Properties

Degree $88$
Conductor $5.215\times 10^{92}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·5-s + 4·7-s + 6·9-s + 4·11-s − 4·13-s − 16·15-s + 4·19-s + 16·21-s − 324·23-s + 6·25-s + 92·27-s − 4·29-s + 752·31-s + 16·33-s − 16·35-s − 4·37-s − 16·39-s − 4·41-s − 804·43-s − 24·45-s + 8·49-s + 748·53-s − 16·55-s + 16·57-s − 1.37e3·59-s − 1.82e3·61-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.357·5-s + 0.215·7-s + 2/9·9-s + 0.109·11-s − 0.0853·13-s − 0.275·15-s + 0.0482·19-s + 0.166·21-s − 2.93·23-s + 0.0479·25-s + 0.655·27-s − 0.0256·29-s + 4.35·31-s + 0.0844·33-s − 0.0772·35-s − 0.0177·37-s − 0.0656·39-s − 0.0152·41-s − 2.85·43-s − 0.0795·45-s + 0.0233·49-s + 1.93·53-s − 0.0392·55-s + 0.0371·57-s − 3.02·59-s − 3.83·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{308}\right)^{s/2} \, \Gamma_{\C}(s)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{308}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(88\)
Conductor: \(2^{308}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{128} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((88,\ 2^{308} ,\ ( \ : [3/2]^{44} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.0992502\)
\(L(\frac12)\) \(\approx\) \(0.0992502\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 4 T + 10 T^{2} - 4 p^{3} T^{3} + 386 T^{4} + 1580 T^{5} + 2366 p T^{6} - 8660 p^{3} T^{7} + 658999 T^{8} - 485864 T^{9} - 8122372 T^{10} + 10277128 T^{11} - 534666356 T^{12} + 539695448 T^{13} + 2665132220 p^{2} T^{14} - 4094593064 p^{3} T^{15} - 6882609583 p T^{16} + 4302754699868 T^{17} - 5691650579006 T^{18} + 28254024166324 T^{19} + 5553406688666 T^{20} - 536311964237468 p T^{21} + 2201118904348706 p^{2} T^{22} + 397658961952900 p^{2} T^{23} - 608237435070227579 T^{24} + 1192756547548600736 T^{25} - 585500912335712816 T^{26} - 9126666242746265440 p T^{27} + 54156918341790038672 T^{28} - \)\(15\!\cdots\!88\)\( T^{29} - \)\(40\!\cdots\!00\)\( p T^{30} + \)\(53\!\cdots\!40\)\( p^{2} T^{31} - \)\(18\!\cdots\!66\)\( T^{32} + \)\(33\!\cdots\!44\)\( T^{33} + \)\(49\!\cdots\!20\)\( T^{34} - \)\(21\!\cdots\!20\)\( T^{35} + \)\(46\!\cdots\!20\)\( T^{36} + \)\(22\!\cdots\!92\)\( T^{37} - \)\(49\!\cdots\!28\)\( p^{3} T^{38} + \)\(18\!\cdots\!88\)\( p^{2} T^{39} + \)\(38\!\cdots\!62\)\( p^{2} T^{40} - \)\(47\!\cdots\!60\)\( T^{41} - \)\(63\!\cdots\!96\)\( T^{42} - \)\(98\!