Dirichlet series
| L(s) = 1 | − 10·3-s − 42·5-s + 86·7-s + 50·9-s + 536·13-s + 420·15-s − 1.82e3·17-s − 2.51e3·19-s − 860·21-s + 7.64e3·23-s + 5.45e3·25-s + 3.11e3·27-s − 3.61e3·35-s − 7.62e3·37-s − 5.36e3·39-s − 2.12e4·41-s − 2.00e4·43-s − 2.10e3·45-s − 2.52e4·47-s + 3.69e3·49-s + 1.82e4·51-s + 1.28e4·53-s + 2.51e4·57-s + 1.42e5·59-s − 2.05e4·61-s + 4.30e3·63-s − 2.25e4·65-s + ⋯ |
| L(s) = 1 | − 0.641·3-s − 0.751·5-s + 0.663·7-s + 0.205·9-s + 0.879·13-s + 0.481·15-s − 1.53·17-s − 1.59·19-s − 0.425·21-s + 3.01·23-s + 1.74·25-s + 0.822·27-s − 0.498·35-s − 0.915·37-s − 0.564·39-s − 1.97·41-s − 1.64·43-s − 0.154·45-s − 1.67·47-s + 0.220·49-s + 0.984·51-s + 0.628·53-s + 1.02·57-s + 5.33·59-s − 0.707·61-s + 0.136·63-s − 0.660·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{80} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.53580\times 10^{22}\) |
| Root analytic conductor: | \(5.06570\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{80} \cdot 5^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(55.47637648\) |
| \(L(\frac12)\) | \(\approx\) | \(55.47637648\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 42 T - 3688 T^{2} - 67274 p T^{3} - 269188 p T^{4} + 29298562 p^{2} T^{5} + 4265352 p^{5} T^{6} - 108153202 p^{5} T^{7} - 119685282 p^{7} T^{8} - 108153202 p^{10} T^{9} + 4265352 p^{15} T^{10} + 29298562 p^{17} T^{11} - 269188 p^{21} T^{12} - 67274 p^{26} T^{13} - 3688 p^{30} T^{14} + 42 p^{35} T^{15} + p^{40} T^{16} \) | |
| good | 3 | \( 1 + 10 T + 50 T^{2} - 346 p^{2} T^{3} - 71776 T^{4} - 1530706 T^{5} - 6869762 T^{6} - 94172042 p T^{7} - 1414085636 T^{8} + 89177429938 T^{9} + 2520680189450 T^{10} + 16643146834714 p T^{11} + 56290902591328 p^{2} T^{12} - 8762550581434 p^{4} T^{13} - 1398712181501050 p^{4} T^{14} - 7474668267920306 p^{5} T^{15} - 60297727631649098 p^{6} T^{16} - 7474668267920306 p^{10} T^{17} - 1398712181501050 p^{14} T^{18} - 8762550581434 p^{19} T^{19} + 56290902591328 p^{22} T^{20} + 16643146834714 p^{26} T^{21} + 2520680189450 p^{30} T^{22} + 89177429938 p^{35} T^{23} - 1414085636 p^{40} T^{24} - 94172042 p^{46} T^{25} - 6869762 p^{50} T^{26} - 1530706 p^{55} T^{27} - 71776 p^{60} T^{28} - 346 p^{67} T^{29} + 50 p^{70} T^{30} + 10 p^{75} T^{31} + p^{80} T^{32} \) |
| 7 | \( 1 - 86 T + 3698 T^{2} + 2808286 T^{3} - 810527584 T^{4} - 2504717314 T^{5} + 7155971823534 T^{6} - 2324030762443574 T^{7} + 166847822527644348 T^{8} + 29386477550704152562 T^{9} - \)\(38\!\cdots\!46\)\( T^{10} + \)\(51\!\cdots\!62\)\( T^{11} + \)\(38\!\cdots\!24\)\( T^{12} - \)\(97\!\cdots\!26\)\( T^{13} + \)\(45\!\cdots\!90\)\( T^{14} + \)\(35\!\cdots\!26\)\( T^{15} - \)\(24\!\cdots\!62\)\( T^{16} + \)\(35\!