Properties

Degree $32$
Conductor $2.815\times 10^{14}$
Sign $1$
Motivic weight $17$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 270·2-s + 2.27e4·4-s + 1.15e7·7-s + 7.68e6·8-s + 7.31e8·9-s + 3.11e9·14-s + 8.63e9·16-s − 7.48e9·17-s + 1.97e11·18-s + 7.46e11·23-s + 5.19e12·25-s + 2.62e11·28-s − 3.18e11·31-s + 2.25e12·32-s − 2.02e12·34-s + 1.66e13·36-s + 7.48e12·41-s + 2.01e14·46-s − 3.76e14·47-s − 1.73e15·49-s + 1.40e15·50-s + 8.86e13·56-s − 8.61e13·62-s + 8.43e15·63-s + 4.44e14·64-s − 1.70e14·68-s + 9.02e15·71-s + ⋯
L(s)  = 1  + 0.745·2-s + 0.173·4-s + 0.755·7-s + 0.162·8-s + 5.66·9-s + 0.563·14-s + 0.502·16-s − 0.260·17-s + 4.22·18-s + 1.98·23-s + 6.81·25-s + 0.131·28-s − 0.0671·31-s + 0.362·32-s − 0.194·34-s + 0.982·36-s + 0.146·41-s + 1.48·46-s − 2.30·47-s − 7.43·49-s + 5.08·50-s + 0.122·56-s − 0.0500·62-s + 4.28·63-s + 0.197·64-s − 0.0451·68-s + 1.65·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48}\)
Sign: $1$
Motivic weight: \(17\)
Character: induced by $\chi_{8} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} ,\ ( \ : [17/2]^{16} ),\ 1 )\)

Particular Values

\(L(9)\) \(\approx\) \(179.270\)
\(L(\frac12)\) \(\approx\) \(179.270\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 135 p T + 6271 p^{3} T^{2} - 235875 p^{6} T^{3} - 3534589 p^{10} T^{4} + 30772125 p^{15} T^{5} - 163124025 p^{21} T^{6} + 3654732165 p^{28} T^{7} - 6134904205 p^{36} T^{8} + 3654732165 p^{45} T^{9} - 163124025 p^{55} T^{10} + 30772125 p^{66} T^{11} - 3534589 p^{78} T^{12} - 235875 p^{91} T^{13} + 6271 p^{105} T^{14} - 135 p^{120} T^{15} + p^{136} T^{16} \)
good3 \( 1 - 731794256 T^{2} + 31354739233887608 p^{2} T^{4} - \)\(14\!\cdots\!92\)\( p^{12} T^{6} + \)\(25\!\cdots\!00\)\( p^{10} T^{8} - \)\(50\!\cdots\!96\)\( p^{14} T^{10} + \)\(31\!\cdots\!00\)\( p^{21} T^{12} - \)\(12\!\cdots\!84\)\( p^{22} T^{14} + \)\(23\!\cdots\!46\)\( p^{30} T^{16} - \)\(12\!\cdots\!84\)\( p^{56} T^{18} + \)\(31\!\cdots\!00\)\( p^{89} T^{20} - \)\(50\!\cdots\!96\)\( p^{116} T^{22} + \)\(25\!\cdots\!00\)\( p^{146} T^{24} - \)\(14\!\cdots\!92\)\( p^{182} T^{26} + 31354739233887608 p^{206} T^{28} - 731794256 p^{238} T^{30} + p^{272} T^{32} \)
5 \( 1 - 5198674409168 T^{2} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(97\!\cdots\!96\)\( p^{2} T^{6} + \)\(53\!\cdots\!88\)\( p^{4} T^{8} - \)\(96\!\cdots\!88\)\( p^{8} T^{10} + \)\(14\!\cdots\!16\)\( p^{12} T^{12} - \)\(20\!\cdots\!92\)\( p^{16} T^{14} + \)\(26\!\cdots\!66\)\( p^{20} T^{16} - \)\(20\!\cdots\!92\)\( p^{50} T^{18} + \)\(14\!\cdots\!16\)\( p^{80} T^{20} - \)\(96\!\cdots\!88\)\( p^{110} T^{22} + \)\(53\!\cdots\!88\)\( p^{140} T^{24} - \)\(97\!\cdots\!96\)\( p^{172} T^{26} + \)\(13\!\cdots\!56\)\( p^{204} T^{28} - 5198674409168 p^{238} T^{30} + p^{272} T^{32} \)
7 \( ( 1 - 5764800 T + 915215692443448 T^{2} - \)\(93\!\cdots\!60\)\( p T^{3} + \)\(10\!\cdots\!76\)\( p^{2} T^{4} - \)\(10\!\cdots\!60\)\( p^{3} T^{5} + \)\(11\!\cdots\!80\)\( p^{5} T^{6} - \)\(20\!\cdots\!60\)\( p^{8} T^{7} + \)\(12\!\cdots\!70\)\( p^{9} T^{8} - \)\(20\!\cdots\!60\)\( p^{25} T^{9} + \)\(11\!\cdots\!80\)\( p^{39} T^{10} - \)\(10\!\cdots\!60\)\( p^{54} T^{11} + \)\(10\!\cdots\!76\)\( p^{70} T^{12} - \)\(93\!\cdots\!60\)\( p^{86} T^{13} + 915215692443448 p^{102} T^{14} - 5764800 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