\cdots\!04\)\( T^{43} - \)\(17\!\cdots\!96\)\( T^{44} - \)\(98\!\cdots\!04\)\( p^{3} T^{45} - \)\(63\!\cdots\!96\)\( p^{6} T^{46} - \)\(47\!\cdots\!60\)\( p^{9} T^{47} + \)\(38\!\cdots\!62\)\( p^{14} T^{48} + \)\(18\!\cdots\!88\)\( p^{17} T^{49} - \)\(49\!\cdots\!28\)\( p^{21} T^{50} + \)\(22\!\cdots\!92\)\( p^{21} T^{51} + \)\(46\!\cdots\!20\)\( p^{24} T^{52} - \)\(21\!\cdots\!20\)\( p^{27} T^{53} + \)\(49\!\cdots\!20\)\( p^{30} T^{54} + \)\(33\!\cdots\!44\)\( p^{33} T^{55} - \)\(18\!\cdots\!66\)\( p^{36} T^{56} + \)\(53\!\cdots\!40\)\( p^{41} T^{57} - \)\(40\!\cdots\!00\)\( p^{43} T^{58} - \)\(15\!\cdots\!88\)\( p^{45} T^{59} + 54156918341790038672 p^{48} T^{60} - 9126666242746265440 p^{52} T^{61} - 585500912335712816 p^{54} T^{62} + 1192756547548600736 p^{57} T^{63} - 608237435070227579 p^{60} T^{64} + 397658961952900 p^{65} T^{65} + 2201118904348706 p^{68} T^{66} - 536311964237468 p^{70} T^{67} + 5553406688666 p^{72} T^{68} + 28254024166324 p^{75} T^{69} - 5691650579006 p^{78} T^{70} + 4302754699868 p^{81} T^{71} - 6882609583 p^{85} T^{72} - 4094593064 p^{90} T^{73} + 2665132220 p^{92} T^{74} + 539695448 p^{93} T^{75} - 534666356 p^{96} T^{76} + 10277128 p^{99} T^{77} - 8122372 p^{102} T^{78} - 485864 p^{105} T^{79} + 658999 p^{108} T^{80} - 8660 p^{114} T^{81} + 2366 p^{115} T^{82} + 1580 p^{117} T^{83} + 386 p^{120} T^{84} - 4 p^{126} T^{85} + 10 p^{126} T^{86} - 4 p^{129} T^{87} + p^{132} T^{88} \)
5 \( 1 + 4 T + 2 p T^{2} - 1916 T^{3} - 1542 p T^{4} + 15364 p T^{5} + 2827674 T^{6} - 7602444 T^{7} + 22028731 p T^{8} + 817630328 T^{9} + 10475950188 p T^{10} - 23955308072 p T^{11} - 5891982633492 T^{12} - 2590962517736 p^{2} T^{13} + 319583994778876 T^{14} + 7497592498833496 T^{15} + 114117715932647043 T^{16} + 125602966689615364 p T^{17} + 11222953712560097346 T^{18} - 1748848153952447276 T^{19} - \)\(10\!\cdots\!74\)\( T^{20} - \)\(10\!\cdots\!24\)\( T^{21} - \)\(69\!\cdots\!82\)\( T^{22} + \)\(17\!\cdots\!12\)\( T^{23} + \)\(33\!\cdots\!37\)\( p T^{24} + \)\(29\!\cdots\!52\)\( T^{25} + \)\(32\!\cdots\!56\)\( T^{26} + \)\(23\!\cdots\!48\)\( T^{27} - \)\(42\!\cdots\!52\)\( p T^{28} - \)\(21\!\cdots\!08\)\( T^{29} - \)\(23\!\cdots\!96\)\( T^{30} + \)\(21\!\cdots\!12\)\( p^{2} T^{31} + \)\(24\!\cdots\!74\)\( T^{32} + \)\(37\!\cdots\!16\)\( T^{33} + \)\(83\!\cdots\!24\)\( T^{34} + \)\(83\!