 72992271740543089094 p T^{19} + \)\(85\!\cdots\!34\)\( T^{20} - \)\(40\!\cdots\!54\)\( T^{21} + \)\(68\!\cdots\!91\)\( T^{22} - \)\(30\!\cdots\!80\)\( T^{23} + \)\(50\!\cdots\!95\)\( T^{24} - \)\(21\!\cdots\!88\)\( T^{25} + \)\(50\!\cdots\!95\)\( p T^{26} - \)\(30\!\cdots\!80\)\( p^{2} T^{27} + \)\(68\!\cdots\!91\)\( p^{3} T^{28} - \)\(40\!\cdots\!54\)\( p^{4} T^{29} + \)\(85\!\cdots\!34\)\( p^{5} T^{30} - 72992271740543089094 p^{7} T^{31} + \)\(96\!\cdots\!64\)\( p^{7} T^{32} - 54306715984986083932 p^{8} T^{33} + 9824055413571677714 p^{9} T^{34} - 541678108764113347 p^{10} T^{35} + 89135724733892179 p^{11} T^{36} - 4772209295069194 p^{12} T^{37} + 709658568161000 p^{13} T^{38} - 36343469865942 p^{14} T^{39} + 4854489879083 p^{15} T^{40} - 232475561272 p^{16} T^{41} + 27698657384 p^{17} T^{42} - 1198818882 p^{18} T^{43} + 126171703 p^{19} T^{44} - 4671988 p^{20} T^{45} + 427655 p^{21} T^{46} - 12219 p^{22} T^{47} + 947 p^{23} T^{48} - 16 p^{24} T^{49} + p^{25} T^{50} \)
71 \( 1 - 60 T + 2583 T^{2} - 80165 T^{3} + 2094718 T^{4} - 46255645 T^{5} + 910141275 T^{6} - 15998939286 T^{7} + 258263963279 T^{8} - 3835559674937 T^{9} + 53370583899922 T^{10} - 9813926067349 p T^{11} + 8656040077695160 T^{12} - 102403043364143445 T^{13} + 1166846291653235229 T^{14} - 12800336895026782613 T^{15} + \)\(13\!\cdots\!24\)\( T^{16} - \)\(14\!\cdots\!53\)\( T^{17} + \)\(14\!\cdots\!40\)\( T^{18} - \)\(13\!\cdots\!08\)\( T^{19} + \)\(13\!\cdots\!05\)\( T^{20} - \)\(12\!\cdots\!90\)\( T^{21} + \)\(11\!\cdots\!95\)\( T^{22} - \)\(10\!\cdots\!93\)\( T^{23} + \)\(87\!\cdots\!05\)\( T^{24} - \)\(74\!\cdots\!92\)\( T^{25} + \)\(87\!\cdots\!05\)\( p T^{26} - \)\(10\!\cdots\!93\)\( p^{2} T^{27} + \)\(11\!\cdots\!95\)\( p^{3} T^{28} - \)\(12\!\cdots\!90\)\( p^{4} T^{29} + \)\(13\!\cdots\!05\)\( p^{5} T^{30} - \)\(13\!\cdots\!08\)\( p^{6} T^{31} + \)\(14\!\cdots\!40\)\( p^{7} T^{32} - \)\(14\!\cdots\!53\)\( p^{8} T^{33} + \)\(13\!\cdots\!24\)\( p^{9} T^{34} - 12800336895026782613 p^{10} T^{35} + 1166846291653235229 p^{11} T^{36} - 102403043364143445 p^{12} T^{37} + 8656040077695160 p^{13} T^{38} - 9813926067349 p^{15} T^{39} + 53370583899922 p^{15} T^{40} - 3835559674937 p^{16} T^{41} + 258263963279 p^{17} T^{42} - 15998939286 p^{18} T^{43} + 910141275 p^{19} T^{44} - 46255645 p^{20} T^{45} + 2094718 p^{21} T^{46} - 80165 p^{22} T^{47} + 2583 p^{23} T^{48} - 60 p^{24} T^{49} + p^{25} T^{50} \)
