Dirichlet series
| L(s) = 1 | − 8·2-s − 12·3-s + 24·4-s + 15·5-s + 96·6-s + 46·7-s − 32·8-s + 54·9-s − 120·10-s − 83·11-s − 288·12-s + 22·13-s − 368·14-s − 180·15-s + 16·16-s − 79·17-s − 432·18-s − 15·19-s + 360·20-s − 552·21-s + 664·22-s − 143·23-s + 384·24-s + 10·25-s − 176·26-s − 108·27-s + 1.10e3·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 3·4-s + 1.34·5-s + 6.53·6-s + 2.48·7-s − 1.41·8-s + 2·9-s − 3.79·10-s − 2.27·11-s − 6.92·12-s + 0.469·13-s − 7.02·14-s − 3.09·15-s + 1/4·16-s − 1.12·17-s − 5.65·18-s − 0.181·19-s + 4.02·20-s − 5.73·21-s + 6.43·22-s − 1.29·23-s + 3.26·24-s + 2/25·25-s − 1.32·26-s − 0.769·27-s + 7.45·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.41682\times 10^{15}\) |
| Root analytic conductor: | \(2.97494\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.004585273164\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004585273164\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{4} \) |
| 3 | \( ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{4} \) | |
| 5 | \( 1 - 3 p T + 43 p T^{2} - 36 p^{3} T^{3} + 11516 p T^{4} - 25629 p^{2} T^{5} + 78496 p^{3} T^{6} - 191994 p^{4} T^{7} + 358861 p^{5} T^{8} - 191994 p^{7} T^{9} + 78496 p^{9} T^{10} - 25629 p^{11} T^{11} + 11516 p^{13} T^{12} - 36 p^{18} T^{13} + 43 p^{19} T^{14} - 3 p^{22} T^{15} + p^{24} T^{16} \) | |
| good | 7 | \( ( 1 - 23 T + 181 p T^{2} - 346 p^{2} T^{3} + 90348 p T^{4} - 6303478 T^{5} + 39104630 p T^{6} - 3056032827 T^{7} + 112722035404 T^{8} - 3056032827 p^{3} T^{9} + 39104630 p^{7} T^{10} - 6303478 p^{9} T^{11} + 90348 p^{13} T^{12} - 346 p^{17} T^{13} + 181 p^{19} T^{14} - 23 p^{21} T^{15} + p^{24} T^{16} )^{2} \) |
| 11 | \( 1 + 83 T + 2131 T^{2} + 46325 T^{3} + 4980270 T^{4} + 219524939 T^{5} + 3816268172 T^{6} + 100157669234 T^{7} + 7409673674820 T^{8} + 25268743328145 p T^{9} + 4888950571730110 T^{10} + 48722406629213165 T^{11} + 7133107727485187410 T^{12} + \)\(33\!\cdots\!70\)\( T^{13} + \)\(48\!\cdots\!70\)\( T^{14} + \)\(27\!\cdots\!75\)\( T^{15} + \)\(20\!\cdots\!60\)\( T^{16} + \)\(27\!\cdots\!75\)\( p^{3} T^{17} + \)\(48\!\cdots\!70\)\( p^{6} T^{18} + \)\(33\!\cdots\!70\)\( p^{9} T^{19} + 7133107727485187410 p^{12} T^{20} + 48722406629213165 p^{15} T^{21} + 4888950571730110 p^{18} T^{22} + 25268743328145 p^{22} T^{23} + 7409673674820 p^{24} T^{24} + 100157669234 p^{27} T^{25} + 3816268172 p^{30} T^{26} + 219524939 p^{33} T^{27} + 4980270 p^{36} T^{28} + 46325 p^{39} T^{29} + 2131 p^{42} T^{30} + 83 p^{45} T^{31} + p^{48} T^{32} \) | |