11 \( 1 - 3877608573076448976 T^{2} + \)\(79\!\cdots\!84\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!28\)\( p^{2} T^{8} - \)\(75\!\cdots\!92\)\( p^{4} T^{10} + \)\(46\!\cdots\!36\)\( p^{6} T^{12} - \)\(24\!\cdots\!88\)\( p^{8} T^{14} + \)\(10\!\cdots\!22\)\( p^{10} T^{16} - \)\(24\!\cdots\!88\)\( p^{42} T^{18} + \)\(46\!\cdots\!36\)\( p^{74} T^{20} - \)\(75\!\cdots\!92\)\( p^{106} T^{22} + \)\(10\!\cdots\!28\)\( p^{138} T^{24} - \)\(11\!\cdots\!04\)\( p^{170} T^{26} + \)\(79\!\cdots\!84\)\( p^{204} T^{28} - 3877608573076448976 p^{238} T^{30} + p^{272} T^{32} \)
13 \( 1 - 63371249746529137488 T^{2} + \)\(20\!\cdots\!16\)\( T^{4} - \)\(28\!\cdots\!52\)\( p^{2} T^{6} + \)\(29\!\cdots\!88\)\( p^{4} T^{8} - \)\(25\!\cdots\!24\)\( p^{6} T^{10} + \)\(18\!\cdots\!84\)\( p^{8} T^{12} - \)\(11\!\cdots\!44\)\( p^{10} T^{14} + \)\(64\!\cdots\!02\)\( p^{12} T^{16} - \)\(11\!\cdots\!44\)\( p^{44} T^{18} + \)\(18\!\cdots\!84\)\( p^{76} T^{20} - \)\(25\!\cdots\!24\)\( p^{108} T^{22} + \)\(29\!\cdots\!88\)\( p^{140} T^{24} - \)\(28\!\cdots\!52\)\( p^{172} T^{26} + \)\(20\!\cdots\!16\)\( p^{204} T^{28} - 63371249746529137488 p^{238} T^{30} + p^{272} T^{32} \)
17 \( ( 1 + 220268400 p T + \)\(34\!\cdots\!44\)\( T^{2} - \)\(60\!\cdots\!00\)\( p T^{3} + \)\(63\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!00\)\( p T^{5} + \)\(81\!\cdots\!48\)\( T^{6} - \)\(30\!\cdots\!00\)\( p T^{7} + \)\(77\!\cdots\!98\)\( T^{8} - \)\(30\!\cdots\!00\)\( p^{18} T^{9} + \)\(81\!\cdots\!48\)\( p^{34} T^{10} - \)\(21\!\cdots\!00\)\( p^{52} T^{11} + \)\(63\!\cdots\!60\)\( p^{68} T^{12} - \)\(60\!\cdots\!00\)\( p^{86} T^{13} + \)\(34\!\cdots\!44\)\( p^{102} T^{14} + 220268400 p^{120} T^{15} + p^{136} T^{16} )^{2} \)
19 \( 1 - \)\(50\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{10} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{14} + \)\(94\!\cdots\!82\)\( T^{16} - \)\(15\!\cdots\!44\)\( p^{34} T^{18} + \)\(22\!\cdots\!64\)\( p^{68} T^{20} - \)\(27\!\cdots\!68\)\( p^{102} T^{22} + \)\(27\!\cdots\!00\)\( p^{136} T^{24} - \)\(21\!\cdots\!48\)\( p^{170} T^{26} + \)\(12\!\cdots\!60\)\( p^{204} T^{28} - \)\(50\!\cdots\!16\)\( p^{238} T^{30} + p^{272} T^{32} \)
23 \( ( 1 - 373422672960 T + \)\(64\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} - \)\(41\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!92\)\( T^{6} - \)\(69\!\cdots\!60\)\( T^{7} + \)\(63\!\cdots\!74\)\( T^{8} - \)\(69\!\cdots\!60\)\( p^{17} T^{9} + \)\(40\!\cdots\!92\)\( p^{34} T^{10} - \)\(41\!\cdots\!20\)\( p^{51} T^{11} + \)\(19\!\cdots\!52\)\( p^{68} T^{12} - \)\(17\!\cdots\!80\)\( p^{85} T^{13} + \)\(64\!\cdots\!76\)\( p^{102} T^{14} - 373422672960 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
29 \( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(75\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(81\!\cdots\!06\)\( T^{16} - \)\(10\!\cdots\!80\)\( p^{34} T^{18} + \)\(11\!\cdots\!20\)\( p^{68} T^{20} - \)\(10\!\cdots\!80\)\( p^{102} T^{22} + \)\(75\!\cdots\!16\)\( p^{136} T^{24} - \)\(44\!\cdots\!20\)\( p^{170} T^{26} + \)\(20\!\cdots\!80\)\( p^{204} T^{28} - \)\(62\!\cdots\!20\)\( p^{238} T^{30} + p^{272} T^{32} \)
