Dirichlet series
| L(s) = 1 | + 32·2-s + 12·3-s + 544·4-s + 384·6-s + 56·7-s + 6.52e3·8-s − 38·9-s + 72·11-s + 6.52e3·12-s + 202·13-s + 1.79e3·14-s + 6.20e4·16-s + 216·17-s − 1.21e3·18-s + 100·19-s + 672·21-s + 2.30e3·22-s + 292·23-s + 7.83e4·24-s + 6.46e3·26-s − 950·27-s + 3.04e4·28-s + 400·29-s + 102·31-s + 4.96e5·32-s + 864·33-s + 6.91e3·34-s + ⋯ |
| L(s) = 1 | + 11.3·2-s + 2.30·3-s + 68·4-s + 26.1·6-s + 3.02·7-s + 288.·8-s − 1.40·9-s + 1.97·11-s + 157.·12-s + 4.30·13-s + 34.2·14-s + 969·16-s + 3.08·17-s − 15.9·18-s + 1.20·19-s + 6.98·21-s + 22.3·22-s + 2.64·23-s + 666.·24-s + 48.7·26-s − 6.77·27-s + 205.·28-s + 2.56·29-s + 0.590·31-s + 2.74e3·32-s + 4.55·33-s + 34.8·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{64}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.66332\times 10^{29}\) |
| Root analytic conductor: | \(8.58792\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{64} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(3.844224694\times10^{8}\) |
| \(L(\frac12)\) | \(\approx\) | \(3.844224694\times10^{8}\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T )^{16} \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 4 p T + 182 T^{2} - 1690 T^{3} + 16510 T^{4} - 131476 T^{5} + 1045592 T^{6} - 273986 p^{3} T^{7} + 51688750 T^{8} - 36889040 p^{2} T^{9} + 2112671518 T^{10} - 12560243876 T^{11} + 24776211497 p T^{12} - 46159842770 p^{2} T^{13} + 85955268980 p^{3} T^{14} - 12332435536022 T^{15} + 65639327016064 T^{16} - 12332435536022 p^{3} T^{17} + 85955268980 p^{9} T^{18} - 46159842770 p^{11} T^{19} + 24776211497 p^{13} T^{20} - 12560243876 p^{15} T^{21} + 2112671518 p^{18} T^{22} - 36889040 p^{23} T^{23} + 51688750 p^{24} T^{24} - 273986 p^{30} T^{25} + 1045592 p^{30} T^{26} - 131476 p^{33} T^{27} + 16510 p^{36} T^{28} - 1690 p^{39} T^{29} + 182 p^{42} T^{30} - 4 p^{46} T^{31} + p^{48} T^{32} \) |
| 7 | \( 1 - 8 p T + 3793 T^{2} - 150520 T^{3} + 6317810 T^{4} - 200782108 T^{5} + 937720684 p T^{6} - 178260474444 T^{7} + 4949706556555 T^{8} - 119487021698360 T^{9} + 2948949619922842 T^{10} - 64875175199063832 T^{11} + 1460374206166236106 T^{12} - 29709051367933713220 T^{13} + 88379568379782030375 p T^{14} - \)\(11\!\cdots\!44\)\( T^{15} + \)\(22\!\cdots\!44\)\( T^{16} - \)\(11\!\cdots\!44\)\( p^{3} T^{17} + 88379568379782030375 p^{7} T^{18} - 29709051367933713220 p^{9} T^{19} + 1460374206166236106 p^{12} T^{20} - 64875175199063832 p^{15} T^{21} + 2948949619922842 p^{18} T^{22} - 119487021698360 p^{21} T^{23} + 4949706556555 p^{24} T^{24} - 178260474444 p^{27} T^{25} + 937720684 p^{31} T^{26} - 200782108 p^{33} T^{27} + 6317810 p^{36} T^{28} - 150520 p^{39} T^{29} + 3793 p^{42} T^{30} - 8 p^{46} T^{31} + p^{48} T^{32} \) | |