73 \( 1 + 14 T + 911 T^{2} + 11262 T^{3} + 415042 T^{4} + 4706800 T^{5} + 127667307 T^{6} + 1357038156 T^{7} + 29923897648 T^{8} + 301827856504 T^{9} + 5697697588450 T^{10} + 54916014021861 T^{11} + 916006659862177 T^{12} + 8467500787206482 T^{13} + 127501989538955282 T^{14} + 1132219965834306216 T^{15} + 15629781340150684005 T^{16} + \)\(13\!\cdots\!03\)\( T^{17} + \)\(17\!\cdots\!15\)\( T^{18} + \)\(13\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!20\)\( T^{20} + \)\(13\!\cdots\!62\)\( T^{21} + \)\(14\!\cdots\!24\)\( T^{22} + \)\(11\!\cdots\!77\)\( T^{23} + \)\(11\!\cdots\!58\)\( T^{24} + \)\(85\!\cdots\!78\)\( T^{25} + \)\(11\!\cdots\!58\)\( p T^{26} + \)\(11\!\cdots\!77\)\( p^{2} T^{27} + \)\(14\!\cdots\!24\)\( p^{3} T^{28} + \)\(13\!\cdots\!62\)\( p^{4} T^{29} + \)\(16\!\cdots\!20\)\( p^{5} T^{30} + \)\(13\!\cdots\!60\)\( p^{6} T^{31} + \)\(17\!\cdots\!15\)\( p^{7} T^{32} + \)\(13\!\cdots\!03\)\( p^{8} T^{33} + 15629781340150684005 p^{9} T^{34} + 1132219965834306216 p^{10} T^{35} + 127501989538955282 p^{11} T^{36} + 8467500787206482 p^{12} T^{37} + 916006659862177 p^{13} T^{38} + 54916014021861 p^{14} T^{39} + 5697697588450 p^{15} T^{40} + 301827856504 p^{16} T^{41} + 29923897648 p^{17} T^{42} + 1357038156 p^{18} T^{43} + 127667307 p^{19} T^{44} + 4706800 p^{20} T^{45} + 415042 p^{21} T^{46} + 11262 p^{22} T^{47} + 911 p^{23} T^{48} + 14 p^{24} T^{49} + p^{25} T^{50} \)
79 \( 1 - 7 T + 986 T^{2} - 4553 T^{3} + 475483 T^{4} - 1210693 T^{5} + 152162571 T^{6} - 99068615 T^{7} + 36748385826 T^{8} + 40647273025 T^{9} + 7192542232533 T^{10} + 19419484266014 T^{11} + 1193442343869684 T^{12} + 4854876619516354 T^{13} + 172986168303868615 T^{14} + 889551037146469336 T^{15} + 22334581234563623250 T^{16} + \)\(13\!\cdots\!04\)\( T^{17} + \)\(25\!\cdots\!39\)\( T^{18} + \)\(16\!\cdots\!77\)\( T^{19} + \)\(27\!\cdots\!14\)\( T^{20} + \)\(17\!\cdots\!27\)\( T^{21} + \)\(26\!\cdots\!55\)\( T^{22} + \)\(16\!\cdots\!93\)\( T^{23} + \)\(22\!\cdots\!25\)\( T^{24} + \)\(13\!\cdots\!52\)\( T^{25} + \)\(22\!\cdots\!25\)\( p T^{26} + \)\(16\!\cdots\!93\)\( p^{2} T^{27} + \)\(26\!\cdots\!55\)\( p^{3} T^{28} + \)\(17\!\cdots\!27\)\( p^{4} T^{29} + \)\(27\!\cdots\!14\)\( p^{5} T^{30} + \)\(16\!\cdots\!77\)\( p^{6} T^{31} + \)\(25\!\cdots\!39\)\( p^{7} T^{32} + \)\(13\!\cdots\!04\)\( p^{8} T^{33} + 22334581234563623250 p^{9} T^{34} + 889551037146469336 p^{10} T^{35} + 172986168303868615 p^{11} T^{36} + 4854876619516354 p^{12} T^{37} + 1193442343869684 p^{13} T^{38} + 19419484266014 p^{14} T^{39} + 7192542232533 p^{15} T^{40} + 40647273025 p^{16} T^{41} + 36748385826 p^{17} T^{42} - 99068615 p^{18} T^{43} + 152162571 p^{19} T^{44} - 1210693 p^{20} T^{45} + 475483 p^{21} T^{46} - 4553 p^{22} T^{47} + 986 p^{23} T^{48} - 7 p^{24} T^{49} + p^{25} T^{50} \)
83 \( 1 - 23 T + 937 T^{2} - 14915 T^{3} + 362666 T^{4} - 4502908 T^{5} + 86718085 T^{6} - 10970129 p T^{7} + 15698151794 T^{8} - 147334229824 T^{9} + 2399434157193 T^{10} - 20746177631634 T^{11} + 325042513848580 T^{12} - 2627321425854298 T^{13} + 39904514916911519 T^{14} - 304202883872281777 T^{15} + 4503807826008953087 T^{16} - 32603094140662462387 T^{17} + \)\(47\!\cdots\!53\)\( T^{18} - \)\(32\!\cdots\!26\)\( T^{19} + \)\(46\!