| 13 | \( 1 - 22 T - 8233 T^{2} + 135730 T^{3} + 32280125 T^{4} - 931341006 T^{5} - 64913082488 T^{6} + 5039072939438 T^{7} + 38938197973430 T^{8} - 18402018257260710 T^{9} + 296409225739147903 T^{10} + 46568068796243862704 T^{11} - \)\(16\!\cdots\!14\)\( T^{12} - \)\(83\!\cdots\!60\)\( T^{13} + \)\(54\!\cdots\!90\)\( T^{14} + \)\(73\!\cdots\!28\)\( T^{15} - \)\(13\!\cdots\!11\)\( T^{16} + \)\(73\!\cdots\!28\)\( p^{3} T^{17} + \)\(54\!\cdots\!90\)\( p^{6} T^{18} - \)\(83\!\cdots\!60\)\( p^{9} T^{19} - \)\(16\!\cdots\!14\)\( p^{12} T^{20} + 46568068796243862704 p^{15} T^{21} + 296409225739147903 p^{18} T^{22} - 18402018257260710 p^{21} T^{23} + 38938197973430 p^{24} T^{24} + 5039072939438 p^{27} T^{25} - 64913082488 p^{30} T^{26} - 931341006 p^{33} T^{27} + 32280125 p^{36} T^{28} + 135730 p^{39} T^{29} - 8233 p^{42} T^{30} - 22 p^{45} T^{31} + p^{48} T^{32} \) | |
| 17 | \( 1 + 79 T + 13253 T^{2} + 1431530 T^{3} + 132103900 T^{4} + 13646894087 T^{5} + 1283825852483 T^{6} + 6299111717903 p T^{7} + 9723144385868115 T^{8} + 829392753779236860 T^{9} + 63950188092072426467 T^{10} + \)\(52\!\cdots\!63\)\( T^{11} + \)\(40\!\cdots\!56\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{13} + \)\(23\!\cdots\!30\)\( T^{14} + \)\(16\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!79\)\( T^{16} + \)\(16\!\cdots\!66\)\( p^{3} T^{17} + \)\(23\!\cdots\!30\)\( p^{6} T^{18} + \)\(29\!\cdots\!10\)\( p^{9} T^{19} + \)\(40\!\cdots\!56\)\( p^{12} T^{20} + \)\(52\!\cdots\!63\)\( p^{15} T^{21} + 63950188092072426467 p^{18} T^{22} + 829392753779236860 p^{21} T^{23} + 9723144385868115 p^{24} T^{24} + 6299111717903 p^{28} T^{25} + 1283825852483 p^{30} T^{26} + 13646894087 p^{33} T^{27} + 132103900 p^{36} T^{28} + 1431530 p^{39} T^{29} + 13253 p^{42} T^{30} + 79 p^{45} T^{31} + p^{48} T^{32} \) | |
| 19 | \( 1 + 15 T - 26911 T^{2} + 134795 T^{3} + 334581480 T^{4} - 6246163335 T^{5} - 2441902724295 T^{6} + 83122703225735 T^{7} + 15092172749204005 T^{8} - 718284879818480945 T^{9} - \)\(11\!\cdots\!23\)\( T^{10} + \)\(49\!\cdots\!05\)\( T^{11} + \)\(43\!\cdots\!52\)\( p T^{12} - \)\(24\!\cdots\!40\)\( T^{13} - \)\(46\!\cdots\!10\)\( T^{14} + \)\(56\!\cdots\!70\)\( T^{15} + \)\(26\!\cdots\!65\)\( T^{16} + \)\(56\!\cdots\!70\)\( p^{3} T^{17} - \)\(46\!\cdots\!10\)\( p^{6} T^{18} - \)\(24\!\cdots\!40\)\( p^{9} T^{19} + \)\(43\!\cdots\!52\)\( p^{13} T^{20} + \)\(49\!\cdots\!05\)\( p^{15} T^{21} - \)\(11\!\cdots\!23\)\( p^{18} T^{22} - 718284879818480945 p^{21} T^{23} + 15092172749204005 p^{24} T^{24} + 83122703225735 p^{27} T^{25} - 2441902724295 p^{30} T^{26} - 6246163335 p^{33} T^{27} + 334581480 p^{36} T^{28} + 134795 p^{39} T^{29} - 26911 p^{42} T^{30} + 15 p^{45} T^{31} + p^{48} T^{32} \) | |
| 23 | \( 1 + 143 T - 39148 T^{2} - 4449205 T^{3} + 970599945 T^{4} + 76383676149 T^{5} - 15610293398058 T^{6} - 29434057471659 p T^{7} + 224832517917945450 T^{8} + 5977086678069106955 T^{9} - \)\(30\!\cdots\!02\)\( T^{10} - \)\(81\!\cdots\!81\)\( T^{11} + \)\(44\!\cdots\!76\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} - \)\(58\!\cdots\!80\)\( T^{14} - \)\(73\!\cdots\!42\)\( T^{15} + \)\(75\!\cdots\!19\)\( T^{16} - \)\(73\!\cdots\!42\)\( p^{3} T^{17} - \)\(58\!\cdots\!80\)\( p^{6} T^{18} + \)\(14\!\cdots\!40\)\( p^{9} T^{19} + \)\(44\!\cdots\!76\)\( p^{12} T^{20} - \)\(81\!\cdots\!81\)\( p^{15} T^{21} - \)\(30\!\cdots\!02\)\( p^{18} T^{22} + 5977086678069106955 p^{21} T^{23} + 224832517917945450 p^{24} T^{24} - 29434057471659 p^{28} T^{25} - 15610293398058 p^{30} T^{26} + 76383676149 p^{33} T^{27} + 970599945 p^{36} T^{28} - 4449205 p^{39} T^{29} - 39148 p^{42} T^{30} + 143 p^{45} T^{31} + p^{48} T^{32} \) | |
