Dirichlet series
| L(s) = 1 | − 4·3-s + 8·4-s − 4·7-s + 6·9-s − 32·12-s + 20·13-s + 24·16-s + 80·19-s + 16·21-s + 108·23-s + 20·25-s − 44·27-s − 32·28-s − 72·29-s − 16·31-s + 48·36-s − 88·37-s − 80·39-s + 108·41-s + 92·43-s + 216·47-s − 96·48-s + 162·49-s + 160·52-s − 320·57-s + 144·59-s − 76·61-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 2·4-s − 4/7·7-s + 2/3·9-s − 8/3·12-s + 1.53·13-s + 3/2·16-s + 4.21·19-s + 0.761·21-s + 4.69·23-s + 4/5·25-s − 1.62·27-s − 8/7·28-s − 2.48·29-s − 0.516·31-s + 4/3·36-s − 2.37·37-s − 2.05·39-s + 2.63·41-s + 2.13·43-s + 4.59·47-s − 2·48-s + 3.30·49-s + 3.07·52-s − 5.61·57-s + 2.44·59-s − 1.24·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.71096\times 10^{6}\) |
| Root analytic conductor: | \(1.56598\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(18.02900885\) |
| \(L(\frac12)\) | \(\approx\) | \(18.02900885\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 3 | \( 1 + 4 T + 10 T^{2} + 20 p T^{3} + 38 p T^{4} - 8 p^{3} T^{5} - 68 p^{2} T^{6} - 160 p^{3} T^{7} - 107 p^{5} T^{8} - 160 p^{5} T^{9} - 68 p^{6} T^{10} - 8 p^{9} T^{11} + 38 p^{9} T^{12} + 20 p^{11} T^{13} + 10 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) | |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) | |
| good | 7 | \( 1 + 4 T - 146 T^{2} + 320 T^{3} + 15746 T^{4} - 100264 T^{5} - 935388 T^{6} + 12289104 T^{7} + 21877533 T^{8} - 912057060 T^{9} + 2014672488 T^{10} + 47681034540 T^{11} - 281856111090 T^{12} - 1778441869140 T^{13} + 2843171454954 p T^{14} + 30529486370016 T^{15} - 1095742037511180 T^{16} + 30529486370016 p^{2} T^{17} + 2843171454954 p^{5} T^{18} - 1778441869140 p^{6} T^{19} - 281856111090 p^{8} T^{20} + 47681034540 p^{10} T^{21} + 2014672488 p^{12} T^{22} - 912057060 p^{14} T^{23} + 21877533 p^{16} T^{24} + 12289104 p^{18} T^{25} - 935388 p^{20} T^{26} - 100264 p^{22} T^{27} + 15746 p^{24} T^{28} + 320 p^{26} T^{29} - 146 p^{28} T^{30} + 4 p^{30} T^{31} + p^{32} T^{32} \) |
| 11 | \( 1 + 404 T^{2} + 88392 T^{4} - 214488 T^{5} + 12254632 T^{6} - 79906176 T^{7} + 1148253362 T^{8} - 16233244992 T^{9} + 93947279244 T^{10} - 1877958262872 T^{11} + 13060815389440 T^{12} - 107465447856312 T^{13} + 2613420437071676 T^{14} + 2033213734736064 T^{15} + 381638304187428051 T^{16} + 2033213734736064 p^{2} T^{17} + 2613420437071676 p^{4} T^{18} - 107465447856312 p^{6} T^{19} + 13060815389440 p^{8} T^{20} - 1877958262872 p^{10} T^{21} + 93947279244 p^{12} T^{22} - 16233244992 p^{14} T^{23} + 1148253362 p^{16} T^{24} - 79906176 p^{18} T^{25} + 12254632 p^{20} T^{26} - 214488 p^{22} T^{27} + 88392 p^{24} T^{28} + 404 p^{28} T^{30} + p^{32} T^{32} \) | |