\cdots\!01\)\( T^{20} - \)\(30\!\cdots\!18\)\( T^{21} + \)\(43\!\cdots\!15\)\( T^{22} - \)\(27\!\cdots\!69\)\( T^{23} + \)\(37\!\cdots\!33\)\( T^{24} - \)\(23\!\cdots\!04\)\( T^{25} + \)\(37\!\cdots\!33\)\( p T^{26} - \)\(27\!\cdots\!69\)\( p^{2} T^{27} + \)\(43\!\cdots\!15\)\( p^{3} T^{28} - \)\(30\!\cdots\!18\)\( p^{4} T^{29} + \)\(46\!\cdots\!01\)\( p^{5} T^{30} - \)\(32\!\cdots\!26\)\( p^{6} T^{31} + \)\(47\!\cdots\!53\)\( p^{7} T^{32} - 32603094140662462387 p^{8} T^{33} + 4503807826008953087 p^{9} T^{34} - 304202883872281777 p^{10} T^{35} + 39904514916911519 p^{11} T^{36} - 2627321425854298 p^{12} T^{37} + 325042513848580 p^{13} T^{38} - 20746177631634 p^{14} T^{39} + 2399434157193 p^{15} T^{40} - 147334229824 p^{16} T^{41} + 15698151794 p^{17} T^{42} - 10970129 p^{19} T^{43} + 86718085 p^{19} T^{44} - 4502908 p^{20} T^{45} + 362666 p^{21} T^{46} - 14915 p^{22} T^{47} + 937 p^{23} T^{48} - 23 p^{24} T^{49} + p^{25} T^{50} \)
89 \( 1 - 10 T + 1307 T^{2} - 13846 T^{3} + 872154 T^{4} - 9477825 T^{5} + 393640106 T^{6} - 4288523400 T^{7} + 134280384088 T^{8} - 1443433313832 T^{9} + 36697340148124 T^{10} - 384966058385607 T^{11} + 8322573580832396 T^{12} - 84547086532939664 T^{13} + 1602960197333801904 T^{14} - 15680870326932680769 T^{15} + \)\(26\!\cdots\!70\)\( T^{16} - \)\(24\!\cdots\!04\)\( T^{17} + \)\(38\!\cdots\!56\)\( T^{18} - \)\(34\!\cdots\!41\)\( T^{19} + \)\(49\!\cdots\!94\)\( T^{20} - \)\(42\!\cdots\!55\)\( T^{21} + \)\(55\!\cdots\!58\)\( T^{22} - \)\(44\!\cdots\!21\)\( T^{23} + \)\(55\!\cdots\!46\)\( T^{24} - \)\(42\!\cdots\!00\)\( T^{25} + \)\(55\!\cdots\!46\)\( p T^{26} - \)\(44\!\cdots\!21\)\( p^{2} T^{27} + \)\(55\!\cdots\!58\)\( p^{3} T^{28} - \)\(42\!\cdots\!55\)\( p^{4} T^{29} + \)\(49\!\cdots\!94\)\( p^{5} T^{30} - \)\(34\!\cdots\!41\)\( p^{6} T^{31} + \)\(38\!\cdots\!56\)\( p^{7} T^{32} - \)\(24\!\cdots\!04\)\( p^{8} T^{33} + \)\(26\!\cdots\!70\)\( p^{9} T^{34} - 15680870326932680769 p^{10} T^{35} + 1602960197333801904 p^{11} T^{36} - 84547086532939664 p^{12} T^{37} + 8322573580832396 p^{13} T^{38} - 384966058385607 p^{14} T^{39} + 36697340148124 p^{15} T^{40} - 1443433313832 p^{16} T^{41} + 134280384088 p^{17} T^{42} - 4288523400 p^{18} T^{43} + 393640106 p^{19} T^{44} - 9477825 p^{20} T^{45} + 872154 p^{21} T^{46} - 13846 p^{22} T^{47} + 1307 p^{23} T^{48} - 10 p^{24} T^{49} + p^{25} T^{50} \)
97 \( 1 + 10 T + 751 T^{2} + 7197 T^{3} + 304528 T^{4} + 2583430 T^{5} + 84992870 T^{6} + 610714298 T^{7} + 18110856994 T^{8} + 104249619854 T^{9} + 3132110088404 T^{10} + 13067547455318 T^{11} + 458918378583482 T^{12} + 1086956428100898 T^{13} + 58831856054248674 T^{14} + 16364624495243206 T^{15} + 6766031235960187400 T^{16} - 14532562789305690374 T^{17} + \)\(71\!\cdots\!54\)\( T^{18} - \)\(34\!\cdots\!45\)\( T^{19} + \)\(70\!\cdots\!58\)\( T^{20} - \)\(54\!