| 29 | \( 1 - 255 T + 12174 T^{2} + 4692930 T^{3} - 197902415 T^{4} + 77636219010 T^{5} - 67519657328140 T^{6} + 9799994553632640 T^{7} + 775272839415067910 T^{8} - \)\(23\!\cdots\!80\)\( T^{9} + \)\(23\!\cdots\!82\)\( T^{10} - \)\(38\!\cdots\!35\)\( T^{11} + \)\(17\!\cdots\!88\)\( T^{12} - \)\(80\!\cdots\!10\)\( T^{13} - \)\(37\!\cdots\!70\)\( T^{14} + \)\(40\!\cdots\!30\)\( T^{15} + \)\(33\!\cdots\!05\)\( T^{16} + \)\(40\!\cdots\!30\)\( p^{3} T^{17} - \)\(37\!\cdots\!70\)\( p^{6} T^{18} - \)\(80\!\cdots\!10\)\( p^{9} T^{19} + \)\(17\!\cdots\!88\)\( p^{12} T^{20} - \)\(38\!\cdots\!35\)\( p^{15} T^{21} + \)\(23\!\cdots\!82\)\( p^{18} T^{22} - \)\(23\!\cdots\!80\)\( p^{21} T^{23} + 775272839415067910 p^{24} T^{24} + 9799994553632640 p^{27} T^{25} - 67519657328140 p^{30} T^{26} + 77636219010 p^{33} T^{27} - 197902415 p^{36} T^{28} + 4692930 p^{39} T^{29} + 12174 p^{42} T^{30} - 255 p^{45} T^{31} + p^{48} T^{32} \) | |
| 31 | \( 1 + 378 T + 114781 T^{2} + 24745335 T^{3} + 6454636985 T^{4} + 43573938429 p T^{5} + 280096152458152 T^{6} + 51120991195516749 T^{7} + 10803844060837049370 T^{8} + \)\(20\!\cdots\!70\)\( T^{9} + \)\(39\!\cdots\!35\)\( T^{10} + \)\(23\!\cdots\!40\)\( p T^{11} + \)\(14\!\cdots\!35\)\( T^{12} + \)\(25\!\cdots\!95\)\( T^{13} + \)\(46\!\cdots\!45\)\( T^{14} + \)\(80\!\cdots\!50\)\( T^{15} + \)\(14\!\cdots\!60\)\( T^{16} + \)\(80\!\cdots\!50\)\( p^{3} T^{17} + \)\(46\!\cdots\!45\)\( p^{6} T^{18} + \)\(25\!\cdots\!95\)\( p^{9} T^{19} + \)\(14\!\cdots\!35\)\( p^{12} T^{20} + \)\(23\!\cdots\!40\)\( p^{16} T^{21} + \)\(39\!\cdots\!35\)\( p^{18} T^{22} + \)\(20\!\cdots\!70\)\( p^{21} T^{23} + 10803844060837049370 p^{24} T^{24} + 51120991195516749 p^{27} T^{25} + 280096152458152 p^{30} T^{26} + 43573938429 p^{34} T^{27} + 6454636985 p^{36} T^{28} + 24745335 p^{39} T^{29} + 114781 p^{42} T^{30} + 378 p^{45} T^{31} + p^{48} T^{32} \) | |
| 37 | \( 1 + 514 T + 34978 T^{2} - 40887955 T^{3} - 9850861885 T^{4} - 826553105378 T^{5} - 303817369039032 T^{6} - 85180199573646609 T^{7} + 28253290189207526495 T^{8} + \)\(12\!\cdots\!35\)\( T^{9} + \)\(14\!\cdots\!07\)\( T^{10} + \)\(55\!\cdots\!28\)\( T^{11} + \)\(55\!\cdots\!96\)\( T^{12} + \)\(75\!\cdots\!20\)\( T^{13} - \)\(45\!\cdots\!50\)\( T^{14} - \)\(97\!\cdots\!59\)\( T^{15} - \)\(76\!\cdots\!56\)\( T^{16} - \)\(97\!\cdots\!59\)\( p^{3} T^{17} - \)\(45\!\cdots\!50\)\( p^{6} T^{18} + \)\(75\!\cdots\!20\)\( p^{9} T^{19} + \)\(55\!\cdots\!96\)\( p^{12} T^{20} + \)\(55\!\cdots\!28\)\( p^{15} T^{21} + \)\(14\!\cdots\!07\)\( p^{18} T^{22} + \)\(12\!\cdots\!35\)\( p^{21} T^{23} + 28253290189207526495 p^{24} T^{24} - 85180199573646609 p^{27} T^{25} - 303817369039032 p^{30} T^{26} - 826553105378 p^{33} T^{27} - 9850861885 p^{36} T^{28} - 40887955 p^{39} T^{29} + 34978 p^{42} T^{30} + 514 p^{45} T^{31} + p^{48} T^{32} \) | |