| 11 | \( 1 - 72 T + 10801 T^{2} - 585490 T^{3} + 54002030 T^{4} - 2413069636 T^{5} + 175687447872 T^{6} - 6755893064946 T^{7} + 427224655766015 T^{8} - 14488113801807310 T^{9} + 840594153299418530 T^{10} - 25676652625663168240 T^{11} + \)\(14\!\cdots\!30\)\( T^{12} - \)\(39\!\cdots\!10\)\( T^{13} + \)\(21\!\cdots\!65\)\( T^{14} - \)\(56\!\cdots\!80\)\( T^{15} + \)\(29\!\cdots\!80\)\( T^{16} - \)\(56\!\cdots\!80\)\( p^{3} T^{17} + \)\(21\!\cdots\!65\)\( p^{6} T^{18} - \)\(39\!\cdots\!10\)\( p^{9} T^{19} + \)\(14\!\cdots\!30\)\( p^{12} T^{20} - 25676652625663168240 p^{15} T^{21} + 840594153299418530 p^{18} T^{22} - 14488113801807310 p^{21} T^{23} + 427224655766015 p^{24} T^{24} - 6755893064946 p^{27} T^{25} + 175687447872 p^{30} T^{26} - 2413069636 p^{33} T^{27} + 54002030 p^{36} T^{28} - 585490 p^{39} T^{29} + 10801 p^{42} T^{30} - 72 p^{45} T^{31} + p^{48} T^{32} \) | |
| 13 | \( 1 - 202 T + 39747 T^{2} - 5174290 T^{3} + 628525215 T^{4} - 62647129196 T^{5} + 5851267321207 T^{6} - 481890697519562 T^{7} + 37488973651649420 T^{8} - 2657845742463856980 T^{9} + 13781381934776090711 p T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(66\!\cdots\!46\)\( T^{12} - \)\(37\!\cdots\!10\)\( T^{13} + \)\(19\!\cdots\!50\)\( T^{14} - \)\(77\!\cdots\!74\)\( p T^{15} + \)\(48\!\cdots\!19\)\( T^{16} - \)\(77\!\cdots\!74\)\( p^{4} T^{17} + \)\(19\!\cdots\!50\)\( p^{6} T^{18} - \)\(37\!\cdots\!10\)\( p^{9} T^{19} + \)\(66\!\cdots\!46\)\( p^{12} T^{20} - \)\(11\!\cdots\!96\)\( p^{15} T^{21} + 13781381934776090711 p^{19} T^{22} - 2657845742463856980 p^{21} T^{23} + 37488973651649420 p^{24} T^{24} - 481890697519562 p^{27} T^{25} + 5851267321207 p^{30} T^{26} - 62647129196 p^{33} T^{27} + 628525215 p^{36} T^{28} - 5174290 p^{39} T^{29} + 39747 p^{42} T^{30} - 202 p^{45} T^{31} + p^{48} T^{32} \) | |
| 17 | \( 1 - 216 T + 72173 T^{2} - 11987540 T^{3} + 2359831680 T^{4} - 322681810268 T^{5} + 47793298253348 T^{6} - 5592038613663804 T^{7} + 681201596043536400 T^{8} - 69833777900733003540 T^{9} + \)\(73\!\cdots\!47\)\( T^{10} - \)\(66\!\cdots\!12\)\( T^{11} + \)\(61\!\cdots\!51\)\( T^{12} - \)\(50\!\cdots\!20\)\( T^{13} + \)\(41\!\cdots\!20\)\( T^{14} - \)\(17\!\cdots\!52\)\( p T^{15} + \)\(22\!\cdots\!44\)\( T^{16} - \)\(17\!\cdots\!52\)\( p^{4} T^{17} + \)\(41\!\cdots\!20\)\( p^{6} T^{18} - \)\(50\!\cdots\!20\)\( p^{9} T^{19} + \)\(61\!\cdots\!51\)\( p^{12} T^{20} - \)\(66\!\cdots\!12\)\( p^{15} T^{21} + \)\(73\!\cdots\!47\)\( p^{18} T^{22} - 69833777900733003540 p^{21} T^{23} + 681201596043536400 p^{24} T^{24} - 5592038613663804 p^{27} T^{25} + 47793298253348 p^{30} T^{26} - 322681810268 p^{33} T^{27} + 2359831680 p^{36} T^{28} - 11987540 p^{39} T^{29} + 72173 p^{42} T^{30} - 216 p^{45} T^{31} + p^{48} T^{32} \) | |
| 19 | \( 1 - 100 T + 62499 T^{2} - 5941070 T^{3} + 1954971040 T^{4} - 179860138490 T^{5} + 40914426675740 T^{6} - 3666018129319910 T^{7} + 643347824274058240 T^{8} - 56071268116571276530 T^{9} + \)\(80\!\cdots\!37\)\( T^{10} - \)\(35\!\cdots\!00\)\( p T^{11} + \)\(83\!\cdots\!43\)\( T^{12} - \)\(67\!\cdots\!60\)\( T^{13} + \)\(72\!\cdots\!20\)\( T^{14} - \)\(55\!\cdots\!20\)\( T^{15} + \)\(54\!\cdots\!20\)\( T^{16} - \)\(55\!\cdots\!20\)\( p^{3} T^{17} + \)\(72\!\cdots\!20\)\( p^{6} T^{18} - \)\(67\!\cdots\!60\)\( p^{9} T^{19} + \)\(83\!\cdots\!43\)\( p^{12} T^{20} - \)\(35\!\cdots\!00\)\( p^{16} T^{21} + \)\(80\!\cdots\!37\)\( p^{18} T^{22} - 56071268116571276530 p^{21} T^{23} + 643347824274058240 p^{24} T^{24} - 3666018129319910 p^{27} T^{25} + 40914426675740 p^{30} T^{26} - 179860138490 p^{33} T^{27} + 1954971040 p^{36} T^{28} - 5941070 p^{39} T^{29} + 62499 p^{42} T^{30} - 100 p^{45} T^{31} + p^{48} T^{32} \) | |