31 \( ( 1 + 5144834816 p T + \)\(10\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!12\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!28\)\( T^{7} + \)\(49\!\cdots\!98\)\( T^{8} + \)\(39\!\cdots\!28\)\( p^{17} T^{9} + \)\(18\!\cdots\!80\)\( p^{34} T^{10} + \)\(11\!\cdots\!72\)\( p^{51} T^{11} + \)\(51\!\cdots\!12\)\( p^{68} T^{12} + \)\(16\!\cdots\!28\)\( p^{85} T^{13} + \)\(10\!\cdots\!04\)\( p^{102} T^{14} + 5144834816 p^{120} T^{15} + p^{136} T^{16} )^{2} \)
37 \( 1 - \)\(39\!\cdots\!44\)\( T^{2} + \)\(80\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!64\)\( T^{8} - \)\(90\!\cdots\!04\)\( T^{10} + \)\(60\!\cdots\!24\)\( T^{12} - \)\(34\!\cdots\!60\)\( T^{14} + \)\(17\!\cdots\!90\)\( T^{16} - \)\(34\!\cdots\!60\)\( p^{34} T^{18} + \)\(60\!\cdots\!24\)\( p^{68} T^{20} - \)\(90\!\cdots\!04\)\( p^{102} T^{22} + \)\(11\!\cdots\!64\)\( p^{136} T^{24} - \)\(10\!\cdots\!72\)\( p^{170} T^{26} + \)\(80\!\cdots\!76\)\( p^{204} T^{28} - \)\(39\!\cdots\!44\)\( p^{238} T^{30} + p^{272} T^{32} \)
41 \( ( 1 - 3741125768016 T + \)\(70\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!00\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!42\)\( T^{8} + \)\(61\!\cdots\!84\)\( p^{17} T^{9} + \)\(99\!\cdots\!52\)\( p^{34} T^{10} + \)\(14\!\cdots\!12\)\( p^{51} T^{11} + \)\(29\!\cdots\!00\)\( p^{68} T^{12} - \)\(14\!\cdots\!84\)\( p^{85} T^{13} + \)\(70\!\cdots\!80\)\( p^{102} T^{14} - 3741125768016 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
43 \( 1 - \)\(41\!\cdots\!32\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(97\!\cdots\!68\)\( T^{10} + \)\(74\!\cdots\!32\)\( T^{12} - \)\(50\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!30\)\( T^{16} - \)\(50\!\cdots\!72\)\( p^{34} T^{18} + \)\(74\!\cdots\!32\)\( p^{68} T^{20} - \)\(97\!\cdots\!68\)\( p^{102} T^{22} + \)\(11\!\cdots\!40\)\( p^{136} T^{24} - \)\(11\!\cdots\!28\)\( p^{170} T^{26} + \)\(84\!\cdots\!12\)\( p^{204} T^{28} - \)\(41\!\cdots\!32\)\( p^{238} T^{30} + p^{272} T^{32} \)
47 \( ( 1 + 188349402410880 T + \)\(19\!\cdots\!72\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!52\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!84\)\( T^{6} + \)\(99\!\cdots\!60\)\( T^{7} + \)\(28\!\cdots\!54\)\( T^{8} + \)\(99\!\cdots\!60\)\( p^{17} T^{9} + \)\(87\!\cdots\!84\)\( p^{34} T^{10} + \)\(23\!\cdots\!00\)\( p^{51} T^{11} + \)\(17\!\cdots\!52\)\( p^{68} T^{12} + \)\(31\!\cdots\!00\)\( p^{85} T^{13} + \)\(19\!\cdots\!72\)\( p^{102} T^{14} + 188349402410880 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
53 \( 1 - \)\(12\!\cdots\!88\)\( T^{2} + \)\(92\!\cdots\!64\)\( T^{4} - \)\(47\!\cdots\!52\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{8} - \)\(62\!\cdots\!40\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{16} - \)\(43\!\cdots\!64\)\( p^{34} T^{18} + \)\(17\!\cdots\!08\)\( p^{68} T^{20} - \)\(62\!\cdots\!40\)\( p^{102} T^{22} + \)\(18\!\cdots\!96\)\( p^{136} T^{24} - \)\(47\!\cdots\!52\)\( p^{170} T^{26} + \)\(92\!\cdots\!64\)\( p^{204} T^{28} - \)\(12\!\cdots\!88\)\( p^{238} T^{30} + p^{272} T^{32} \)