| 41 | \( 1 + 918 T + 332221 T^{2} + 31400740 T^{3} - 19545308845 T^{4} - 9610122317136 T^{5} - 1869108009716788 T^{6} + 135769977213045144 T^{7} + \)\(24\!\cdots\!20\)\( T^{8} + \)\(91\!\cdots\!70\)\( T^{9} + \)\(15\!\cdots\!85\)\( T^{10} - \)\(21\!\cdots\!10\)\( T^{11} - \)\(18\!\cdots\!90\)\( T^{12} - \)\(44\!\cdots\!30\)\( T^{13} - \)\(88\!\cdots\!30\)\( T^{14} + \)\(27\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!85\)\( T^{16} + \)\(27\!\cdots\!00\)\( p^{3} T^{17} - \)\(88\!\cdots\!30\)\( p^{6} T^{18} - \)\(44\!\cdots\!30\)\( p^{9} T^{19} - \)\(18\!\cdots\!90\)\( p^{12} T^{20} - \)\(21\!\cdots\!10\)\( p^{15} T^{21} + \)\(15\!\cdots\!85\)\( p^{18} T^{22} + \)\(91\!\cdots\!70\)\( p^{21} T^{23} + \)\(24\!\cdots\!20\)\( p^{24} T^{24} + 135769977213045144 p^{27} T^{25} - 1869108009716788 p^{30} T^{26} - 9610122317136 p^{33} T^{27} - 19545308845 p^{36} T^{28} + 31400740 p^{39} T^{29} + 332221 p^{42} T^{30} + 918 p^{45} T^{31} + p^{48} T^{32} \) | |
| 43 | \( ( 1 - 506 T + 529623 T^{2} - 217946122 T^{3} + 131977207041 T^{4} - 44544422354176 T^{5} + 19671375227058705 T^{6} - 5479302172478432416 T^{7} + \)\(19\!\cdots\!04\)\( T^{8} - 5479302172478432416 p^{3} T^{9} + 19671375227058705 p^{6} T^{10} - 44544422354176 p^{9} T^{11} + 131977207041 p^{12} T^{12} - 217946122 p^{15} T^{13} + 529623 p^{18} T^{14} - 506 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 47 | \( 1 + 209 T - 475992 T^{2} - 73076000 T^{3} + 111624408835 T^{4} + 6572303077612 T^{5} - 18637926631261012 T^{6} + 1150860426442005456 T^{7} + \)\(25\!\cdots\!40\)\( T^{8} - \)\(41\!\cdots\!70\)\( T^{9} - \)\(27\!\cdots\!88\)\( T^{10} + \)\(62\!\cdots\!83\)\( T^{11} + \)\(21\!\cdots\!16\)\( T^{12} - \)\(56\!\cdots\!30\)\( T^{13} - \)\(10\!\cdots\!30\)\( T^{14} + \)\(23\!\cdots\!76\)\( T^{15} + \)\(56\!\cdots\!49\)\( T^{16} + \)\(23\!\cdots\!76\)\( p^{3} T^{17} - \)\(10\!\cdots\!30\)\( p^{6} T^{18} - \)\(56\!\cdots\!30\)\( p^{9} T^{19} + \)\(21\!\cdots\!16\)\( p^{12} T^{20} + \)\(62\!\cdots\!83\)\( p^{15} T^{21} - \)\(27\!\cdots\!88\)\( p^{18} T^{22} - \)\(41\!\cdots\!70\)\( p^{21} T^{23} + \)\(25\!\cdots\!40\)\( p^{24} T^{24} + 1150860426442005456 p^{27} T^{25} - 18637926631261012 p^{30} T^{26} + 6572303077612 p^{33} T^{27} + 111624408835 p^{36} T^{28} - 73076000 p^{39} T^{29} - 475992 p^{42} T^{30} + 209 p^{45} T^{31} + p^{48} T^{32} \) | |