| 23 | \( 1 - 292 T + 128627 T^{2} - 26421680 T^{3} + 7024883945 T^{4} - 1172902402876 T^{5} + 242834291333827 T^{6} - 35473964950750932 T^{7} + 6259074241000358225 T^{8} - \)\(82\!\cdots\!20\)\( T^{9} + \)\(12\!\cdots\!48\)\( T^{10} - \)\(15\!\cdots\!36\)\( T^{11} + \)\(22\!\cdots\!26\)\( T^{12} - \)\(11\!\cdots\!20\)\( p T^{13} + \)\(33\!\cdots\!70\)\( T^{14} - \)\(35\!\cdots\!92\)\( T^{15} + \)\(43\!\cdots\!14\)\( T^{16} - \)\(35\!\cdots\!92\)\( p^{3} T^{17} + \)\(33\!\cdots\!70\)\( p^{6} T^{18} - \)\(11\!\cdots\!20\)\( p^{10} T^{19} + \)\(22\!\cdots\!26\)\( p^{12} T^{20} - \)\(15\!\cdots\!36\)\( p^{15} T^{21} + \)\(12\!\cdots\!48\)\( p^{18} T^{22} - \)\(82\!\cdots\!20\)\( p^{21} T^{23} + 6259074241000358225 p^{24} T^{24} - 35473964950750932 p^{27} T^{25} + 242834291333827 p^{30} T^{26} - 1172902402876 p^{33} T^{27} + 7024883945 p^{36} T^{28} - 26421680 p^{39} T^{29} + 128627 p^{42} T^{30} - 292 p^{45} T^{31} + p^{48} T^{32} \) | |
| 29 | \( 1 - 400 T + 271059 T^{2} - 82043350 T^{3} + 32707890495 T^{4} - 8021155972900 T^{5} + 2406118644462195 T^{6} - 495795701767372100 T^{7} + \)\(12\!\cdots\!20\)\( T^{8} - \)\(21\!\cdots\!50\)\( T^{9} + \)\(46\!\cdots\!67\)\( T^{10} - \)\(72\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!18\)\( T^{12} - \)\(19\!\cdots\!00\)\( T^{13} + \)\(37\!\cdots\!10\)\( T^{14} - \)\(48\!\cdots\!00\)\( T^{15} + \)\(91\!\cdots\!35\)\( T^{16} - \)\(48\!\cdots\!00\)\( p^{3} T^{17} + \)\(37\!\cdots\!10\)\( p^{6} T^{18} - \)\(19\!\cdots\!00\)\( p^{9} T^{19} + \)\(14\!\cdots\!18\)\( p^{12} T^{20} - \)\(72\!\cdots\!00\)\( p^{15} T^{21} + \)\(46\!\cdots\!67\)\( p^{18} T^{22} - \)\(21\!\cdots\!50\)\( p^{21} T^{23} + \)\(12\!\cdots\!20\)\( p^{24} T^{24} - 495795701767372100 p^{27} T^{25} + 2406118644462195 p^{30} T^{26} - 8021155972900 p^{33} T^{27} + 32707890495 p^{36} T^{28} - 82043350 p^{39} T^{29} + 271059 p^{42} T^{30} - 400 p^{45} T^{31} + p^{48} T^{32} \) | |
| 31 | \( 1 - 102 T + 204311 T^{2} - 20042720 T^{3} + 21433721485 T^{4} - 1900395373036 T^{5} + 1528704883073127 T^{6} - 118346387424117906 T^{7} + 2696669049822108215 p T^{8} - \)\(54\!\cdots\!60\)\( T^{9} + \)\(37\!\cdots\!80\)\( T^{10} - \)\(20\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!30\)\( T^{12} - \)\(67\!\cdots\!60\)\( T^{13} + \)\(50\!\cdots\!90\)\( T^{14} - \)\(20\!\cdots\!80\)\( T^{15} + \)\(15\!\cdots\!30\)\( T^{16} - \)\(20\!\cdots\!80\)\( p^{3} T^{17} + \)\(50\!\cdots\!90\)\( p^{6} T^{18} - \)\(67\!\cdots\!60\)\( p^{9} T^{19} + \)\(14\!\cdots\!30\)\( p^{12} T^{20} - \)\(20\!\cdots\!40\)\( p^{15} T^{21} + \)\(37\!\cdots\!80\)\( p^{18} T^{22} - \)\(54\!\cdots\!60\)\( p^{21} T^{23} + 2696669049822108215 p^{25} T^{24} - 118346387424117906 p^{27} T^{25} + 1528704883073127 p^{30} T^{26} - 1900395373036 p^{33} T^{27} + 21433721485 p^{36} T^{28} - 20042720 p^{39} T^{29} + 204311 p^{42} T^{30} - 102 p^{45} T^{31} + p^{48} T^{32} \) | |