| 13 | \( 1 - 20 T - 440 T^{2} + 15224 T^{3} + 2792 T^{4} - 4384708 T^{5} + 37559376 T^{6} + 543700836 T^{7} - 10574796078 T^{8} + 20303152224 T^{9} + 1137996863304 T^{10} - 20871655435884 T^{11} + 117112910158848 T^{12} + 303351707609364 p T^{13} - 426096362180904 p^{2} T^{14} - 21482973020381472 p T^{15} + 15670006860879007155 T^{16} - 21482973020381472 p^{3} T^{17} - 426096362180904 p^{6} T^{18} + 303351707609364 p^{7} T^{19} + 117112910158848 p^{8} T^{20} - 20871655435884 p^{10} T^{21} + 1137996863304 p^{12} T^{22} + 20303152224 p^{14} T^{23} - 10574796078 p^{16} T^{24} + 543700836 p^{18} T^{25} + 37559376 p^{20} T^{26} - 4384708 p^{22} T^{27} + 2792 p^{24} T^{28} + 15224 p^{26} T^{29} - 440 p^{28} T^{30} - 20 p^{30} T^{31} + p^{32} T^{32} \) | |
| 17 | \( 1 - 2704 T^{2} + 3597072 T^{4} - 3145908176 T^{6} + 2042929880828 T^{8} - 1054356753438384 T^{10} + 450684258093903856 T^{12} - \)\(16\!\cdots\!80\)\( T^{14} + \)\(50\!\cdots\!18\)\( T^{16} - \)\(16\!\cdots\!80\)\( p^{4} T^{18} + 450684258093903856 p^{8} T^{20} - 1054356753438384 p^{12} T^{22} + 2042929880828 p^{16} T^{24} - 3145908176 p^{20} T^{26} + 3597072 p^{24} T^{28} - 2704 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - 40 T + 2316 T^{2} - 55232 T^{3} + 1965752 T^{4} - 36048288 T^{5} + 1102408996 T^{6} - 17932429144 T^{7} + 471934498878 T^{8} - 17932429144 p^{2} T^{9} + 1102408996 p^{4} T^{10} - 36048288 p^{6} T^{11} + 1965752 p^{8} T^{12} - 55232 p^{10} T^{13} + 2316 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 108 T + 7862 T^{2} - 429192 T^{3} + 19576578 T^{4} - 772306704 T^{5} + 27124132948 T^{6} - 863106175800 T^{7} + 25183628943629 T^{8} - 680940319078332 T^{9} + 17164978367140392 T^{10} - 406729799107094700 T^{11} + 9130559797431098062 T^{12} - \)\(19\!\cdots\!68\)\( T^{13} + \)\(41\!\cdots\!10\)\( T^{14} - \)\(89\!\cdots\!40\)\( T^{15} + \)\(20\!\cdots\!20\)\( T^{16} - \)\(89\!\cdots\!40\)\( p^{2} T^{17} + \)\(41\!\cdots\!10\)\( p^{4} T^{18} - \)\(19\!\cdots\!68\)\( p^{6} T^{19} + 9130559797431098062 p^{8} T^{20} - 406729799107094700 p^{10} T^{21} + 17164978367140392 p^{12} T^{22} - 680940319078332 p^{14} T^{23} + 25183628943629 p^{16} T^{24} - 863106175800 p^{18} T^{25} + 27124132948 p^{20} T^{26} - 772306704 p^{22} T^{27} + 19576578 p^{24} T^{28} - 429192 p^{26} T^{29} + 7862 p^{28} T^{30} - 108 p^{30} T^{31} + p^{32} T^{32} \) | |