| 53 | \( 1 + 1738 T + 1211032 T^{2} + 263538985 T^{3} - 187516548715 T^{4} - 167565205042176 T^{5} - 49773191933381108 T^{6} + 1868145018506317003 T^{7} + \)\(60\!\cdots\!25\)\( T^{8} + \)\(19\!\cdots\!65\)\( T^{9} + \)\(45\!\cdots\!93\)\( T^{10} + \)\(23\!\cdots\!74\)\( T^{11} + \)\(10\!\cdots\!66\)\( T^{12} + \)\(86\!\cdots\!70\)\( T^{13} - \)\(12\!\cdots\!90\)\( T^{14} - \)\(72\!\cdots\!47\)\( T^{15} - \)\(27\!\cdots\!36\)\( T^{16} - \)\(72\!\cdots\!47\)\( p^{3} T^{17} - \)\(12\!\cdots\!90\)\( p^{6} T^{18} + \)\(86\!\cdots\!70\)\( p^{9} T^{19} + \)\(10\!\cdots\!66\)\( p^{12} T^{20} + \)\(23\!\cdots\!74\)\( p^{15} T^{21} + \)\(45\!\cdots\!93\)\( p^{18} T^{22} + \)\(19\!\cdots\!65\)\( p^{21} T^{23} + \)\(60\!\cdots\!25\)\( p^{24} T^{24} + 1868145018506317003 p^{27} T^{25} - 49773191933381108 p^{30} T^{26} - 167565205042176 p^{33} T^{27} - 187516548715 p^{36} T^{28} + 263538985 p^{39} T^{29} + 1211032 p^{42} T^{30} + 1738 p^{45} T^{31} + p^{48} T^{32} \) | |
| 59 | \( 1 - 695 T + 650109 T^{2} - 649504360 T^{3} + 323116464260 T^{4} - 187937135311545 T^{5} + 115051021927005445 T^{6} - 29063741037236700480 T^{7} + \)\(10\!\cdots\!85\)\( T^{8} - \)\(44\!\cdots\!15\)\( T^{9} - \)\(21\!\cdots\!48\)\( T^{10} + \)\(63\!\cdots\!85\)\( T^{11} - \)\(44\!\cdots\!22\)\( T^{12} + \)\(28\!\cdots\!45\)\( T^{13} + \)\(42\!\cdots\!55\)\( T^{14} - \)\(47\!\cdots\!35\)\( T^{15} - \)\(64\!\cdots\!90\)\( T^{16} - \)\(47\!\cdots\!35\)\( p^{3} T^{17} + \)\(42\!\cdots\!55\)\( p^{6} T^{18} + \)\(28\!\cdots\!45\)\( p^{9} T^{19} - \)\(44\!\cdots\!22\)\( p^{12} T^{20} + \)\(63\!\cdots\!85\)\( p^{15} T^{21} - \)\(21\!\cdots\!48\)\( p^{18} T^{22} - \)\(44\!\cdots\!15\)\( p^{21} T^{23} + \)\(10\!\cdots\!85\)\( p^{24} T^{24} - 29063741037236700480 p^{27} T^{25} + 115051021927005445 p^{30} T^{26} - 187937135311545 p^{33} T^{27} + 323116464260 p^{36} T^{28} - 649504360 p^{39} T^{29} + 650109 p^{42} T^{30} - 695 p^{45} T^{31} + p^{48} T^{32} \) | |
| 61 | \( 1 - 352 T - 988859 T^{2} + 383977960 T^{3} + 370714761035 T^{4} - 163175493974896 T^{5} - 46548262854333468 T^{6} + 31613097768425914484 T^{7} - \)\(89\!\cdots\!80\)\( T^{8} - \)\(23\!\cdots\!30\)\( T^{9} + \)\(38\!\cdots\!85\)\( T^{10} + \)\(32\!\cdots\!90\)\( T^{11} - \)\(40\!\cdots\!90\)\( T^{12} - \)\(21\!\cdots\!30\)\( T^{13} - \)\(41\!\cdots\!30\)\( T^{14} + \)\(31\!\cdots\!00\)\( T^{15} + \)\(20\!\cdots\!85\)\( T^{16} + \)\(31\!\cdots\!00\)\( p^{3} T^{17} - \)\(41\!\cdots\!30\)\( p^{6} T^{18} - \)\(21\!\cdots\!30\)\( p^{9} T^{19} - \)\(40\!\cdots\!90\)\( p^{12} T^{20} + \)\(32\!\cdots\!90\)\( p^{15} T^{21} + \)\(38\!\cdots\!85\)\( p^{18} T^{22} - \)\(23\!\cdots\!30\)\( p^{21} T^{23} - \)\(89\!\cdots\!80\)\( p^{24} T^{24} + 31613097768425914484 p^{27} T^{25} - 46548262854333468 p^{30} T^{26} - 163175493974896 p^{33} T^{27} + 370714761035 p^{36} T^{28} + 383977960 p^{39} T^{29} - 988859 p^{42} T^{30} - 352 p^{45} T^{31} + p^{48} T^{32} \) | |