\cdots\!26\)\( p^{5} T^{17} + \)\(45\!\cdots\!90\)\( p^{10} T^{18} - \)\(97\!\cdots\!26\)\( p^{15} T^{19} + \)\(38\!\cdots\!24\)\( p^{20} T^{20} + \)\(51\!\cdots\!62\)\( p^{25} T^{21} - \)\(38\!\cdots\!46\)\( p^{30} T^{22} + 29386477550704152562 p^{35} T^{23} + 166847822527644348 p^{40} T^{24} - 2324030762443574 p^{45} T^{25} + 7155971823534 p^{50} T^{26} - 2504717314 p^{55} T^{27} - 810527584 p^{60} T^{28} + 2808286 p^{65} T^{29} + 3698 p^{70} T^{30} - 86 p^{75} T^{31} + p^{80} T^{32} \) | |
| 11 | \( 1 - 580860 T^{2} + 233332093584 T^{4} - 74638580678134772 T^{6} + \)\(20\!\cdots\!20\)\( T^{8} - \)\(46\!\cdots\!24\)\( T^{10} + \)\(96\!\cdots\!08\)\( T^{12} - \)\(18\!\cdots\!84\)\( T^{14} + \)\(30\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!84\)\( p^{10} T^{18} + \)\(96\!\cdots\!08\)\( p^{20} T^{20} - \)\(46\!\cdots\!24\)\( p^{30} T^{22} + \)\(20\!\cdots\!20\)\( p^{40} T^{24} - 74638580678134772 p^{50} T^{26} + 233332093584 p^{60} T^{28} - 580860 p^{70} T^{30} + p^{80} T^{32} \) | |
| 13 | \( 1 - 536 T + 143648 T^{2} - 293170824 T^{3} - 185935117448 T^{4} + 112653613917608 T^{5} + 9301436713751904 T^{6} + 69204174166800470584 T^{7} + \)\(13\!\cdots\!60\)\( T^{8} - \)\(25\!\cdots\!64\)\( T^{9} - \)\(50\!\cdots\!68\)\( T^{10} - \)\(90\!\cdots\!08\)\( T^{11} + \)\(40\!\cdots\!76\)\( T^{12} + \)\(23\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!08\)\( T^{14} + \)\(14\!\cdots\!44\)\( T^{15} - \)\(96\!\cdots\!10\)\( T^{16} + \)\(14\!\cdots\!44\)\( p^{5} T^{17} + \)\(13\!\cdots\!08\)\( p^{10} T^{18} + \)\(23\!\cdots\!68\)\( p^{15} T^{19} + \)\(40\!\cdots\!76\)\( p^{20} T^{20} - \)\(90\!\cdots\!08\)\( p^{25} T^{21} - \)\(50\!\cdots\!68\)\( p^{30} T^{22} - \)\(25\!\cdots\!64\)\( p^{35} T^{23} + \)\(13\!\cdots\!60\)\( p^{40} T^{24} + 69204174166800470584 p^{45} T^{25} + 9301436713751904 p^{50} T^{26} + 112653613917608 p^{55} T^{27} - 185935117448 p^{60} T^{28} - 293170824 p^{65} T^{29} + 143648 p^{70} T^{30} - 536 p^{75} T^{31} + p^{80} T^{32} \) | |
| 17 | \( 1 + 1828 T + 1670792 T^{2} - 327758324 T^{3} + 3015753425928 T^{4} + 10493189333351412 T^{5} + 14196566162828930648 T^{6} + \)\(64\!\cdots\!44\)\( T^{7} + \)\(60\!\cdots\!96\)\( T^{8} + \)\(26\!\cdots\!20\)\( T^{9} + \)\(48\!\cdots\!00\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!88\)\( T^{12} + \)\(53\!\cdots\!44\)\( T^{13} + \)\(10\!\cdots\!76\)\( T^{14} + \)\(94\!\cdots\!68\)\( T^{15} + \)\(94\!\cdots\!74\)\( T^{16} + \)\(94\!\cdots\!68\)\( p^{5} T^{17} + \)\(10\!\cdots\!76\)\( p^{10} T^{18} + \)\(53\!\cdots\!44\)\( p^{15} T^{19} + \)\(30\!\cdots\!88\)\( p^{20} T^{20} + \)\(40\!\cdots\!40\)\( p^{25} T^{21} + \)\(48\!\cdots\!00\)\( p^{30} T^{22} + \)\(26\!\cdots\!20\)\( p^{35} T^{23} + \)\(60\!\cdots\!96\)\( p^{40} T^{24} + \)\(64\!\cdots\!44\)\( p^{45} T^{25} + 14196566162828930648 p^{50} T^{26} + 10493189333351412 p^{55} T^{27} + 3015753425928 p^{60} T^{28} - 327758324 p^{65} T^{29} + 1670792 p^{70} T^{30} + 1828 p^{75} T^{31} + p^{80} T^{32} \) | |