| 37 | \( 1 - 646 T + 451973 T^{2} - 184421990 T^{3} + 75174798015 T^{4} - 22444096316738 T^{5} + 6766961418265313 T^{6} - 1613693017428207254 T^{7} + \)\(41\!\cdots\!40\)\( T^{8} - \)\(88\!\cdots\!10\)\( T^{9} + \)\(22\!\cdots\!97\)\( T^{10} - \)\(12\!\cdots\!06\)\( p T^{11} + \)\(11\!\cdots\!46\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{13} + \)\(49\!\cdots\!90\)\( T^{14} - \)\(84\!\cdots\!44\)\( T^{15} + \)\(21\!\cdots\!39\)\( T^{16} - \)\(84\!\cdots\!44\)\( p^{3} T^{17} + \)\(49\!\cdots\!90\)\( p^{6} T^{18} - \)\(21\!\cdots\!00\)\( p^{9} T^{19} + \)\(11\!\cdots\!46\)\( p^{12} T^{20} - \)\(12\!\cdots\!06\)\( p^{16} T^{21} + \)\(22\!\cdots\!97\)\( p^{18} T^{22} - \)\(88\!\cdots\!10\)\( p^{21} T^{23} + \)\(41\!\cdots\!40\)\( p^{24} T^{24} - 1613693017428207254 p^{27} T^{25} + 6766961418265313 p^{30} T^{26} - 22444096316738 p^{33} T^{27} + 75174798015 p^{36} T^{28} - 184421990 p^{39} T^{29} + 451973 p^{42} T^{30} - 646 p^{45} T^{31} + p^{48} T^{32} \) | |
| 41 | \( 1 - 532 T + 713996 T^{2} - 342661760 T^{3} + 259655890900 T^{4} - 112748491742316 T^{5} + 62811070209601982 T^{6} - 24770640704267111436 T^{7} + \)\(11\!\cdots\!40\)\( T^{8} - \)\(40\!\cdots\!60\)\( T^{9} + \)\(15\!\cdots\!30\)\( T^{10} - \)\(51\!\cdots\!40\)\( T^{11} + \)\(17\!\cdots\!05\)\( T^{12} - \)\(52\!\cdots\!60\)\( T^{13} + \)\(16\!\cdots\!40\)\( T^{14} - \)\(43\!\cdots\!80\)\( T^{15} + \)\(12\!\cdots\!80\)\( T^{16} - \)\(43\!\cdots\!80\)\( p^{3} T^{17} + \)\(16\!\cdots\!40\)\( p^{6} T^{18} - \)\(52\!\cdots\!60\)\( p^{9} T^{19} + \)\(17\!\cdots\!05\)\( p^{12} T^{20} - \)\(51\!\cdots\!40\)\( p^{15} T^{21} + \)\(15\!\cdots\!30\)\( p^{18} T^{22} - \)\(40\!\cdots\!60\)\( p^{21} T^{23} + \)\(11\!\cdots\!40\)\( p^{24} T^{24} - 24770640704267111436 p^{27} T^{25} + 62811070209601982 p^{30} T^{26} - 112748491742316 p^{33} T^{27} + 259655890900 p^{36} T^{28} - 342661760 p^{39} T^{29} + 713996 p^{42} T^{30} - 532 p^{45} T^{31} + p^{48} T^{32} \) | |
| 43 | \( 1 - 902 T + 1049057 T^{2} - 753094950 T^{3} + 536512457350 T^{4} - 313549212610726 T^{5} + 173970320866417052 T^{6} - 85827622932674777032 T^{7} + \)\(40\!\cdots\!75\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{9} + \)\(69\!\cdots\!38\)\( T^{10} - \)\(26\!\cdots\!26\)\( T^{11} + \)\(94\!\cdots\!66\)\( T^{12} - \)\(32\!\cdots\!50\)\( T^{13} + \)\(10\!\cdots\!25\)\( T^{14} - \)\(31\!\cdots\!02\)\( T^{15} + \)\(91\!\cdots\!04\)\( T^{16} - \)\(31\!\cdots\!02\)\( p^{3} T^{17} + \)\(10\!\cdots\!25\)\( p^{6} T^{18} - \)\(32\!\cdots\!50\)\( p^{9} T^{19} + \)\(94\!\cdots\!66\)\( p^{12} T^{20} - \)\(26\!\cdots\!26\)\( p^{15} T^{21} + \)\(69\!\cdots\!38\)\( p^{18} T^{22} - \)\(17\!\cdots\!00\)\( p^{21} T^{23} + \)\(40\!\cdots\!75\)\( p^{24} T^{24} - 85827622932674777032 p^{27} T^{25} + 173970320866417052 p^{30} T^{26} - 313549212610726 p^{33} T^{27} + 536512457350 p^{36} T^{28} - 753094950 p^{39} T^{29} + 1049057 p^{42} T^{30} - 902 p^{45} T^{31} + p^{48} T^{32} \) | |