| 29 | \( 1 + 72 T + 4868 T^{2} + 226080 T^{3} + 9315966 T^{4} + 320687784 T^{5} + 9327003208 T^{6} + 247496300712 T^{7} + 4917458817737 T^{8} + 88011447124536 T^{9} - 25776577235064 T^{10} - 58639910604255000 T^{11} - 3830026844576274050 T^{12} - \)\(17\!\cdots\!56\)\( T^{13} - \)\(63\!\cdots\!24\)\( T^{14} - \)\(22\!\cdots\!72\)\( T^{15} - \)\(65\!\cdots\!16\)\( T^{16} - \)\(22\!\cdots\!72\)\( p^{2} T^{17} - \)\(63\!\cdots\!24\)\( p^{4} T^{18} - \)\(17\!\cdots\!56\)\( p^{6} T^{19} - 3830026844576274050 p^{8} T^{20} - 58639910604255000 p^{10} T^{21} - 25776577235064 p^{12} T^{22} + 88011447124536 p^{14} T^{23} + 4917458817737 p^{16} T^{24} + 247496300712 p^{18} T^{25} + 9327003208 p^{20} T^{26} + 320687784 p^{22} T^{27} + 9315966 p^{24} T^{28} + 226080 p^{26} T^{29} + 4868 p^{28} T^{30} + 72 p^{30} T^{31} + p^{32} T^{32} \) | |
| 31 | \( 1 + 16 T - 1484 T^{2} + 13712 T^{3} + 428600 T^{4} - 59823736 T^{5} + 1382601960 T^{6} + 50763585168 T^{7} - 1957641873390 T^{8} + 30863201756928 T^{9} + 1995366851958156 T^{10} - 75213916040906472 T^{11} - 711367308837542976 T^{12} + 51859590456216526104 T^{13} - \)\(16\!\cdots\!64\)\( T^{14} - \)\(12\!\cdots\!04\)\( T^{15} + \)\(28\!\cdots\!87\)\( T^{16} - \)\(12\!\cdots\!04\)\( p^{2} T^{17} - \)\(16\!\cdots\!64\)\( p^{4} T^{18} + 51859590456216526104 p^{6} T^{19} - 711367308837542976 p^{8} T^{20} - 75213916040906472 p^{10} T^{21} + 1995366851958156 p^{12} T^{22} + 30863201756928 p^{14} T^{23} - 1957641873390 p^{16} T^{24} + 50763585168 p^{18} T^{25} + 1382601960 p^{20} T^{26} - 59823736 p^{22} T^{27} + 428600 p^{24} T^{28} + 13712 p^{26} T^{29} - 1484 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} \) | |
| 37 | \( ( 1 + 44 T + 6264 T^{2} + 196852 T^{3} + 16543112 T^{4} + 409697244 T^{5} + 28023661432 T^{6} + 621670925924 T^{7} + 39838114215198 T^{8} + 621670925924 p^{2} T^{9} + 28023661432 p^{4} T^{10} + 409697244 p^{6} T^{11} + 16543112 p^{8} T^{12} + 196852 p^{10} T^{13} + 6264 p^{12} T^{14} + 44 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 108 T + 10568 T^{2} - 721440 T^{3} + 42436782 T^{4} - 1967806764 T^{5} + 72001786144 T^{6} - 1523454495228 T^{7} - 28249267979503 T^{8} + 5485010695922556 T^{9} - 358071512610352464 T^{10} + 16383393892680422724 T^{11} - \)\(54\!\cdots\!98\)\( T^{12} + \)\(23\!\cdots\!16\)\( p T^{13} + \)\(20\!\cdots\!28\)\( T^{14} - \)\(71\!\cdots\!76\)\( p T^{15} + \)\(15\!\cdots\!44\)\( T^{16} - \)\(71\!\cdots\!76\)\( p^{3} T^{17} + \)\(20\!\cdots\!28\)\( p^{4} T^{18} + \)\(23\!\cdots\!16\)\( p^{7} T^{19} - \)\(54\!\cdots\!98\)\( p^{8} T^{20} + 16383393892680422724 p^{10} T^{21} - 358071512610352464 p^{12} T^{22} + 5485010695922556 p^{14} T^{23} - 28249267979503 p^{16} T^{24} - 1523454495228 p^{18} T^{25} + 72001786144 p^{20} T^{26} - 1967806764 p^{22} T^{27} + 42436782 p^{24} T^{28} - 721440 p^{26} T^{29} + 10568 p^{28} T^{30} - 108 p^{30} T^{31} + p^{32} T^{32} \) | |