| 19 | \( ( 1 + 1256 T + 11933496 T^{2} + 17157951112 T^{3} + 74779232061244 T^{4} + 104521619172501960 T^{5} + \)\(31\!\cdots\!88\)\( T^{6} + \)\(20\!\cdots\!56\)\( p T^{7} + \)\(93\!\cdots\!18\)\( T^{8} + \)\(20\!\cdots\!56\)\( p^{6} T^{9} + \)\(31\!\cdots\!88\)\( p^{10} T^{10} + 104521619172501960 p^{15} T^{11} + 74779232061244 p^{20} T^{12} + 17157951112 p^{25} T^{13} + 11933496 p^{30} T^{14} + 1256 p^{35} T^{15} + p^{40} T^{16} )^{2} \) | |
| 23 | \( 1 - 7642 T + 29200082 T^{2} - 114423930750 T^{3} + 421188911449792 T^{4} - 1070557300674963934 T^{5} + \)\(24\!\cdots\!34\)\( T^{6} - \)\(58\!\cdots\!94\)\( T^{7} + \)\(77\!\cdots\!36\)\( T^{8} + \)\(18\!\cdots\!62\)\( T^{9} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(97\!\cdots\!06\)\( T^{11} - \)\(29\!\cdots\!68\)\( T^{12} + \)\(18\!\cdots\!70\)\( T^{13} + \)\(73\!\cdots\!74\)\( T^{14} - \)\(37\!\cdots\!54\)\( T^{15} + \)\(12\!\cdots\!78\)\( T^{16} - \)\(37\!\cdots\!54\)\( p^{5} T^{17} + \)\(73\!\cdots\!74\)\( p^{10} T^{18} + \)\(18\!\cdots\!70\)\( p^{15} T^{19} - \)\(29\!\cdots\!68\)\( p^{20} T^{20} + \)\(97\!\cdots\!06\)\( p^{25} T^{21} - \)\(19\!\cdots\!06\)\( p^{30} T^{22} + \)\(18\!\cdots\!62\)\( p^{35} T^{23} + \)\(77\!\cdots\!36\)\( p^{40} T^{24} - \)\(58\!\cdots\!94\)\( p^{45} T^{25} + \)\(24\!\cdots\!34\)\( p^{50} T^{26} - 1070557300674963934 p^{55} T^{27} + 421188911449792 p^{60} T^{28} - 114423930750 p^{65} T^{29} + 29200082 p^{70} T^{30} - 7642 p^{75} T^{31} + p^{80} T^{32} \) | |
| 29 | \( 1 - 129450464 T^{2} + 9016787664231832 T^{4} - \)\(15\!\cdots\!64\)\( p T^{6} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(53\!\cdots\!48\)\( T^{10} + \)\(14\!\cdots\!16\)\( T^{12} - \)\(34\!\cdots\!32\)\( T^{14} + \)\(74\!\cdots\!82\)\( T^{16} - \)\(34\!\cdots\!32\)\( p^{10} T^{18} + \)\(14\!\cdots\!16\)\( p^{20} T^{20} - \)\(53\!\cdots\!48\)\( p^{30} T^{22} + \)\(16\!\cdots\!60\)\( p^{40} T^{24} - \)\(15\!\cdots\!64\)\( p^{51} T^{26} + 9016787664231832 p^{60} T^{28} - 129450464 p^{70} T^{30} + p^{80} T^{32} \) | |
| 31 | \( 1 - 221015276 T^{2} + 25619057922480912 T^{4} - \)\(20\!\cdots\!04\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{8} - \)\(64\!\cdots\!72\)\( T^{10} + \)\(27\!\cdots\!96\)\( T^{12} - \)\(98\!\cdots\!48\)\( T^{14} + \)\(30\!\cdots\!82\)\( T^{16} - \)\(98\!\cdots\!48\)\( p^{10} T^{18} + \)\(27\!\cdots\!96\)\( p^{20} T^{20} - \)\(64\!\cdots\!72\)\( p^{30} T^{22} + \)\(12\!\cdots\!00\)\( p^{40} T^{24} - \)\(20\!\cdots\!04\)\( p^{50} T^{26} + 25619057922480912 p^{60} T^{28} - 221015276 p^{70} T^{30} + p^{80} T^{32} \) | |