\cdots\!08\)\( T^{35} + \)\(82\!\cdots\!12\)\( T^{36} - \)\(78\!\cdots\!16\)\( T^{37} - \)\(92\!\cdots\!76\)\( p T^{38} + \)\(10\!\cdots\!76\)\( T^{39} + \)\(16\!\cdots\!54\)\( T^{40} + \)\(41\!\cdots\!76\)\( T^{41} + \)\(19\!\cdots\!92\)\( T^{42} + \)\(18\!\cdots\!16\)\( T^{43} + \)\(35\!\cdots\!32\)\( T^{44} + \)\(18\!\cdots\!16\)\( p^{3} T^{45} + \)\(19\!\cdots\!92\)\( p^{6} T^{46} + \)\(41\!\cdots\!76\)\( p^{9} T^{47} + \)\(16\!\cdots\!54\)\( p^{12} T^{48} + \)\(10\!\cdots\!76\)\( p^{15} T^{49} - \)\(92\!\cdots\!76\)\( p^{19} T^{50} - \)\(78\!\cdots\!16\)\( p^{21} T^{51} + \)\(82\!\cdots\!12\)\( p^{24} T^{52} + \)\(83\!\cdots\!08\)\( p^{27} T^{53} + \)\(83\!\cdots\!24\)\( p^{30} T^{54} + \)\(37\!\cdots\!16\)\( p^{33} T^{55} + \)\(24\!\cdots\!74\)\( p^{36} T^{56} + \)\(21\!\cdots\!12\)\( p^{41} T^{57} - \)\(23\!\cdots\!96\)\( p^{42} T^{58} - \)\(21\!\cdots\!08\)\( p^{45} T^{59} - \)\(42\!\cdots\!52\)\( p^{49} T^{60} + \)\(23\!\cdots\!48\)\( p^{51} T^{61} + \)\(32\!\cdots\!56\)\( p^{54} T^{62} + \)\(29\!\cdots\!52\)\( p^{57} T^{63} + \)\(33\!\cdots\!37\)\( p^{61} T^{64} + \)\(17\!\cdots\!12\)\( p^{63} T^{65} - \)\(69\!\cdots\!82\)\( p^{66} T^{66} - \)\(10\!\cdots\!24\)\( p^{69} T^{67} - \)\(10\!\cdots\!74\)\( p^{72} T^{68} - 1748848153952447276 p^{75} T^{69} + 11222953712560097346 p^{78} T^{70} + 125602966689615364 p^{82} T^{71} + 114117715932647043 p^{84} T^{72} + 7497592498833496 p^{87} T^{73} + 319583994778876 p^{90} T^{74} - 2590962517736 p^{95} T^{75} - 5891982633492 p^{96} T^{76} - 23955308072 p^{100} T^{77} + 10475950188 p^{103} T^{78} + 817630328 p^{105} T^{79} + 22028731 p^{109} T^{80} - 7602444 p^{111} T^{81} + 2827674 p^{114} T^{82} + 15364 p^{118} T^{83} - 1542 p^{121} T^{84} - 1916 p^{123} T^{85} + 2 p^{127} T^{86} + 4 p^{129} T^{87} + p^{132} T^{88} \)
7 \( 1 - 4 T + 8 T^{2} - 2076 p T^{3} - 114774 T^{4} + 4205300 T^{5} + 89686504 T^{6} + 377449452 p T^{7} - 28540556025 T^{8} - 1212316300936 T^{9} - 14182125548208 T^{10} - 19352038018744 p T^{11} + 12302061087893372 T^{12} + 143478432603579336 T^{13} + 88644906624772656 p T^{14} - 42804055517561905208 T^{15} - \)\(15\!\cdots\!81\)\( T^{16} - \)\(63\!\cdots\!00\)\( p T^{17} + \)\(19\!\cdots\!60\)\( T^{18} + \)\(50\!\cdots\!16\)\( T^{19} + \)\(12\!\cdots\!18\)\( T^{20} - \)\(11\!\cdots\!64\)\( T^{21} - \)\(13\!\cdots\!36\)\( T^{22} - \)\(52\!\cdots\!80\)\( T^{23} - \)\(52\!