| 43 | \( 1 - 92 T - 1568 T^{2} + 254840 T^{3} + 8150816 T^{4} - 537369844 T^{5} - 20488290624 T^{6} - 7659542484 T^{7} + 57044470643010 T^{8} + 1574216574050400 T^{9} - 44174083476013248 T^{10} - 5569275917834463324 T^{11} - \)\(11\!\cdots\!52\)\( T^{12} + \)\(10\!\cdots\!60\)\( T^{13} + \)\(53\!\cdots\!40\)\( T^{14} - \)\(74\!\cdots\!88\)\( T^{15} - \)\(12\!\cdots\!09\)\( T^{16} - \)\(74\!\cdots\!88\)\( p^{2} T^{17} + \)\(53\!\cdots\!40\)\( p^{4} T^{18} + \)\(10\!\cdots\!60\)\( p^{6} T^{19} - \)\(11\!\cdots\!52\)\( p^{8} T^{20} - 5569275917834463324 p^{10} T^{21} - 44174083476013248 p^{12} T^{22} + 1574216574050400 p^{14} T^{23} + 57044470643010 p^{16} T^{24} - 7659542484 p^{18} T^{25} - 20488290624 p^{20} T^{26} - 537369844 p^{22} T^{27} + 8150816 p^{24} T^{28} + 254840 p^{26} T^{29} - 1568 p^{28} T^{30} - 92 p^{30} T^{31} + p^{32} T^{32} \) | |
| 47 | \( 1 - 216 T + 33146 T^{2} - 3800304 T^{3} + 367425930 T^{4} - 30405141828 T^{5} + 2232801084508 T^{6} - 146263160342844 T^{7} + 8667022533254405 T^{8} - 463891709259983880 T^{9} + 22451590784326612344 T^{10} - \)\(97\!\cdots\!64\)\( T^{11} + \)\(37\!\cdots\!06\)\( T^{12} - \)\(12\!\cdots\!48\)\( T^{13} + \)\(31\!\cdots\!18\)\( T^{14} - \)\(58\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!08\)\( T^{16} - \)\(58\!\cdots\!40\)\( p^{2} T^{17} + \)\(31\!\cdots\!18\)\( p^{4} T^{18} - \)\(12\!\cdots\!48\)\( p^{6} T^{19} + \)\(37\!\cdots\!06\)\( p^{8} T^{20} - \)\(97\!\cdots\!64\)\( p^{10} T^{21} + 22451590784326612344 p^{12} T^{22} - 463891709259983880 p^{14} T^{23} + 8667022533254405 p^{16} T^{24} - 146263160342844 p^{18} T^{25} + 2232801084508 p^{20} T^{26} - 30405141828 p^{22} T^{27} + 367425930 p^{24} T^{28} - 3800304 p^{26} T^{29} + 33146 p^{28} T^{30} - 216 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 - 18832 T^{2} + 191515128 T^{4} - 1380332557232 T^{6} + 7788690946630556 T^{8} - 36282818472706231440 T^{10} + \)\(14\!\cdots\!92\)\( T^{12} - \)\(49\!\cdots\!44\)\( T^{14} + \)\(14\!\cdots\!66\)\( T^{16} - \)\(49\!\cdots\!44\)\( p^{4} T^{18} + \)\(14\!\cdots\!92\)\( p^{8} T^{20} - 36282818472706231440 p^{12} T^{22} + 7788690946630556 p^{16} T^{24} - 1380332557232 p^{20} T^{26} + 191515128 p^{24} T^{28} - 18832 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 144 T + 21200 T^{2} - 2057472 T^{3} + 196896468 T^{4} - 15378312528 T^{5} + 1060803715552 T^{6} - 65273054006736 T^{7} + 3198924129557930 T^{8} - 139199140969147872 T^{9} + 2224697091366962544 T^{10} + \)\(16\!\cdots\!20\)\( T^{11} - \)\(28\!\cdots\!88\)\( T^{12} + \)\(25\!\cdots\!40\)\( T^{13} - \)\(17\!\cdots\!48\)\( T^{14} + \)\(12\!\cdots\!92\)\( T^{15} - \)\(69\!\cdots\!17\)\( T^{16} + \)\(12\!\cdots\!92\)\( p^{2} T^{17} - \)\(17\!\cdots\!48\)\( p^{4} T^{18} + \)\(25\!\cdots\!40\)\( p^{6} T^{19} - \)\(28\!\cdots\!88\)\( p^{8} T^{20} + \)\(16\!\cdots\!20\)\( p^{10} T^{21} + 2224697091366962544 p^{12} T^{22} - 139199140969147872 p^{14} T^{23} + 3198924129557930 p^{16} T^{24} - 65273054006736 p^{18} T^{25} + 1060803715552 p^{20} T^{26} - 15378312528 p^{22} T^{27} + 196896468 p^{24} T^{28} - 2057472 p^{26} T^{29} + 21200 p^{28} T^{30} - 144 p^{30} T^{31} + p^{32} T^{32} \) | |