| 47 | \( 1 - 776 T + 1170028 T^{2} - 662371700 T^{3} + 581637045080 T^{4} - 259934123575328 T^{5} + 172130204692934318 T^{6} - 62961059654174142404 T^{7} + \)\(34\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(52\!\cdots\!62\)\( T^{10} - \)\(13\!\cdots\!12\)\( T^{11} + \)\(64\!\cdots\!21\)\( T^{12} - \)\(13\!\cdots\!40\)\( T^{13} + \)\(67\!\cdots\!60\)\( T^{14} - \)\(12\!\cdots\!44\)\( T^{15} + \)\(69\!\cdots\!64\)\( T^{16} - \)\(12\!\cdots\!44\)\( p^{3} T^{17} + \)\(67\!\cdots\!60\)\( p^{6} T^{18} - \)\(13\!\cdots\!40\)\( p^{9} T^{19} + \)\(64\!\cdots\!21\)\( p^{12} T^{20} - \)\(13\!\cdots\!12\)\( p^{15} T^{21} + \)\(52\!\cdots\!62\)\( p^{18} T^{22} - \)\(10\!\cdots\!60\)\( p^{21} T^{23} + \)\(34\!\cdots\!20\)\( p^{24} T^{24} - 62961059654174142404 p^{27} T^{25} + 172130204692934318 p^{30} T^{26} - 259934123575328 p^{33} T^{27} + 581637045080 p^{36} T^{28} - 662371700 p^{39} T^{29} + 1170028 p^{42} T^{30} - 776 p^{45} T^{31} + p^{48} T^{32} \) | |
| 53 | \( 1 - 632 T + 1219952 T^{2} - 699643440 T^{3} + 762331283840 T^{4} - 404226560019396 T^{5} + 324207600797101862 T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!20\)\( T^{8} - \)\(48\!\cdots\!80\)\( T^{9} + \)\(27\!\cdots\!18\)\( T^{10} - \)\(11\!\cdots\!36\)\( T^{11} + \)\(60\!\cdots\!61\)\( T^{12} - \)\(24\!\cdots\!60\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} - \)\(42\!\cdots\!12\)\( T^{15} + \)\(18\!\cdots\!04\)\( T^{16} - \)\(42\!\cdots\!12\)\( p^{3} T^{17} + \)\(11\!\cdots\!00\)\( p^{6} T^{18} - \)\(24\!\cdots\!60\)\( p^{9} T^{19} + \)\(60\!\cdots\!61\)\( p^{12} T^{20} - \)\(11\!\cdots\!36\)\( p^{15} T^{21} + \)\(27\!\cdots\!18\)\( p^{18} T^{22} - \)\(48\!\cdots\!80\)\( p^{21} T^{23} + \)\(10\!\cdots\!20\)\( p^{24} T^{24} - \)\(16\!\cdots\!92\)\( p^{27} T^{25} + 324207600797101862 p^{30} T^{26} - 404226560019396 p^{33} T^{27} + 762331283840 p^{36} T^{28} - 699643440 p^{39} T^{29} + 1219952 p^{42} T^{30} - 632 p^{45} T^{31} + p^{48} T^{32} \) | |
| 59 | \( 1 - 1000 T + 2196014 T^{2} - 1924892980 T^{3} + 2393168358835 T^{4} - 1846293063107160 T^{5} + 1699729484781572760 T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(88\!\cdots\!35\)\( T^{8} - \)\(54\!\cdots\!20\)\( T^{9} + \)\(35\!\cdots\!32\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!93\)\( T^{12} - \)\(60\!\cdots\!40\)\( T^{13} + \)\(30\!\cdots\!30\)\( T^{14} - \)\(14\!\cdots\!80\)\( T^{15} + \)\(69\!\cdots\!80\)\( T^{16} - \)\(14\!\cdots\!80\)\( p^{3} T^{17} + \)\(30\!\cdots\!30\)\( p^{6} T^{18} - \)\(60\!\cdots\!40\)\( p^{9} T^{19} + \)\(11\!\cdots\!93\)\( p^{12} T^{20} - \)\(20\!\cdots\!00\)\( p^{15} T^{21} + \)\(35\!\cdots\!32\)\( p^{18} T^{22} - \)\(54\!\cdots\!20\)\( p^{21} T^{23} + \)\(88\!\cdots\!35\)\( p^{24} T^{24} - \)\(11\!\cdots\!40\)\( p^{27} T^{25} + 1699729484781572760 p^{30} T^{26} - 1846293063107160 p^{33} T^{27} + 2393168358835 p^{36} T^{28} - 1924892980 p^{39} T^{29} + 2196014 p^{42} T^{30} - 1000 p^{45} T^{31} + p^{48} T^{32} \) | |