| 37 | \( 1 + 7620 T + 29032200 T^{2} + 1965688293660 T^{3} + 22987009602091912 T^{4} + 46723925283632703348 T^{5} + \)\(16\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!40\)\( T^{7} + \)\(98\!\cdots\!88\)\( T^{8} + \)\(66\!\cdots\!16\)\( T^{9} + \)\(19\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(55\!\cdots\!44\)\( T^{12} + \)\(97\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!44\)\( T^{14} + \)\(70\!\cdots\!36\)\( T^{15} + \)\(48\!\cdots\!10\)\( T^{16} + \)\(70\!\cdots\!36\)\( p^{5} T^{17} + \)\(11\!\cdots\!44\)\( p^{10} T^{18} + \)\(97\!\cdots\!64\)\( p^{15} T^{19} + \)\(55\!\cdots\!44\)\( p^{20} T^{20} + \)\(13\!\cdots\!00\)\( p^{25} T^{21} + \)\(19\!\cdots\!52\)\( p^{30} T^{22} + \)\(66\!\cdots\!16\)\( p^{35} T^{23} + \)\(98\!\cdots\!88\)\( p^{40} T^{24} + \)\(27\!\cdots\!40\)\( p^{45} T^{25} + \)\(16\!\cdots\!60\)\( p^{50} T^{26} + 46723925283632703348 p^{55} T^{27} + 22987009602091912 p^{60} T^{28} + 1965688293660 p^{65} T^{29} + 29032200 p^{70} T^{30} + 7620 p^{75} T^{31} + p^{80} T^{32} \) | |
| 41 | \( ( 1 + 10642 T + 229001616 T^{2} + 2308310140798 T^{3} + 41488616565167196 T^{4} + 8188562606838649722 p T^{5} + \)\(55\!\cdots\!56\)\( T^{6} + \)\(38\!\cdots\!10\)\( T^{7} + \)\(15\!\cdots\!50\)\( p T^{8} + \)\(38\!\cdots\!10\)\( p^{5} T^{9} + \)\(55\!\cdots\!56\)\( p^{10} T^{10} + 8188562606838649722 p^{16} T^{11} + 41488616565167196 p^{20} T^{12} + 2308310140798 p^{25} T^{13} + 229001616 p^{30} T^{14} + 10642 p^{35} T^{15} + p^{40} T^{16} )^{2} \) | |
| 43 | \( 1 + 20002 T + 200040002 T^{2} + 4643840984270 T^{3} + 57483685077584672 T^{4} - 83126825813520796778 T^{5} - \)\(23\!\cdots\!50\)\( T^{6} - \)\(17\!\cdots\!90\)\( p T^{7} - \)\(32\!\cdots\!64\)\( T^{8} - \)\(35\!\cdots\!86\)\( T^{9} - \)\(28\!\cdots\!02\)\( T^{10} - \)\(51\!\cdots\!50\)\( T^{11} - \)\(19\!\cdots\!52\)\( T^{12} + \)\(54\!\cdots\!78\)\( T^{13} + \)\(53\!\cdots\!62\)\( T^{14} + \)\(11\!\cdots\!58\)\( T^{15} + \)\(24\!\cdots\!66\)\( T^{16} + \)\(11\!\cdots\!58\)\( p^{5} T^{17} + \)\(53\!\cdots\!62\)\( p^{10} T^{18} + \)\(54\!\cdots\!78\)\( p^{15} T^{19} - \)\(19\!\cdots\!52\)\( p^{20} T^{20} - \)\(51\!\cdots\!50\)\( p^{25} T^{21} - \)\(28\!\cdots\!02\)\( p^{30} T^{22} - \)\(35\!\cdots\!86\)\( p^{35} T^{23} - \)\(32\!\cdots\!64\)\( p^{40} T^{24} - \)\(17\!\cdots\!90\)\( p^{46} T^{25} - \)\(23\!\cdots\!50\)\( p^{50} T^{26} - 83126825813520796778 p^{55} T^{27} + 57483685077584672 p^{60} T^{28} + 4643840984270 p^{65} T^{29} + 200040002 p^{70} T^{30} + 20002 p^{75} T^{31} + p^{80} T^{32} \) | |