59 \( 1 - \)\(20\!\cdots\!76\)\( p T^{2} + \)\(69\!\cdots\!76\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(69\!\cdots\!16\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!32\)\( T^{12} - \)\(35\!\cdots\!68\)\( T^{14} + \)\(46\!\cdots\!62\)\( T^{16} - \)\(35\!\cdots\!68\)\( p^{34} T^{18} + \)\(24\!\cdots\!32\)\( p^{68} T^{20} - \)\(14\!\cdots\!00\)\( p^{102} T^{22} + \)\(69\!\cdots\!16\)\( p^{136} T^{24} - \)\(25\!\cdots\!84\)\( p^{170} T^{26} + \)\(69\!\cdots\!76\)\( p^{204} T^{28} - \)\(20\!\cdots\!76\)\( p^{239} T^{30} + p^{272} T^{32} \)
61 \( 1 - \)\(16\!\cdots\!44\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(64\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!36\)\( T^{8} - \)\(82\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(64\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!02\)\( T^{16} - \)\(64\!\cdots\!28\)\( p^{34} T^{18} + \)\(24\!\cdots\!72\)\( p^{68} T^{20} - \)\(82\!\cdots\!80\)\( p^{102} T^{22} + \)\(24\!\cdots\!36\)\( p^{136} T^{24} - \)\(64\!\cdots\!84\)\( p^{170} T^{26} + \)\(12\!\cdots\!56\)\( p^{204} T^{28} - \)\(16\!\cdots\!44\)\( p^{238} T^{30} + p^{272} T^{32} \)
67 \( 1 - \)\(82\!\cdots\!12\)\( T^{2} + \)\(35\!\cdots\!44\)\( T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(47\!\cdots\!60\)\( T^{10} + \)\(75\!\cdots\!48\)\( T^{12} - \)\(10\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!82\)\( T^{16} - \)\(10\!\cdots\!76\)\( p^{34} T^{18} + \)\(75\!\cdots\!48\)\( p^{68} T^{20} - \)\(47\!\cdots\!60\)\( p^{102} T^{22} + \)\(24\!\cdots\!96\)\( p^{136} T^{24} - \)\(10\!\cdots\!88\)\( p^{170} T^{26} + \)\(35\!\cdots\!44\)\( p^{204} T^{28} - \)\(82\!\cdots\!12\)\( p^{238} T^{30} + p^{272} T^{32} \)
71 \( ( 1 - 4512963142788288 T + \)\(90\!\cdots\!40\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!80\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(78\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} + \)\(17\!\cdots\!64\)\( p^{17} T^{9} + \)\(78\!\cdots\!64\)\( p^{34} T^{10} + \)\(14\!\cdots\!12\)\( p^{51} T^{11} + \)\(31\!\cdots\!80\)\( p^{68} T^{12} - \)\(19\!\cdots\!68\)\( p^{85} T^{13} + \)\(90\!\cdots\!40\)\( p^{102} T^{14} - 4512963142788288 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
73 \( ( 1 - 5666001023059280 T + \)\(17\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!64\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!40\)\( p^{17} T^{9} + \)\(61\!\cdots\!08\)\( p^{34} T^{10} - \)\(66\!\cdots\!60\)\( p^{51} T^{11} + \)\(12\!\cdots\!64\)\( p^{68} T^{12} - \)\(10\!\cdots\!40\)\( p^{85} T^{13} + \)\(17\!\cdots\!28\)\( p^{102} T^{14} - 5666001023059280 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
79 \( ( 1 + 22649835696004224 T + \)\(78\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!48\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!08\)\( T^{6} - \)\(78\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!66\)\( T^{8} - \)\(78\!\cdots\!52\)\( p^{17} T^{9} + \)\(30\!\cdots\!08\)\( p^{34} T^{10} + \)\(15\!\cdots\!08\)\( p^{51} T^{11} + \)\(22\!\cdots\!48\)\( p^{68} T^{12} + \)\(10\!\cdots\!12\)\( p^{85} T^{13} + \)\(78\!\cdots\!40\)\( p^{102} T^{14} + 22649835696004224 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