| 61 | \( 1 - 662 T + 2245981 T^{2} - 1218087510 T^{3} + 2428417859875 T^{4} - 1109600424374866 T^{5} + 1706095795992339137 T^{6} - \)\(66\!\cdots\!46\)\( T^{7} + \)\(88\!\cdots\!40\)\( T^{8} - \)\(29\!\cdots\!10\)\( T^{9} + \)\(35\!\cdots\!05\)\( T^{10} - \)\(10\!\cdots\!90\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(31\!\cdots\!60\)\( T^{13} + \)\(34\!\cdots\!90\)\( T^{14} - \)\(82\!\cdots\!80\)\( T^{15} + \)\(83\!\cdots\!55\)\( T^{16} - \)\(82\!\cdots\!80\)\( p^{3} T^{17} + \)\(34\!\cdots\!90\)\( p^{6} T^{18} - \)\(31\!\cdots\!60\)\( p^{9} T^{19} + \)\(12\!\cdots\!30\)\( p^{12} T^{20} - \)\(10\!\cdots\!90\)\( p^{15} T^{21} + \)\(35\!\cdots\!05\)\( p^{18} T^{22} - \)\(29\!\cdots\!10\)\( p^{21} T^{23} + \)\(88\!\cdots\!40\)\( p^{24} T^{24} - \)\(66\!\cdots\!46\)\( p^{27} T^{25} + 1706095795992339137 p^{30} T^{26} - 1109600424374866 p^{33} T^{27} + 2428417859875 p^{36} T^{28} - 1218087510 p^{39} T^{29} + 2245981 p^{42} T^{30} - 662 p^{45} T^{31} + p^{48} T^{32} \) | |
| 67 | \( 1 - 1326 T + 3647353 T^{2} - 3910383290 T^{3} + 6158608849610 T^{4} - 5591628984132978 T^{5} + 6545195005549082128 T^{6} - \)\(51\!\cdots\!44\)\( T^{7} + \)\(49\!\cdots\!95\)\( T^{8} - \)\(35\!\cdots\!00\)\( T^{9} + \)\(29\!\cdots\!22\)\( T^{10} - \)\(18\!\cdots\!82\)\( T^{11} + \)\(13\!\cdots\!46\)\( T^{12} - \)\(81\!\cdots\!10\)\( T^{13} + \)\(54\!\cdots\!05\)\( T^{14} - \)\(29\!\cdots\!14\)\( T^{15} + \)\(17\!\cdots\!84\)\( T^{16} - \)\(29\!\cdots\!14\)\( p^{3} T^{17} + \)\(54\!\cdots\!05\)\( p^{6} T^{18} - \)\(81\!\cdots\!10\)\( p^{9} T^{19} + \)\(13\!\cdots\!46\)\( p^{12} T^{20} - \)\(18\!\cdots\!82\)\( p^{15} T^{21} + \)\(29\!\cdots\!22\)\( p^{18} T^{22} - \)\(35\!\cdots\!00\)\( p^{21} T^{23} + \)\(49\!\cdots\!95\)\( p^{24} T^{24} - \)\(51\!\cdots\!44\)\( p^{27} T^{25} + 6545195005549082128 p^{30} T^{26} - 5591628984132978 p^{33} T^{27} + 6158608849610 p^{36} T^{28} - 3910383290 p^{39} T^{29} + 3647353 p^{42} T^{30} - 1326 p^{45} T^{31} + p^{48} T^{32} \) | |
| 71 | \( 1 - 1292 T + 4697316 T^{2} - 4986685400 T^{3} + 10207045746340 T^{4} - 130517238014576 p T^{5} + 13905564925768620242 T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(13\!\cdots\!40\)\( T^{8} - \)\(95\!\cdots\!60\)\( T^{9} + \)\(99\!\cdots\!30\)\( T^{10} - \)\(63\!\cdots\!40\)\( T^{11} + \)\(58\!\cdots\!05\)\( T^{12} - \)\(33\!\cdots\!60\)\( T^{13} + \)\(27\!\cdots\!40\)\( T^{14} - \)\(14\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!80\)\( T^{16} - \)\(14\!\cdots\!80\)\( p^{3} T^{17} + \)\(27\!\cdots\!40\)\( p^{6} T^{18} - \)\(33\!\cdots\!60\)\( p^{9} T^{19} + \)\(58\!\cdots\!05\)\( p^{12} T^{20} - \)\(63\!\cdots\!40\)\( p^{15} T^{21} + \)\(99\!\cdots\!30\)\( p^{18} T^{22} - \)\(95\!\cdots\!60\)\( p^{21} T^{23} + \)\(13\!\cdots\!40\)\( p^{24} T^{24} - \)\(11\!\cdots\!76\)\( p^{27} T^{25} + 13905564925768620242 p^{30} T^{26} - 130517238014576 p^{34} T^{27} + 10207045746340 p^{36} T^{28} - 4986685400 p^{39} T^{29} + 4697316 p^{42} T^{30} - 1292 p^{45} T^{31} + p^{48} T^{32} \) | |
| 73 | \( 1 - 2272 T + 5567972 T^{2} - 8398360160 T^{3} + 12434621217980 T^{4} - 14383819818282136 T^{5} + 16086893804882013842 T^{6} - \)\(15\!\cdots\!32\)\( T^{7} + \)\(13\!\cdots\!60\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(87\!\cdots\!58\)\( T^{10} - \)\(61\!\cdots\!16\)\( T^{11} + \)\(42\!\cdots\!41\)\( T^{12} - \)\(26\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!52\)\( T^{15} + \)\(67\!\cdots\!04\)\( T^{16} - \)\(10\!\cdots\!52\)\( p^{3} T^{17} + \)\(17\!\cdots\!60\)\( p^{6} T^{18} - \)\(26\!\cdots\!60\)\( p^{9} T^{19} + \)\(42\!\cdots\!41\)\( p^{12} T^{20} - \)\(61\!\cdots\!16\)\( p^{15} T^{21} + \)\(87\!\cdots\!58\)\( p^{18} T^{22} - \)\(11\!\cdots\!00\)\( p^{21} T^{23} + \)\(13\!\cdots\!60\)\( p^{24} T^{24} - \)\(15\!\cdots\!32\)\( p^{27} T^{25} + 16086893804882013842 p^{30} T^{26} - 14383819818282136 p^{33} T^{27} + 12434621217980 p^{36} T^{28} - 8398360160 p^{39} T^{29} + 5567972 p^{42} T^{30} - 2272 p^{45} T^{31} + p^{48} T^{32} \) | |