| 67 | \( 1 + 1554 T + 262833 T^{2} - 850195060 T^{3} - 493453069400 T^{4} + 203098551275722 T^{5} + 296141339334582468 T^{6} + 26380457041566225046 T^{7} - \)\(10\!\cdots\!80\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{9} + \)\(22\!\cdots\!27\)\( T^{10} + \)\(25\!\cdots\!18\)\( T^{11} + \)\(23\!\cdots\!31\)\( T^{12} - \)\(72\!\cdots\!20\)\( T^{13} - \)\(34\!\cdots\!60\)\( T^{14} + \)\(89\!\cdots\!76\)\( T^{15} + \)\(13\!\cdots\!24\)\( T^{16} + \)\(89\!\cdots\!76\)\( p^{3} T^{17} - \)\(34\!\cdots\!60\)\( p^{6} T^{18} - \)\(72\!\cdots\!20\)\( p^{9} T^{19} + \)\(23\!\cdots\!31\)\( p^{12} T^{20} + \)\(25\!\cdots\!18\)\( p^{15} T^{21} + \)\(22\!\cdots\!27\)\( p^{18} T^{22} - \)\(46\!\cdots\!20\)\( p^{21} T^{23} - \)\(10\!\cdots\!80\)\( p^{24} T^{24} + 26380457041566225046 p^{27} T^{25} + 296141339334582468 p^{30} T^{26} + 203098551275722 p^{33} T^{27} - 493453069400 p^{36} T^{28} - 850195060 p^{39} T^{29} + 262833 p^{42} T^{30} + 1554 p^{45} T^{31} + p^{48} T^{32} \) | |
| 71 | \( 1 + 1543 T - 208329 T^{2} - 1005016350 T^{3} + 344257949820 T^{4} + 698351279304729 T^{5} - 62650049213568413 T^{6} - \)\(19\!\cdots\!71\)\( T^{7} + \)\(57\!\cdots\!95\)\( T^{8} + \)\(81\!\cdots\!20\)\( T^{9} + \)\(21\!\cdots\!35\)\( T^{10} - \)\(47\!\cdots\!35\)\( T^{11} - \)\(12\!\cdots\!40\)\( T^{12} + \)\(40\!\cdots\!70\)\( T^{13} + \)\(13\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} - \)\(41\!\cdots\!15\)\( T^{16} + \)\(11\!\cdots\!00\)\( p^{3} T^{17} + \)\(13\!\cdots\!20\)\( p^{6} T^{18} + \)\(40\!\cdots\!70\)\( p^{9} T^{19} - \)\(12\!\cdots\!40\)\( p^{12} T^{20} - \)\(47\!\cdots\!35\)\( p^{15} T^{21} + \)\(21\!\cdots\!35\)\( p^{18} T^{22} + \)\(81\!\cdots\!20\)\( p^{21} T^{23} + \)\(57\!\cdots\!95\)\( p^{24} T^{24} - \)\(19\!\cdots\!71\)\( p^{27} T^{25} - 62650049213568413 p^{30} T^{26} + 698351279304729 p^{33} T^{27} + 344257949820 p^{36} T^{28} - 1005016350 p^{39} T^{29} - 208329 p^{42} T^{30} + 1543 p^{45} T^{31} + p^{48} T^{32} \) | |
| 73 | \( 1 + 778 T - 500693 T^{2} - 273706210 T^{3} + 585980941270 T^{4} + 176624269337404 T^{5} - 366545805128711908 T^{6} - 53805977380721679492 T^{7} + \)\(19\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!10\)\( T^{9} - \)\(88\!\cdots\!67\)\( T^{10} + \)\(63\!\cdots\!34\)\( T^{11} + \)\(35\!\cdots\!51\)\( T^{12} - \)\(30\!\cdots\!00\)\( p T^{13} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(10\!\cdots\!28\)\( T^{15} + \)\(50\!\cdots\!04\)\( T^{16} + \)\(10\!\cdots\!28\)\( p^{3} T^{17} - \)\(11\!\cdots\!20\)\( p^{6} T^{18} - \)\(30\!\cdots\!00\)\( p^{10} T^{19} + \)\(35\!\cdots\!51\)\( p^{12} T^{20} + \)\(63\!\cdots\!34\)\( p^{15} T^{21} - \)\(88\!\cdots\!67\)\( p^{18} T^{22} - \)\(19\!\cdots\!10\)\( p^{21} T^{23} + \)\(19\!\cdots\!70\)\( p^{24} T^{24} - 53805977380721679492 p^{27} T^{25} - 366545805128711908 p^{30} T^{26} + 176624269337404 p^{33} T^{27} + 585980941270 p^{36} T^{28} - 273706210 p^{39} T^{29} - 500693 p^{42} T^{30} + 778 p^{45} T^{31} + p^{48} T^{32} \) | |