\cdots\!75\)\( T^{24} + \)\(90\!\cdots\!24\)\( T^{25} + \)\(10\!\cdots\!92\)\( T^{26} + \)\(37\!\cdots\!52\)\( T^{27} + \)\(50\!\cdots\!08\)\( T^{28} - \)\(10\!\cdots\!32\)\( T^{29} - \)\(12\!\cdots\!04\)\( p T^{30} - \)\(60\!\cdots\!00\)\( T^{31} + \)\(26\!\cdots\!82\)\( T^{32} + \)\(69\!\cdots\!00\)\( T^{33} + \)\(27\!\cdots\!08\)\( T^{34} - \)\(24\!\cdots\!72\)\( T^{35} - \)\(18\!\cdots\!32\)\( T^{36} - \)\(97\!\cdots\!60\)\( T^{37} + \)\(28\!\cdots\!76\)\( T^{38} + \)\(14\!\cdots\!16\)\( T^{39} - \)\(73\!\cdots\!78\)\( T^{40} - \)\(40\!\cdots\!36\)\( p^{2} T^{41} - \)\(63\!\cdots\!52\)\( T^{42} + \)\(98\!\cdots\!36\)\( T^{43} + \)\(14\!\cdots\!68\)\( T^{44} + \)\(98\!\cdots\!36\)\( p^{3} T^{45} - \)\(63\!\cdots\!52\)\( p^{6} T^{46} - \)\(40\!\cdots\!36\)\( p^{11} T^{47} - \)\(73\!\cdots\!78\)\( p^{12} T^{48} + \)\(14\!\cdots\!16\)\( p^{15} T^{49} + \)\(28\!\cdots\!76\)\( p^{18} T^{50} - \)\(97\!\cdots\!60\)\( p^{21} T^{51} - \)\(18\!\cdots\!32\)\( p^{24} T^{52} - \)\(24\!\cdots\!72\)\( p^{27} T^{53} + \)\(27\!\cdots\!08\)\( p^{30} T^{54} + \)\(69\!\cdots\!00\)\( p^{33} T^{55} + \)\(26\!\cdots\!82\)\( p^{36} T^{56} - \)\(60\!\cdots\!00\)\( p^{39} T^{57} - \)\(12\!\cdots\!04\)\( p^{43} T^{58} - \)\(10\!\cdots\!32\)\( p^{45} T^{59} + \)\(50\!\cdots\!08\)\( p^{48} T^{60} + \)\(37\!\cdots\!52\)\( p^{51} T^{61} + \)\(10\!\cdots\!92\)\( p^{54} T^{62} + \)\(90\!\cdots\!24\)\( p^{57} T^{63} - \)\(52\!\cdots\!75\)\( p^{60} T^{64} - \)\(52\!\cdots\!80\)\( p^{63} T^{65} - \)\(13\!\cdots\!36\)\( p^{66} T^{66} - \)\(11\!\cdots\!64\)\( p^{69} T^{67} + \)\(12\!\cdots\!18\)\( p^{72} T^{68} + \)\(50\!\cdots\!16\)\( p^{75} T^{69} + \)\(19\!\cdots\!60\)\( p^{78} T^{70} - \)\(63\!\cdots\!00\)\( p^{82} T^{71} - \)\(15\!\cdots\!81\)\( p^{84} T^{72} - 42804055517561905208 p^{87} T^{73} + 88644906624772656 p^{91} T^{74} + 143478432603579336 p^{93} T^{75} + 12302061087893372 p^{96} T^{76} - 19352038018744 p^{100} T^{77} - 14182125548208 p^{102} T^{78} - 1212316300936 p^{105} T^{79} - 28540556025 p^{108} T^{80} + 377449452 p^{112} T^{81} + 89686504 p^{114} T^{82} + 4205300 p^{117} T^{83} - 114774 p^{120} T^{84} - 2076 p^{124} T^{85} + 8 p^{126} T^{86} - 4 p^{129} T^{87} + p^{132} T^{88} \)