| 61 | \( 1 + 76 T - 9320 T^{2} - 1083808 T^{3} + 11094062 T^{4} + 5581041404 T^{5} + 159798669120 T^{6} - 12074870136564 T^{7} - 621037375385967 T^{8} + 17944339734614148 T^{9} + 17394375872241456 T^{10} - \)\(15\!\cdots\!20\)\( T^{11} + \)\(94\!\cdots\!74\)\( T^{12} + \)\(95\!\cdots\!52\)\( T^{13} + \)\(38\!\cdots\!52\)\( T^{14} - \)\(19\!\cdots\!84\)\( T^{15} - \)\(25\!\cdots\!08\)\( T^{16} - \)\(19\!\cdots\!84\)\( p^{2} T^{17} + \)\(38\!\cdots\!52\)\( p^{4} T^{18} + \)\(95\!\cdots\!52\)\( p^{6} T^{19} + \)\(94\!\cdots\!74\)\( p^{8} T^{20} - \)\(15\!\cdots\!20\)\( p^{10} T^{21} + 17394375872241456 p^{12} T^{22} + 17944339734614148 p^{14} T^{23} - 621037375385967 p^{16} T^{24} - 12074870136564 p^{18} T^{25} + 159798669120 p^{20} T^{26} + 5581041404 p^{22} T^{27} + 11094062 p^{24} T^{28} - 1083808 p^{26} T^{29} - 9320 p^{28} T^{30} + 76 p^{30} T^{31} + p^{32} T^{32} \) | |
| 67 | \( 1 - 56 T - 23798 T^{2} + 937736 T^{3} + 330095450 T^{4} - 8222399476 T^{5} - 3112158934692 T^{6} + 32243942523060 T^{7} + 22217313567549861 T^{8} + 48648332394643920 T^{9} - \)\(12\!\cdots\!28\)\( T^{10} - \)\(14\!\cdots\!88\)\( T^{11} + \)\(57\!\cdots\!10\)\( T^{12} + \)\(86\!\cdots\!72\)\( T^{13} - \)\(35\!\cdots\!10\)\( p T^{14} - \)\(18\!\cdots\!40\)\( T^{15} + \)\(10\!\cdots\!68\)\( T^{16} - \)\(18\!\cdots\!40\)\( p^{2} T^{17} - \)\(35\!\cdots\!10\)\( p^{5} T^{18} + \)\(86\!\cdots\!72\)\( p^{6} T^{19} + \)\(57\!\cdots\!10\)\( p^{8} T^{20} - \)\(14\!\cdots\!88\)\( p^{10} T^{21} - \)\(12\!\cdots\!28\)\( p^{12} T^{22} + 48648332394643920 p^{14} T^{23} + 22217313567549861 p^{16} T^{24} + 32243942523060 p^{18} T^{25} - 3112158934692 p^{20} T^{26} - 8222399476 p^{22} T^{27} + 330095450 p^{24} T^{28} + 937736 p^{26} T^{29} - 23798 p^{28} T^{30} - 56 p^{30} T^{31} + p^{32} T^{32} \) | |
| 71 | \( 1 - 29320 T^{2} + 456322080 T^{4} - 4832484884696 T^{6} + 38491279852604636 T^{8} - \)\(24\!\cdots\!16\)\( T^{10} + \)\(13\!\cdots\!24\)\( T^{12} - \)\(62\!\cdots\!08\)\( T^{14} + \)\(30\!\cdots\!54\)\( T^{16} - \)\(62\!\cdots\!08\)\( p^{4} T^{18} + \)\(13\!\cdots\!24\)\( p^{8} T^{20} - \)\(24\!