| 47 | \( 1 + 25298 T + 319994402 T^{2} + 6541896537590 T^{3} - 11741966210019840 T^{4} - \)\(87\!\cdots\!98\)\( T^{5} + \)\(31\!\cdots\!26\)\( T^{6} - \)\(28\!\cdots\!26\)\( T^{7} + \)\(89\!\cdots\!80\)\( T^{8} + \)\(49\!\cdots\!50\)\( T^{9} - \)\(48\!\cdots\!58\)\( T^{10} + \)\(14\!\cdots\!46\)\( T^{11} + \)\(91\!\cdots\!24\)\( T^{12} - \)\(38\!\cdots\!50\)\( T^{13} + \)\(47\!\cdots\!30\)\( T^{14} + \)\(60\!\cdots\!90\)\( T^{15} + \)\(93\!\cdots\!70\)\( T^{16} + \)\(60\!\cdots\!90\)\( p^{5} T^{17} + \)\(47\!\cdots\!30\)\( p^{10} T^{18} - \)\(38\!\cdots\!50\)\( p^{15} T^{19} + \)\(91\!\cdots\!24\)\( p^{20} T^{20} + \)\(14\!\cdots\!46\)\( p^{25} T^{21} - \)\(48\!\cdots\!58\)\( p^{30} T^{22} + \)\(49\!\cdots\!50\)\( p^{35} T^{23} + \)\(89\!\cdots\!80\)\( p^{40} T^{24} - \)\(28\!\cdots\!26\)\( p^{45} T^{25} + \)\(31\!\cdots\!26\)\( p^{50} T^{26} - \)\(87\!\cdots\!98\)\( p^{55} T^{27} - 11741966210019840 p^{60} T^{28} + 6541896537590 p^{65} T^{29} + 319994402 p^{70} T^{30} + 25298 p^{75} T^{31} + p^{80} T^{32} \) | |
| 53 | \( 1 - 12852 T + 82586952 T^{2} + 12779415797620 T^{3} - 141875027604640888 T^{4} - \)\(55\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!20\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} - \)\(86\!\cdots\!84\)\( T^{8} + \)\(14\!\cdots\!16\)\( T^{9} + \)\(64\!\cdots\!88\)\( T^{10} - \)\(74\!\cdots\!80\)\( T^{11} + \)\(67\!\cdots\!08\)\( T^{12} + \)\(29\!\cdots\!72\)\( T^{13} - \)\(53\!\cdots\!88\)\( T^{14} - \)\(64\!\cdots\!48\)\( T^{15} + \)\(25\!\cdots\!06\)\( T^{16} - \)\(64\!\cdots\!48\)\( p^{5} T^{17} - \)\(53\!\cdots\!88\)\( p^{10} T^{18} + \)\(29\!\cdots\!72\)\( p^{15} T^{19} + \)\(67\!\cdots\!08\)\( p^{20} T^{20} - \)\(74\!\cdots\!80\)\( p^{25} T^{21} + \)\(64\!\cdots\!88\)\( p^{30} T^{22} + \)\(14\!\cdots\!16\)\( p^{35} T^{23} - \)\(86\!\cdots\!84\)\( p^{40} T^{24} - \)\(16\!\cdots\!60\)\( p^{45} T^{25} + \)\(16\!\cdots\!20\)\( p^{50} T^{26} - \)\(55\!\cdots\!72\)\( p^{55} T^{27} - 141875027604640888 p^{60} T^{28} + 12779415797620 p^{65} T^{29} + 82586952 p^{70} T^{30} - 12852 p^{75} T^{31} + p^{80} T^{32} \) | |
| 59 | \( ( 1 - 71352 T + 5147553272 T^{2} - 263083260595160 T^{3} + 11526051017122699388 T^{4} - \)\(44\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} - \)\(46\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(46\!\cdots\!12\)\( p^{5} T^{9} + \)\(15\!\cdots\!16\)\( p^{10} T^{10} - \)\(44\!\cdots\!28\)\( p^{15} T^{11} + 11526051017122699388 p^{20} T^{12} - 263083260595160 p^{25} T^{13} + 5147553272 p^{30} T^{14} - 71352 p^{35} T^{15} + p^{40} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 10282 T + 3727559576 T^{2} + 11244934877598 T^{3} + 6582098174228731436 T^{4} - \)\(31\!