11 \( 1 - 4T + 1.37e3T^{2} - 2.47e3T^{3} + 9.26e5T^{4} - 4.17e7T^{5} - 1.85e9T^{6} + 1.42e10T^{7} + 6.88e12T^{8} - 9.55e11T^{9} + 1.47e16T^{10} + 1.39e17T^{11} + 1.25e19T^{12} - 3.72e20T^{13} - 7.02e21T^{14} - 1.70e23T^{15} + 2.06e25T^{16} + 2.03e26T^{17} + 7.00e28T^{18} + 1.67e30T^{19} + 5.93e31T^{20} - 1.77e32T^{21} - 1.67e34T^{22} - 1.40e36T^{23} + 7.02e37T^{24} + 9.91e38T^{25} + 2.36e41T^{26} + 8.20e42T^{27} + 2.42e44T^{28} + 4.65e45T^{29} - 1.25e47T^{30} - 1.73e48T^{31} + 1.21e50T^{32} + 4.24e51T^{33} + 6.27e53T^{34} + 3.59e55T^{35} + 9.65e56T^{36}+O(T^{37}) \)
13 \( 1 + 4T + 10T^{2} - 2.96e5T^{3} - 1.18e6T^{4} - 8.31e8T^{5} + 2.00e10T^{6} - 4.54e11T^{7} + 2.13e14T^{8} + 1.94e15T^{9} + 5.20e17T^{10} - 9.02e18T^{11} + 2.32e20T^{12} - 8.92e22T^{13} - 1.65e24T^{14} - 2.31e26T^{15} + 5.20e25T^{16} - 1.27e29T^{17} + 2.28e31T^{18} + 6.74e32T^{19} + 7.89e34T^{20} + 1.82e36T^{21} + 9.76e37T^{22} - 2.12e39T^{23} - 1.59e41T^{24} - 2.12e43T^{25} - 1.00e45T^{26} - 4.93e46T^{27} - 1.14e48T^{28} - 3.60e48T^{29} + 3.85e51T^{30} + 3.09e53T^{31} + 1.75e55T^{32} + 6.97e56T^{33} + 1.94e58T^{34}+O(T^{35}) \)
17 \( 1 - 1.10e5T^{2} + 6.21e9T^{4} - 2.33e14T^{6} + 6.62e18T^{8} - 1.51e23T^{10} + 2.87e27T^{12} - 4.69e31T^{14} + 6.71e35T^{16} - 8.53e39T^{18} + 9.73e43T^{20} - 1.00e48T^{22} + 9.51e51T^{24} - 8.27e55T^{26} + 6.65e59T^{28} - 4.97e63T^{30}+O(T^{32}) \)
19 \( 1 - 4T - 3.92e3T^{2} - 2.06e6T^{3} + 1.60e7T^{4} + 3.10e9T^{5} + 2.02e12T^{6} - 8.87e13T^{7} + 1.78e15T^{8} - 8.35e17T^{9} + 1.50e20T^{10} - 5.54e21T^{11} + 1.25e23T^{12} - 1.26e26T^{13} + 6.80e27T^{14} - 8.80e28T^{15} + 5.44e31T^{16} - 6.22e33T^{17} + 1.51e35T^{18} - 1.66e37T^{19} + 3.54e39T^{20} - 1.04e41T^{21} + 9.29e42T^{22} - 1.19e45T^{23} + 5.81e46T^{24} - 3.00e48T^{25} + 4.63e50T^{26} - 1.59e52T^{27} - 2.69e54T^{28} - 2.89e56T^{29} + 2.23e57T^{30}+O(T^{31}) \)
23 \( 1 + 324T + 5.24e4T^{2} + 1.17e7T^{3} + 2.40e9T^{4} + 3.25e11T^{5} + 4.78e13T^{6} + 7.88e15T^{7} + 1.07e18T^{8} + 1.53e20T^{9} + 2.30e22T^{10} + 3.40e24T^{11} + 5.02e26T^{12} + 6.79e28T^{13} + 9.22e30T^{14} + 1.29e33T^{15} + 1.64e35T^{16} + 2.10e37T^{17} + 2.85e39T^{18} + 3.71e41T^{19} + 4.82e43T^{20} + 6.24e45T^{21} + 7.94e47T^{22} + 1.01e50T^{23} + 1.24e52T^{24} + 1.49e54T^{25} + 1.88e56T^{26} + 2.28e58T^{27} + 2.75e60T^{28}+O(T^{29}) \)