| 79 | \( 1 - 920 T + 525809 T^{2} - 278706170 T^{3} + 308776177695 T^{4} + 63942976170010 T^{5} - 129966518173800480 T^{6} + 12078879224764750140 T^{7} + \)\(57\!\cdots\!70\)\( T^{8} - \)\(17\!\cdots\!30\)\( T^{9} + \)\(82\!\cdots\!67\)\( T^{10} - \)\(83\!\cdots\!40\)\( T^{11} + \)\(38\!\cdots\!18\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!10\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{15} - \)\(68\!\cdots\!65\)\( T^{16} + \)\(15\!\cdots\!30\)\( p^{3} T^{17} + \)\(15\!\cdots\!10\)\( p^{6} T^{18} - \)\(10\!\cdots\!60\)\( p^{9} T^{19} + \)\(38\!\cdots\!18\)\( p^{12} T^{20} - \)\(83\!\cdots\!40\)\( p^{15} T^{21} + \)\(82\!\cdots\!67\)\( p^{18} T^{22} - \)\(17\!\cdots\!30\)\( p^{21} T^{23} + \)\(57\!\cdots\!70\)\( p^{24} T^{24} + 12078879224764750140 p^{27} T^{25} - 129966518173800480 p^{30} T^{26} + 63942976170010 p^{33} T^{27} + 308776177695 p^{36} T^{28} - 278706170 p^{39} T^{29} + 525809 p^{42} T^{30} - 920 p^{45} T^{31} + p^{48} T^{32} \) | |
| 83 | \( 1 - 2967 T + 3285857 T^{2} - 889502295 T^{3} - 1458233474800 T^{4} + 2342356175857219 T^{5} - 2887246304185396218 T^{6} + \)\(27\!\cdots\!48\)\( T^{7} - \)\(10\!\cdots\!70\)\( T^{8} - \)\(95\!\cdots\!35\)\( T^{9} + \)\(16\!\cdots\!28\)\( T^{10} - \)\(15\!\cdots\!81\)\( T^{11} + \)\(11\!\cdots\!06\)\( T^{12} - \)\(42\!\cdots\!60\)\( T^{13} - \)\(32\!\cdots\!10\)\( T^{14} + \)\(58\!\cdots\!53\)\( T^{15} - \)\(50\!\cdots\!96\)\( T^{16} + \)\(58\!\cdots\!53\)\( p^{3} T^{17} - \)\(32\!\cdots\!10\)\( p^{6} T^{18} - \)\(42\!\cdots\!60\)\( p^{9} T^{19} + \)\(11\!\cdots\!06\)\( p^{12} T^{20} - \)\(15\!\cdots\!81\)\( p^{15} T^{21} + \)\(16\!\cdots\!28\)\( p^{18} T^{22} - \)\(95\!\cdots\!35\)\( p^{21} T^{23} - \)\(10\!\cdots\!70\)\( p^{24} T^{24} + \)\(27\!\cdots\!48\)\( p^{27} T^{25} - 2887246304185396218 p^{30} T^{26} + 2342356175857219 p^{33} T^{27} - 1458233474800 p^{36} T^{28} - 889502295 p^{39} T^{29} + 3285857 p^{42} T^{30} - 2967 p^{45} T^{31} + p^{48} T^{32} \) | |
| 89 | \( 1 + 2605 T + 2992849 T^{2} + 2136344045 T^{3} + 1262225805360 T^{4} + 23016854835835 p T^{5} + 3568950052505913070 T^{6} + \)\(32\!\cdots\!10\)\( T^{7} + \)\(14\!\cdots\!60\)\( T^{8} + \)\(13\!\cdots\!05\)\( T^{9} + \)\(40\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!35\)\( T^{11} + \)\(51\!\cdots\!58\)\( T^{12} - \)\(50\!\cdots\!90\)\( T^{13} - \)\(76\!\cdots\!20\)\( T^{14} - \)\(30\!\cdots\!05\)\( T^{15} - \)\(25\!\cdots\!40\)\( T^{16} - \)\(30\!\cdots\!05\)\( p^{3} T^{17} - \)\(76\!\cdots\!20\)\( p^{6} T^{18} - \)\(50\!\cdots\!90\)\( p^{9} T^{19} + \)\(51\!\cdots\!58\)\( p^{12} T^{20} + \)\(10\!\cdots\!35\)\( p^{15} T^{21} + \)\(40\!\cdots\!52\)\( p^{18} T^{22} + \)\(13\!\cdots\!05\)\( p^{21} T^{23} + \)\(14\!\cdots\!60\)\( p^{24} T^{24} + \)\(32\!\cdots\!10\)\( p^{27} T^{25} + 3568950052505913070 p^{30} T^{26} + 23016854835835 p^{34} T^{27} + 1262225805360 p^{36} T^{28} + 2136344045 p^{39} T^{29} + 2992849 p^{42} T^{30} + 2605 p^{45} T^{31} + p^{48} T^{32} \) | |