\cdots\!30\)\( T^{5} + \)\(74\!\cdots\!72\)\( T^{6} - \)\(78\!\cdots\!62\)\( T^{7} + \)\(66\!\cdots\!54\)\( T^{8} - \)\(78\!\cdots\!62\)\( p^{5} T^{9} + \)\(74\!\cdots\!72\)\( p^{10} T^{10} - \)\(31\!\cdots\!30\)\( p^{15} T^{11} + 6582098174228731436 p^{20} T^{12} + 11244934877598 p^{25} T^{13} + 3727559576 p^{30} T^{14} + 10282 p^{35} T^{15} + p^{40} T^{16} )^{2} \) | |
| 67 | \( 1 - 10506 T + 55188018 T^{2} - 64685094733110 T^{3} - 1145101582269918112 T^{4} + \)\(11\!\cdots\!66\)\( T^{5} + \)\(90\!\cdots\!70\)\( T^{6} + \)\(13\!\cdots\!94\)\( T^{7} - \)\(58\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!70\)\( T^{9} + \)\(57\!\cdots\!86\)\( T^{10} + \)\(37\!\cdots\!82\)\( T^{11} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(51\!\cdots\!34\)\( T^{13} + \)\(11\!\cdots\!26\)\( T^{14} - \)\(28\!\cdots\!74\)\( T^{15} - \)\(82\!\cdots\!58\)\( T^{16} - \)\(28\!\cdots\!74\)\( p^{5} T^{17} + \)\(11\!\cdots\!26\)\( p^{10} T^{18} - \)\(51\!\cdots\!34\)\( p^{15} T^{19} + \)\(12\!\cdots\!56\)\( p^{20} T^{20} + \)\(37\!\cdots\!82\)\( p^{25} T^{21} + \)\(57\!\cdots\!86\)\( p^{30} T^{22} - \)\(10\!\cdots\!70\)\( p^{35} T^{23} - \)\(58\!\cdots\!16\)\( p^{40} T^{24} + \)\(13\!\cdots\!94\)\( p^{45} T^{25} + \)\(90\!\cdots\!70\)\( p^{50} T^{26} + \)\(11\!\cdots\!66\)\( p^{55} T^{27} - 1145101582269918112 p^{60} T^{28} - 64685094733110 p^{65} T^{29} + 55188018 p^{70} T^{30} - 10506 p^{75} T^{31} + p^{80} T^{32} \) | |
| 71 | \( 1 - 13607740108 T^{2} + 94972263534983306896 T^{4} - \)\(45\!\cdots\!72\)\( T^{6} + \)\(16\!\cdots\!96\)\( T^{8} - \)\(50\!\cdots\!76\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{12} - \)\(27\!\cdots\!08\)\( T^{14} + \)\(53\!\cdots\!06\)\( T^{16} - \)\(27\!\cdots\!08\)\( p^{10} T^{18} + \)\(12\!\cdots\!40\)\( p^{20} T^{20} - \)\(50\!\cdots\!76\)\( p^{30} T^{22} + \)\(16\!\cdots\!96\)\( p^{40} T^{24} - \)\(45\!\cdots\!72\)\( p^{50} T^{26} + 94972263534983306896 p^{60} T^{28} - 13607740108 p^{70} T^{30} + p^{80} T^{32} \) | |
| 73 | \( 1 - 15432 T + 119073312 T^{2} - 56032493808632 T^{3} + 4235186228799589816 T^{4} + \)\(61\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!24\)\( T^{7} - \)\(89\!\cdots\!60\)\( T^{8} + \)\(71\!\cdots\!68\)\( T^{9} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(11\!\cdots\!04\)\( T^{11} - \)\(29\!\cdots\!68\)\( T^{12} - \)\(26\!\cdots\!56\)\( T^{13} + \)\(53\!\cdots\!44\)\( T^{14} + \)\(14\!\cdots\!88\)\( T^{15} + \)\(30\!\cdots\!30\)\( T^{16} + \)\(14\!\cdots\!88\)\( p^{5} T^{17} + \)\(53\!\cdots\!44\)\( p^{10} T^{18} - \)\(26\!\cdots\!56\)\( p^{15} T^{19} - \)\(29\!\cdots\!68\)\( p^{20} T^{20} + \)\(11\!\cdots\!04\)\( p^{25} T^{21} - \)\(22\!\cdots\!48\)\( p^{30} T^{22} + \)\(71\!\cdots\!68\)\( p^{35} T^{23} - \)\(89\!\cdots\!60\)\( p^{40} T^{24} - \)\(11\!\cdots\!24\)\( p^{45} T^{25} + \)\(12\!\cdots\!48\)\( p^{50} T^{26} + \)\(61\!\cdots\!96\)\( p^{55} T^{27} + 4235186228799589816 p^{60} T^{28} - 56032493808632 p^{65} T^{29} + 119073312 p^{70} T^{30} - 15432 p^{75} T^{31} + p^{80} T^{32} \) | |