| 79 | \( 1 - 320 T + 4361414 T^{2} - 695716460 T^{3} + 8905075330575 T^{4} - 76126507029420 T^{5} + 11547893735920139200 T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(32\!\cdots\!60\)\( T^{9} + \)\(80\!\cdots\!72\)\( T^{10} + \)\(36\!\cdots\!60\)\( T^{11} + \)\(49\!\cdots\!33\)\( T^{12} + \)\(29\!\cdots\!20\)\( T^{13} + \)\(26\!\cdots\!50\)\( T^{14} + \)\(18\!\cdots\!40\)\( T^{15} + \)\(13\!\cdots\!00\)\( T^{16} + \)\(18\!\cdots\!40\)\( p^{3} T^{17} + \)\(26\!\cdots\!50\)\( p^{6} T^{18} + \)\(29\!\cdots\!20\)\( p^{9} T^{19} + \)\(49\!\cdots\!33\)\( p^{12} T^{20} + \)\(36\!\cdots\!60\)\( p^{15} T^{21} + \)\(80\!\cdots\!72\)\( p^{18} T^{22} + \)\(32\!\cdots\!60\)\( p^{21} T^{23} + \)\(10\!\cdots\!75\)\( p^{24} T^{24} + \)\(16\!\cdots\!20\)\( p^{27} T^{25} + 11547893735920139200 p^{30} T^{26} - 76126507029420 p^{33} T^{27} + 8905075330575 p^{36} T^{28} - 695716460 p^{39} T^{29} + 4361414 p^{42} T^{30} - 320 p^{45} T^{31} + p^{48} T^{32} \) | |
| 83 | \( 1 - 2842 T + 9326507 T^{2} - 17809387470 T^{3} + 35328029084085 T^{4} - 52654518439651096 T^{5} + 79669118173514577387 T^{6} - \)\(99\!\cdots\!82\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} - \)\(13\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!88\)\( T^{10} - \)\(14\!\cdots\!26\)\( T^{11} + \)\(13\!\cdots\!26\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!70\)\( T^{14} - \)\(83\!\cdots\!42\)\( T^{15} + \)\(66\!\cdots\!94\)\( T^{16} - \)\(83\!\cdots\!42\)\( p^{3} T^{17} + \)\(10\!\cdots\!70\)\( p^{6} T^{18} - \)\(12\!\cdots\!20\)\( p^{9} T^{19} + \)\(13\!\cdots\!26\)\( p^{12} T^{20} - \)\(14\!\cdots\!26\)\( p^{15} T^{21} + \)\(14\!\cdots\!88\)\( p^{18} T^{22} - \)\(13\!\cdots\!50\)\( p^{21} T^{23} + \)\(12\!\cdots\!45\)\( p^{24} T^{24} - \)\(99\!\cdots\!82\)\( p^{27} T^{25} + 79669118173514577387 p^{30} T^{26} - 52654518439651096 p^{33} T^{27} + 35328029084085 p^{36} T^{28} - 17809387470 p^{39} T^{29} + 9326507 p^{42} T^{30} - 2842 p^{45} T^{31} + p^{48} T^{32} \) | |
| 89 | \( 1 - 2780 T + 9510594 T^{2} - 18653107300 T^{3} + 38542170342335 T^{4} - 59258990758178900 T^{5} + 92645395025398628980 T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!35\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!72\)\( T^{10} - \)\(17\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!93\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!30\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(98\!\cdots\!40\)\( T^{16} - \)\(11\!\cdots\!00\)\( p^{3} T^{17} + \)\(13\!\cdots\!30\)\( p^{6} T^{18} - \)\(14\!\cdots\!00\)\( p^{9} T^{19} + \)\(17\!\cdots\!93\)\( p^{12} T^{20} - \)\(17\!\cdots\!60\)\( p^{15} T^{21} + \)\(18\!\cdots\!72\)\( p^{18} T^{22} - \)\(16\!\cdots\!00\)\( p^{21} T^{23} + \)\(15\!\cdots\!35\)\( p^{24} T^{24} - \)\(11\!\cdots\!00\)\( p^{27} T^{25} + 92645395025398628980 p^{30} T^{26} - 59258990758178900 p^{33} T^{27} + 38542170342335 p^{36} T^{28} - 18653107300 p^{39} T^{29} + 9510594 p^{42} T^{30} - 2780 p^{45} T^{31} + p^{48} T^{32} \) | |