29 \( 1 + 4T - 3.31e4T^{2} + 9.49e6T^{3} + 5.89e8T^{4} - 1.70e11T^{5} + 3.09e13T^{6} + 4.51e15T^{7} + 4.36e17T^{8} - 5.20e19T^{9} + 1.26e22T^{10} + 8.66e24T^{11} - 3.59e26T^{12} + 8.76e26T^{13} + 3.33e31T^{14} + 1.92e33T^{15} + 2.17e35T^{16} + 1.41e37T^{17} + 2.36e40T^{18} + 2.90e42T^{19} - 2.75e44T^{20} + 9.90e46T^{21} + 1.54e49T^{22} + 4.47e50T^{23} + 2.04e53T^{24} + 4.64e55T^{25} + 1.23e58T^{26}+O(T^{27}) \)
31 \( 1 - 752T + 9.74e5T^{2} - 5.72e8T^{3} + 4.38e11T^{4} - 2.16e14T^{5} + 1.25e17T^{6} - 5.40e19T^{7} + 2.60e22T^{8} - 1.01e25T^{9} + 4.26e27T^{10} - 1.51e30T^{11} + 5.72e32T^{12} - 1.87e35T^{13} + 6.51e37T^{14} - 2.00e40T^{15} + 6.43e42T^{16} - 1.85e45T^{17} + 5.60e47T^{18} - 1.53e50T^{19} + 4.35e52T^{20} - 1.13e55T^{21} + 3.06e57T^{22} - 7.61e59T^{23} + 1.95e62T^{24} - 4.66e64T^{25} + 1.14e67T^{26}+O(T^{27}) \)
37 \( 1 + 4T - 5.99e3T^{2} - 3.72e7T^{3} - 1.30e8T^{4} - 3.84e11T^{5} + 1.01e15T^{6} + 2.71e16T^{7} + 2.77e19T^{8} - 2.24e22T^{9} - 7.20e23T^{10} - 9.19e26T^{11} + 4.08e29T^{12} + 1.19e31T^{13} + 2.24e34T^{14} - 6.26e36T^{15} - 6.24e37T^{16} - 4.59e41T^{17} + 8.26e43T^{18} - 2.42e45T^{19} + 7.98e48T^{20} - 9.30e50T^{21} + 1.05e53T^{22} - 1.19e56T^{23} + 8.72e57T^{24}+O(T^{25}) \)
41 \( 1 + 4T + 8T^{2} - 1.23e6T^{3} + 5.23e9T^{4} + 7.99e11T^{5} + 3.91e12T^{6} - 6.03e16T^{7} - 4.00e19T^{8} + 1.37e22T^{9} + 4.29e23T^{10} - 3.27e26T^{11} - 4.32e29T^{12} + 4.64e31T^{13} + 1.13e34T^{14} + 9.25e35T^{15} - 1.14e39T^{16} - 4.26e41T^{17} + 1.08e44T^{18} + 2.72e46T^{19} + 4.67e48T^{20} - 5.18e51T^{21} + 3.17e53T^{22} + 1.83e56T^{23} + 5.56e58T^{24}+O(T^{25}) \)
43 \( 1 + 804T + 2.49e5T^{2} - 1.19e7T^{3} - 3.51e10T^{4} - 1.55e13T^{5} - 3.86e15T^{6} - 6.27e17T^{7} + 2.63e18T^{8} + 7.50e22T^{9} + 5.28e25T^{10} + 1.98e28T^{11} + 4.52e30T^{12} + 4.05e32T^{13} - 1.97e35T^{14} - 1.44e38T^{15} - 5.83e40T^{16} - 1.65e43T^{17} - 2.74e45T^{18} + 2.28e47T^{19} + 3.16e50T^{20} + 1.27e53T^{21} + 3.76e55T^{22} + 8.92e57T^{23}+O(T^{24}) \)
47 \( 1 - 2.43e6T^{2} + 2.97e12T^{4} - 2.42e18T^{6} + 1.48e24T^{8} - 7.27e29T^{10} + 2.96e35T^{12} - 1.03e41T^{14} + 3.17e46T^{16} - 8.61e51T^{18} + 2.10e57T^{20} - 4.66e62T^{22}+O(T^{24}) \)
53 \( 1 - 748T - 3.08e4T^{2} + 2.86e8T^{3} - 1.18e11T^{4} - 9.27e12T^{5} + 2.32e16T^{6} - 6.29e18T^{7} + 2.40e20T^{8} - 3.90e23T^{9} + 3.05e26T^{10} + 1.09e29T^{11} - 1.75e32T^{12} + 6.24e34T^{13} + 3.56e36T^{14} - 1.00e40T^{15} + 3.16e42T^{16} - 3.94e44T^{17} + 1.19e47T^{18} - 7.51e49T^{19} - 9.78e51T^{20} + 3.36e55T^{21} - 1.64e58T^{22}+O(T^{23}) \)