| 79 | \( ( 1 - 79672 T + 15921600120 T^{2} - 883492472350200 T^{3} + \)\(11\!\cdots\!40\)\( T^{4} - \)\(53\!\cdots\!80\)\( T^{5} + \)\(56\!\cdots\!80\)\( T^{6} - \)\(23\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!20\)\( p^{5} T^{9} + \)\(56\!\cdots\!80\)\( p^{10} T^{10} - \)\(53\!\cdots\!80\)\( p^{15} T^{11} + \)\(11\!\cdots\!40\)\( p^{20} T^{12} - 883492472350200 p^{25} T^{13} + 15921600120 p^{30} T^{14} - 79672 p^{35} T^{15} + p^{40} T^{16} )^{2} \) | |
| 83 | \( 1 - 61222 T + 1874066642 T^{2} - 12255516798122 T^{3} - 16226769598202256704 T^{4} + \)\(11\!\cdots\!34\)\( T^{5} + \)\(23\!\cdots\!62\)\( T^{6} - \)\(43\!\cdots\!38\)\( T^{7} + \)\(11\!\cdots\!04\)\( T^{8} + \)\(19\!\cdots\!26\)\( T^{9} - \)\(94\!\cdots\!78\)\( T^{10} + \)\(75\!\cdots\!74\)\( T^{11} + \)\(22\!\cdots\!04\)\( T^{12} - \)\(26\!\cdots\!90\)\( T^{13} + \)\(45\!\cdots\!30\)\( T^{14} + \)\(37\!\cdots\!50\)\( T^{15} - \)\(11\!\cdots\!10\)\( T^{16} + \)\(37\!\cdots\!50\)\( p^{5} T^{17} + \)\(45\!\cdots\!30\)\( p^{10} T^{18} - \)\(26\!\cdots\!90\)\( p^{15} T^{19} + \)\(22\!\cdots\!04\)\( p^{20} T^{20} + \)\(75\!\cdots\!74\)\( p^{25} T^{21} - \)\(94\!\cdots\!78\)\( p^{30} T^{22} + \)\(19\!\cdots\!26\)\( p^{35} T^{23} + \)\(11\!\cdots\!04\)\( p^{40} T^{24} - \)\(43\!\cdots\!38\)\( p^{45} T^{25} + \)\(23\!\cdots\!62\)\( p^{50} T^{26} + \)\(11\!\cdots\!34\)\( p^{55} T^{27} - 16226769598202256704 p^{60} T^{28} - 12255516798122 p^{65} T^{29} + 1874066642 p^{70} T^{30} - 61222 p^{75} T^{31} + p^{80} T^{32} \) | |
| 89 | \( 1 - 59849990032 T^{2} + \)\(17\!\cdots\!44\)\( T^{4} - \)\(33\!\cdots\!56\)\( T^{6} + \)\(45\!\cdots\!84\)\( T^{8} - \)\(48\!\cdots\!60\)\( T^{10} + \)\(42\!\cdots\!32\)\( T^{12} - \)\(30\!\cdots\!72\)\( T^{14} + \)\(18\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!72\)\( p^{10} T^{18} + \)\(42\!\cdots\!32\)\( p^{20} T^{20} - \)\(48\!\cdots\!60\)\( p^{30} T^{22} + \)\(45\!\cdots\!84\)\( p^{40} T^{24} - \)\(33\!\cdots\!56\)\( p^{50} T^{26} + \)\(17\!\cdots\!44\)\( p^{60} T^{28} - 59849990032 p^{70} T^{30} + p^{80} T^{32} \) | |