\cdots\!16\)\( p^{12} T^{22} + 38491279852604636 p^{16} T^{24} - 4832484884696 p^{20} T^{26} + 456322080 p^{24} T^{28} - 29320 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 208 T + 28872 T^{2} - 2710832 T^{3} + 229184924 T^{4} - 17054302608 T^{5} + 1368308092024 T^{6} - 103445322727024 T^{7} + 8054138875878342 T^{8} - 103445322727024 p^{2} T^{9} + 1368308092024 p^{4} T^{10} - 17054302608 p^{6} T^{11} + 229184924 p^{8} T^{12} - 2710832 p^{10} T^{13} + 28872 p^{12} T^{14} - 208 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 - 80 T - 28760 T^{2} + 1507040 T^{3} + 508086404 T^{4} - 13609786000 T^{5} - 6144516450000 T^{6} + 21445168937040 T^{7} + 55831409291528010 T^{8} + 828278678318735040 T^{9} - \)\(37\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!36\)\( T^{12} + \)\(78\!\cdots\!80\)\( T^{13} - \)\(68\!\cdots\!40\)\( T^{14} - \)\(21\!\cdots\!00\)\( T^{15} + \)\(29\!\cdots\!39\)\( T^{16} - \)\(21\!\cdots\!00\)\( p^{2} T^{17} - \)\(68\!\cdots\!40\)\( p^{4} T^{18} + \)\(78\!\cdots\!80\)\( p^{6} T^{19} + \)\(18\!\cdots\!36\)\( p^{8} T^{20} - \)\(11\!\cdots\!00\)\( p^{10} T^{21} - \)\(37\!\cdots\!40\)\( p^{12} T^{22} + 828278678318735040 p^{14} T^{23} + 55831409291528010 p^{16} T^{24} + 21445168937040 p^{18} T^{25} - 6144516450000 p^{20} T^{26} - 13609786000 p^{22} T^{27} + 508086404 p^{24} T^{28} + 1507040 p^{26} T^{29} - 28760 p^{28} T^{30} - 80 p^{30} T^{31} + p^{32} T^{32} \) | |
| 83 | \( 1 - 396 T + 92078 T^{2} - 15763176 T^{3} + 2140505778 T^{4} - 234761054904 T^{5} + 20006783488708 T^{6} - 1142981276912064 T^{7} + 4056709768361981 T^{8} + 9233818211443669380 T^{9} - \)\(14\!\cdots\!40\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} - \)\(79\!\cdots\!38\)\( T^{12} + \)\(49\!\cdots\!08\)\( T^{13} + \)\(58\!\cdots\!06\)\( T^{14} - \)\(91\!\cdots\!52\)\( T^{15} + \)\(89\!\cdots\!76\)\( T^{16} - \)\(91\!\cdots\!52\)\( p^{2} T^{17} + \)\(58\!\cdots\!06\)\( p^{4} T^{18} + \)\(49\!\cdots\!08\)\( p^{6} T^{19} - \)\(79\!\cdots\!38\)\( p^{8} T^{20} + \)\(13\!\cdots\!00\)\( p^{10} T^{21} - \)\(14\!\cdots\!40\)\( p^{12} T^{22} + 9233818211443669380 p^{14} T^{23} + 4056709768361981 p^{16} T^{24} - 1142981276912064 p^{18} T^{25} + 20006783488708 p^{20} T^{26} - 234761054904 p^{22} T^{27} + 2140505778 p^{24} T^{28} - 15763176 p^{26} T^{29} + 92078 p^{28} T^{30} - 396 p^{30} T^{31} + p^{32} T^{32} \) | |
| 89 | \( 1 - 54376 T^{2} + 1558732692 T^{4} - 30956653156304 T^{6} + 478765947608893658 T^{8} - \)\(61\!\cdots\!56\)\( T^{10} + \)\(67\!\cdots\!76\)\( T^{12} - \)\(64\!\cdots\!60\)\( T^{14} + \)\(54\!\cdots\!03\)\( T^{16} - \)\(64\!\cdots\!60\)\( p^{4} T^{18} + \)\(67\!\cdots\!76\)\( p^{8} T^{20} - \)\(61\!\cdots\!56\)\( p^{12} T^{22} + 478765947608893658 p^{16} T^{24} - 30956653156304 p^{20} T^{26} + 1558732692 p^{24} T^{28} - 54376 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( 1 + 16 T - 43400 T^{2} - 2026336 T^{3} + 985562660 T^{4} + 64897485008 T^{5} - 12951436842864 T^{6} - 1104593771689488 T^{7} + 97534185729569802 T^{8} + 10122796114307965248 T^{9} - \)\(24\!