59 \( 1 + 1.37e3T + 1.78e6T^{2} + 1.77e9T^{3} + 1.55e12T^{4} + 1.25e15T^{5} + 9.08e17T^{6} + 6.23e20T^{7} + 3.97e23T^{8} + 2.40e26T^{9} + 1.39e29T^{10} + 7.66e31T^{11} + 4.07e34T^{12} + 2.08e37T^{13} + 1.04e40T^{14} + 5.10e42T^{15} + 2.46e45T^{16} + 1.18e48T^{17} + 5.73e50T^{18} + 2.79e53T^{19} + 1.37e56T^{20} + 6.82e58T^{21} + 3.40e61T^{22}+O(T^{23}) \)
61 \( 1 + 1.82e3T + 2.69e6T^{2} + 2.84e9T^{3} + 2.62e12T^{4} + 2.07e15T^{5} + 1.50e18T^{6} + 1.01e21T^{7} + 6.60e23T^{8} + 4.17e26T^{9} + 2.61e29T^{10} + 1.61e32T^{11} + 9.85e34T^{12} + 5.90e37T^{13} + 3.48e40T^{14} + 2.01e43T^{15} + 1.14e46T^{16} + 6.44e48T^{17} + 3.60e51T^{18} + 1.99e54T^{19} + 1.09e57T^{20} + 5.99e59T^{21}+O(T^{22}) \)
67 \( 1 + 2.03e3T + 3.88e6T^{2} + 5.46e9T^{3} + 6.85e12T^{4} + 7.59e15T^{5} + 7.73e18T^{6} + 7.28e21T^{7} + 6.46e24T^{8} + 5.41e27T^{9} + 4.32e30T^{10} + 3.29e33T^{11} + 2.41e36T^{12} + 1.70e39T^{13} + 1.16e42T^{14} + 7.63e44T^{15} + 4.86e47T^{16} + 3.00e50T^{17} + 1.79e53T^{18} + 1.04e56T^{19} + 5.90e58T^{20} + 3.24e61T^{21}+O(T^{22}) \)
71 \( 1 + 220T + 2.42e4T^{2} - 6.80e7T^{3} - 7.34e11T^{4} - 1.97e14T^{5} - 2.34e16T^{6} + 1.19e19T^{7} + 2.69e23T^{8} + 8.67e25T^{9} + 1.41e28T^{10} + 6.64e30T^{11} - 6.61e34T^{12} - 2.61e37T^{13} - 5.01e39T^{14} - 4.25e42T^{15} + 1.25e46T^{16} + 6.09e48T^{17} + 1.30e51T^{18} + 1.30e54T^{19} - 1.95e57T^{20} - 1.17e60T^{21}+O(T^{22}) \)
73 \( 1 + 4T + 8T^{2} + 1.16e8T^{3} - 1.27e11T^{4} + 5.51e13T^{5} + 6.98e15T^{6} + 3.88e18T^{7} - 6.31e21T^{8} + 1.58e24T^{9} + 2.87e27T^{10} - 1.87e30T^{11} - 2.54e33T^{12} - 2.31e36T^{13} + 2.78e38T^{14} - 1.15e42T^{15} + 3.04e44T^{16} - 3.26e47T^{17} - 1.97e50T^{18} - 4.17e52T^{19} + 2.26e55T^{20} + 1.27e58T^{21}+O(T^{22}) \)
79 \( 1 - 1.08e7T^{2} + 5.88e13T^{4} - 2.14e20T^{6} + 5.89e26T^{8} - 1.30e33T^{10} + 2.40e39T^{12} - 3.82e45T^{14} + 5.33e51T^{16} - 6.64e57T^{18} + 7.48e63T^{20}+O(T^{21}) \)
83 \( 1 + 2.43e3T - 6.53e5T^{2} - 6.88e9T^{3} - 3.88e12T^{4} + 8.45e15T^{5} + 1.00e19T^{6} - 4.41e21T^{7} - 1.18e25T^{8} - 1.29e27T^{9} + 7.86e30T^{10} + 3.32e33T^{11} - 2.61e36T^{12} - 1.32e39T^{13} + 1.59e41T^{14} - 1.26e45T^{15} - 4.44e47T^{16} + 2.03e51T^{17} + 1.21e54T^{18} - 1.33e57T^{19} - 1.18e60T^{20}+O(T^{21}) \)
89 \( 1 + 4T + 8T^{2} - 6.73e8T^{3} + 8.97e11T^{4} + 1.57e15T^{5} + 2.33e17T^{6} + 2.53e20T^{7} - 5.41e23T^{8} + 1.22e27T^{9} + 8.67e29T^{10} + 6.10e32T^{11} + 3.41e35T^{12} + 1.14e38T^{13} + 8.96e41T^{14} + 4.22e44T^{15} + 3.05e47T^{16} + 1.27e50T^{17} + 3.84e53T^{18} + 5.00e56T^{19} + 2.36e59T^{20}+O(T^{21}) \)