| 97 | \( 1 - 3796 T + 15012988 T^{2} - 39094072880 T^{3} + 96554104523990 T^{4} - 196639812569713728 T^{5} + \)\(37\!\cdots\!68\)\( T^{6} - \)\(63\!\cdots\!24\)\( T^{7} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!72\)\( T^{10} - \)\(27\!\cdots\!56\)\( p T^{11} + \)\(32\!\cdots\!16\)\( T^{12} - \)\(37\!\cdots\!80\)\( T^{13} + \)\(41\!\cdots\!00\)\( T^{14} - \)\(42\!\cdots\!64\)\( T^{15} + \)\(41\!\cdots\!49\)\( T^{16} - \)\(42\!\cdots\!64\)\( p^{3} T^{17} + \)\(41\!\cdots\!00\)\( p^{6} T^{18} - \)\(37\!\cdots\!80\)\( p^{9} T^{19} + \)\(32\!\cdots\!16\)\( p^{12} T^{20} - \)\(27\!\cdots\!56\)\( p^{16} T^{21} + \)\(20\!\cdots\!72\)\( p^{18} T^{22} - \)\(14\!\cdots\!40\)\( p^{21} T^{23} + \)\(10\!\cdots\!95\)\( p^{24} T^{24} - \)\(63\!\cdots\!24\)\( p^{27} T^{25} + \)\(37\!\cdots\!68\)\( p^{30} T^{26} - 196639812569713728 p^{33} T^{27} + 96554104523990 p^{36} T^{28} - 39094072880 p^{39} T^{29} + 15012988 p^{42} T^{30} - 3796 p^{45} T^{31} + p^{48} T^{32} \) | |
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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
−2.22022243282390868541617821554, −2.20122151578521860664808096036, −2.14967604261745787925636602504, −2.13141591359017550618981013051, −2.08566130981062361501264725234, −2.07031582633104919069729294518, −1.94616439877025685259137962073, −1.71585516671353219037824173274, −1.68754732245608658036118234769, −1.43313239246847050775726131644, −1.37642287266312885754479557042, −1.25667329965269707724944843048, −1.20028148069636913448149221366, −1.18649181079642255311429683338, −1.11611725617890758826147615256, −1.03754204485759975313681039012, −1.00269062664159924480842669595, −0.937456990268974178105990168758, −0.929658944815369527419241202194, −0.68472086528532450155186025340, −0.68059475045234877564356964650, −0.65866321610745500983917653147, −0.62085412700596674165065272417, −0.47763148946698205106766724056, −0.44440063414311641243957571626, 0.44440063414311641243957571626, 0.47763148946698205106766724056, 0.62085412700596674165065272417, 0.65866321610745500983917653147, 0.68059475045234877564356964650, 0.68472086528532450155186025340, 0.929658944815369527419241202194, 0.937456990268974178105990168758, 1.00269062664159924480842669595, 1.03754204485759975313681039012, 1.11611725617890758826147615256, 1.18649181079642255311429683338, 1.20028148069636913448149221366, 1.25667329965269707724944843048, 1.37642287266312885754479557042, 1.43313239246847050775726131644, 1.68754732245608658036118234769, 1.71585516671353219037824173274, 1.94616439877025685259137962073, 2.07031582633104919069729294518, 2.08566130981062361501264725234, 2.13141591359017550618981013051, 2.14967604261745787925636602504, 2.20122151578521860664808096036, 2.22022243282390868541617821554
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.