| 97 | \( 1 + 17344 T + 150407168 T^{2} - 246974906594432 T^{3} + 52594420471951685112 T^{4} + \)\(73\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!48\)\( T^{6} + \)\(60\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!36\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{9} + \)\(30\!\cdots\!48\)\( T^{10} - \)\(19\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} + \)\(57\!\cdots\!52\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(29\!\cdots\!16\)\( T^{15} + \)\(43\!\cdots\!78\)\( T^{16} + \)\(29\!\cdots\!16\)\( p^{5} T^{17} + \)\(10\!\cdots\!20\)\( p^{10} T^{18} + \)\(57\!\cdots\!52\)\( p^{15} T^{19} + \)\(13\!\cdots\!92\)\( p^{20} T^{20} - \)\(19\!\cdots\!76\)\( p^{25} T^{21} + \)\(30\!\cdots\!48\)\( p^{30} T^{22} + \)\(15\!\cdots\!56\)\( p^{35} T^{23} + \)\(10\!\cdots\!36\)\( p^{40} T^{24} + \)\(60\!\cdots\!32\)\( p^{45} T^{25} + \)\(15\!\cdots\!48\)\( p^{50} T^{26} + \)\(73\!\cdots\!08\)\( p^{55} T^{27} + 52594420471951685112 p^{60} T^{28} - 246974906594432 p^{65} T^{29} + 150407168 p^{70} T^{30} + 17344 p^{75} T^{31} + p^{80} T^{32} \) | |
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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
−2.85362722903485081013931113636, −2.74562200000437408681752524810, −2.43571397938896933413995454360, −2.23456100039570877041808649559, −2.22267968456416505943591538196, −2.21795646826953005670105699888, −2.06227356812062481815239264430, −1.95845931205801844769434723912, −1.92116645520560213442555609935, −1.84885803388222086020906999088, −1.66502583403857000031525412542, −1.63460240928748611426302421262, −1.45736372450880482479177556704, −1.39635354192445867532973689711, −1.35279993853222676703315272509, −0.957063696298394575275555005223, −0.821623201903202778975844262354, −0.75772929975766275804108063018, −0.70417976714304766017299195883, −0.64286379314842526860615844438, −0.64210427217530464440283275340, −0.51720716537245668907318231384, −0.45682829701277687483776477033, −0.24830470077930439217168376090, −0.18558809161459473944060720573, 0.18558809161459473944060720573, 0.24830470077930439217168376090, 0.45682829701277687483776477033, 0.51720716537245668907318231384, 0.64210427217530464440283275340, 0.64286379314842526860615844438, 0.70417976714304766017299195883, 0.75772929975766275804108063018, 0.821623201903202778975844262354, 0.957063696298394575275555005223, 1.35279993853222676703315272509, 1.39635354192445867532973689711, 1.45736372450880482479177556704, 1.63460240928748611426302421262, 1.66502583403857000031525412542, 1.84885803388222086020906999088, 1.92116645520560213442555609935, 1.95845931205801844769434723912, 2.06227356812062481815239264430, 2.21795646826953005670105699888, 2.22267968456416505943591538196, 2.23456100039570877041808649559, 2.43571397938896933413995454360, 2.74562200000437408681752524810, 2.85362722903485081013931113636
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.