97 \( 1 + 8T + 2.23e7T^{2} + 2.42e9T^{3} + 2.48e14T^{4} + 5.37e16T^{5} + 1.83e21T^{6} + 6.02e23T^{7} + 1.00e28T^{8} + 4.48e30T^{9} + 4.39e34T^{10} + 2.47e37T^{11} + 1.60e41T^{12} + 1.08e44T^{13} + 5.01e47T^{14} + 3.92e50T^{15} + 1.38e54T^{16} + 1.20e57T^{17} + 3.39e60T^{18} + 3.21e63T^{19}+O(T^{20}) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{88} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.45435400389015609143760846034, −1.44216417299400771096273970935, −1.44066552882439602836749417894, −1.41140106709282889693970558839, −1.36936912528589012568040150094, −1.35859206581633489657844433672, −1.32312149845876963037472538092, −1.30611018587938653550858511905, −1.04153292066156965486272647081, −0.977646890635642344000520836352, −0.954635136336723371357672091056, −0.951373998866872902390455289672, −0.908750052461166322737982576038, −0.813770039235749775532379016527, −0.66340199389947310529863983884, −0.65273658823773463938822981810, −0.63228437548960960805254311552, −0.54614934153372723013186166382, −0.41484794021145959798859763051, −0.32971439258420958348719613579, −0.25860092197044917566514889711, −0.16440523152190428156906800260, −0.07297010082473211155855719257, −0.07263800893732150677081471978, −0.04837821624346476269289734211, 0.04837821624346476269289734211, 0.07263800893732150677081471978, 0.07297010082473211155855719257, 0.16440523152190428156906800260, 0.25860092197044917566514889711, 0.32971439258420958348719613579, 0.41484794021145959798859763051, 0.54614934153372723013186166382, 0.63228437548960960805254311552, 0.65273658823773463938822981810, 0.66340199389947310529863983884, 0.813770039235749775532379016527, 0.908750052461166322737982576038, 0.951373998866872902390455289672, 0.954635136336723371357672091056, 0.977646890635642344000520836352, 1.04153292066156965486272647081, 1.30611018587938653550858511905, 1.32312149845876963037472538092, 1.35859206581633489657844433672, 1.36936912528589012568040150094, 1.41140106709282889693970558839, 1.44066552882439602836749417894, 1.44216417299400771096273970935, 1.45435400389015609143760846034

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

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