\cdots\!04\)\( T^{21} + \)\(67\!\cdots\!36\)\( T^{22} - \)\(67\!\cdots\!90\)\( T^{23} + \)\(64\!\cdots\!92\)\( T^{24} - \)\(71\!\cdots\!00\)\( T^{25} + \)\(64\!\cdots\!92\)\( p T^{26} - \)\(67\!\cdots\!90\)\( p^{2} T^{27} + \)\(67\!\cdots\!36\)\( p^{3} T^{28} - \)\(54\!\cdots\!04\)\( p^{4} T^{29} + \)\(70\!\cdots\!58\)\( p^{5} T^{30} - \)\(34\!\cdots\!45\)\( p^{6} T^{31} + \)\(71\!\cdots\!54\)\( p^{7} T^{32} - 14532562789305690374 p^{8} T^{33} + 6766031235960187400 p^{9} T^{34} + 16364624495243206 p^{10} T^{35} + 58831856054248674 p^{11} T^{36} + 1086956428100898 p^{12} T^{37} + 458918378583482 p^{13} T^{38} + 13067547455318 p^{14} T^{39} + 3132110088404 p^{15} T^{40} + 104249619854 p^{16} T^{41} + 18110856994 p^{17} T^{42} + 610714298 p^{18} T^{43} + 84992870 p^{19} T^{44} + 2583430 p^{20} T^{45} + 304528 p^{21} T^{46} + 7197 p^{22} T^{47} + 751 p^{23} T^{48} + 10 p^{24} T^{49} + p^{25} T^{50} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{50} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.35426036152705946998885207700, −1.27724393392296465274660646952, −1.22807775338954572762789162271, −1.16230956297795650499637477637, −1.13183107888023680539055164761, −1.11479538680347859819396747523, −1.06090014014777302345335766748, −1.02267452900582628883960278985, −1.00819149862694652931888540769, −0.906321536055679331959831709130, −0.800693965236434549622211149522, −0.78406440423641847536723469655, −0.76985518137393737681112426385, −0.76318629579711004644147789725, −0.75428285198665737373365684782, −0.73359261335936170791267302204, −0.62314679281945546705286803413, −0.61932687048761507787696553104, −0.59001572515744252675893387014, −0.54090751412779429958864545248, −0.49786330155549241147428421295, −0.46471390766637537831850624826, −0.41520401008259682883954334463, −0.32392210636250452958421956677, −0.24581619256385634309094619643, 0.24581619256385634309094619643, 0.32392210636250452958421956677, 0.41520401008259682883954334463, 0.46471390766637537831850624826, 0.49786330155549241147428421295, 0.54090751412779429958864545248, 0.59001572515744252675893387014, 0.61932687048761507787696553104, 0.62314679281945546705286803413, 0.73359261335936170791267302204, 0.75428285198665737373365684782, 0.76318629579711004644147789725, 0.76985518137393737681112426385, 0.78406440423641847536723469655, 0.800693965236434549622211149522, 0.906321536055679331959831709130, 1.00819149862694652931888540769, 1.02267452900582628883960278985, 1.06090014014777302345335766748, 1.11479538680347859819396747523, 1.13183107888023680539055164761, 1.16230956297795650499637477637, 1.22807775338954572762789162271, 1.27724393392296465274660646952, 1.35426036152705946998885207700

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

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