83 \( 1 - \)\(26\!\cdots\!92\)\( T^{2} + \)\(34\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{10} + \)\(71\!\cdots\!12\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(34\!\cdots\!12\)\( p^{34} T^{18} + \)\(71\!\cdots\!12\)\( p^{68} T^{20} - \)\(13\!\cdots\!28\)\( p^{102} T^{22} + \)\(21\!\cdots\!00\)\( p^{136} T^{24} - \)\(29\!\cdots\!68\)\( p^{170} T^{26} + \)\(34\!\cdots\!32\)\( p^{204} T^{28} - \)\(26\!\cdots\!92\)\( p^{238} T^{30} + p^{272} T^{32} \)
89 \( ( 1 + 34939587304383024 T + \)\(39\!\cdots\!00\)\( T^{2} + \)\(75\!\cdots\!32\)\( T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(60\!\cdots\!48\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} + \)\(76\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!66\)\( T^{8} + \)\(76\!\cdots\!28\)\( p^{17} T^{9} + \)\(64\!\cdots\!08\)\( p^{34} T^{10} + \)\(60\!\cdots\!48\)\( p^{51} T^{11} + \)\(62\!\cdots\!48\)\( p^{68} T^{12} + \)\(75\!\cdots\!32\)\( p^{85} T^{13} + \)\(39\!\cdots\!00\)\( p^{102} T^{14} + 34939587304383024 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
97 \( ( 1 - 47796699301090320 T + \)\(35\!\cdots\!20\)\( p T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} - \)\(31\!\cdots\!00\)\( p^{17} T^{9} + \)\(55\!\cdots\!60\)\( p^{34} T^{10} - \)\(32\!\cdots\!40\)\( p^{51} T^{11} + \)\(54\!\cdots\!44\)\( p^{68} T^{12} - \)\(19\!\cdots\!20\)\( p^{85} T^{13} + \)\(35\!\cdots\!20\)\( p^{103} T^{14} - 47796699301090320 p^{119} T^{15} + p^{136} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.42236100638088223131837215769, −3.34416092327149839940736315786, −3.26197578022000518802863522119, −3.15373025124775396266202590568, −2.97717704296000622773981362798, −2.96628385141698808331269133597, −2.80710324285546941622272471045, −2.65632942381159082257042719693, −2.33131756261166549602944838397, −2.12566203620866197403036355806, −2.06939227516634202756925143554, −1.72892071410434151983222169508, −1.65474362341665800312596460035, −1.63999979014838998478265782047, −1.55303549721333137406206445400, −1.40750216629179864349843802112, −1.24440228023970873025986676931, −1.22236482801359392616534505069, −1.12854597882151162986182962752, −0.842591543557848077890502433286, −0.76062552356690262625588702070, −0.71467164609970527284521361178, −0.48336059363789582544197741461, −0.29246881389725511467881259021, −0.20320152641273845558803230875, 0.20320152641273845558803230875, 0.29246881389725511467881259021, 0.48336059363789582544197741461, 0.71467164609970527284521361178, 0.76062552356690262625588702070, 0.842591543557848077890502433286, 1.12854597882151162986182962752, 1.22236482801359392616534505069, 1.24440228023970873025986676931, 1.40750216629179864349843802112, 1.55303549721333137406206445400, 1.63999979014838998478265782047, 1.65474362341665800312596460035, 1.72892071410434151983222169508, 2.06939227516634202756925143554, 2.12566203620866197403036355806, 2.33131756261166549602944838397, 2.65632942381159082257042719693, 2.80710324285546941622272471045, 2.96628385141698808331269133597, 2.97717704296000622773981362798, 3.15373025124775396266202590568, 3.26197578022000518802863522119, 3.34416092327149839940736315786, 3.42236100638088223131837215769

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

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