| 97 | \( 1 - 1251 T - 1544117 T^{2} + 2105987265 T^{3} + 1990526674490 T^{4} - 3856701043597483 T^{5} + 27092960650973008 T^{6} + \)\(37\!\cdots\!96\)\( T^{7} - \)\(61\!\cdots\!20\)\( T^{8} - \)\(42\!\cdots\!25\)\( T^{9} + \)\(23\!\cdots\!02\)\( T^{10} + \)\(19\!\cdots\!83\)\( T^{11} - \)\(16\!\cdots\!14\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{13} + \)\(25\!\cdots\!70\)\( T^{14} - \)\(16\!\cdots\!89\)\( T^{15} - \)\(13\!\cdots\!56\)\( T^{16} - \)\(16\!\cdots\!89\)\( p^{3} T^{17} + \)\(25\!\cdots\!70\)\( p^{6} T^{18} - \)\(16\!\cdots\!40\)\( p^{9} T^{19} - \)\(16\!\cdots\!14\)\( p^{12} T^{20} + \)\(19\!\cdots\!83\)\( p^{15} T^{21} + \)\(23\!\cdots\!02\)\( p^{18} T^{22} - \)\(42\!\cdots\!25\)\( p^{21} T^{23} - \)\(61\!\cdots\!20\)\( p^{24} T^{24} + \)\(37\!\cdots\!96\)\( p^{27} T^{25} + 27092960650973008 p^{30} T^{26} - 3856701043597483 p^{33} T^{27} + 1990526674490 p^{36} T^{28} + 2105987265 p^{39} T^{29} - 1544117 p^{42} T^{30} - 1251 p^{45} T^{31} + p^{48} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.23026723542137874889234416451, −3.12413030831585182135256336205, −3.07066793471123357662495610534, −2.99174513803311828164114153020, −2.84376165920353446523209992180, −2.60387370843089757922783137795, −2.54251902550737608893983255800, −2.24993901962847763860694307091, −2.13201796070724937145546970367, −2.09099952085679018961472068231, −2.06093588460895550948822677029, −1.93220270524547281670977192624, −1.75031635394400088137280190157, −1.71414807594683907677617088271, −1.70114973653886076921567752101, −1.53913261105045436801308615954, −1.24171572037348567457698786732, −1.22486068123036822283335563093, −1.09528497788674332712177344649, −0.836364174382042418765522857434, −0.51775916447468740014259136599, −0.44988538317506098809196430437, −0.34008918723327171879180249980, −0.095975148005085959430155554357, −0.06677363693016618557041513944, 0.06677363693016618557041513944, 0.095975148005085959430155554357, 0.34008918723327171879180249980, 0.44988538317506098809196430437, 0.51775916447468740014259136599, 0.836364174382042418765522857434, 1.09528497788674332712177344649, 1.22486068123036822283335563093, 1.24171572037348567457698786732, 1.53913261105045436801308615954, 1.70114973653886076921567752101, 1.71414807594683907677617088271, 1.75031635394400088137280190157, 1.93220270524547281670977192624, 2.06093588460895550948822677029, 2.09099952085679018961472068231, 2.13201796070724937145546970367, 2.24993901962847763860694307091, 2.54251902550737608893983255800, 2.60387370843089757922783137795, 2.84376165920353446523209992180, 2.99174513803311828164114153020, 3.07066793471123357662495610534, 3.12413030831585182135256336205, 3.23026723542137874889234416451
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.