\cdots\!32\)\( T^{10} - \)\(30\!\cdots\!52\)\( T^{11} - \)\(20\!\cdots\!48\)\( T^{12} - \)\(31\!\cdots\!56\)\( T^{13} + \)\(15\!\cdots\!64\)\( T^{14} + \)\(23\!\cdots\!64\)\( T^{15} - \)\(29\!\cdots\!81\)\( T^{16} + \)\(23\!\cdots\!64\)\( p^{2} T^{17} + \)\(15\!\cdots\!64\)\( p^{4} T^{18} - \)\(31\!\cdots\!56\)\( p^{6} T^{19} - \)\(20\!\cdots\!48\)\( p^{8} T^{20} - \)\(30\!\cdots\!52\)\( p^{10} T^{21} - \)\(24\!\cdots\!32\)\( p^{12} T^{22} + 10122796114307965248 p^{14} T^{23} + 97534185729569802 p^{16} T^{24} - 1104593771689488 p^{18} T^{25} - 12951436842864 p^{20} T^{26} + 64897485008 p^{22} T^{27} + 985562660 p^{24} T^{28} - 2026336 p^{26} T^{29} - 43400 p^{28} T^{30} + 16 p^{30} T^{31} + p^{32} T^{32} \) | |
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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
−3.99953331475481411148553933402, −3.94431200758564569507297587038, −3.93501894858058748752948947561, −3.76203052702036200612623718259, −3.72442411743723926718547485637, −3.61113671084698812846547471540, −3.37351022128007414294537467642, −3.24388627792719964153576609265, −3.22637075865224150355372124084, −2.95687885489487378124008322189, −2.81120408818938242113195256017, −2.75274954092210290517433031312, −2.69074546713504795194217060688, −2.65261523613620927858509476103, −2.54383996294067229419574579918, −2.14522109819184327602136130625, −2.07723002649749592257045673598, −1.82110288423197669961708578448, −1.78109891240610821220384224516, −1.74722718629331814252010034210, −0.952203735634484483835926557326, −0.948152243733705572695541758763, −0.863844211004514106839273906447, −0.839311568507153202490003376927, −0.798742387571258959738370462567, 0.798742387571258959738370462567, 0.839311568507153202490003376927, 0.863844211004514106839273906447, 0.948152243733705572695541758763, 0.952203735634484483835926557326, 1.74722718629331814252010034210, 1.78109891240610821220384224516, 1.82110288423197669961708578448, 2.07723002649749592257045673598, 2.14522109819184327602136130625, 2.54383996294067229419574579918, 2.65261523613620927858509476103, 2.69074546713504795194217060688, 2.75274954092210290517433031312, 2.81120408818938242113195256017, 2.95687885489487378124008322189, 3.22637075865224150355372124084, 3.24388627792719964153576609265, 3.37351022128007414294537467642, 3.61113671084698812846547471540, 3.72442411743723926718547485637, 3.76203052702036200612623718259, 3.93501894858058748752948947561, 3.94431200758564569507297587038, 3.99953331475481411148553933402
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.