Dirichlet series
| L(s) = 1 | + 16·2-s + 128·4-s − 2·5-s + 672·8-s + 64·9-s − 32·10-s + 20·11-s + 4·13-s + 2.54e3·16-s + 36·17-s + 1.02e3·18-s − 16·19-s − 256·20-s + 320·22-s + 16·23-s + 24·25-s + 64·26-s − 20·29-s + 92·31-s + 7.10e3·32-s + 576·34-s + 8.19e3·36-s + 32·37-s − 256·38-s − 1.34e3·40-s − 60·41-s + 52·43-s + ⋯ |
| L(s) = 1 | + 8·2-s + 32·4-s − 2/5·5-s + 84·8-s + 64/9·9-s − 3.19·10-s + 1.81·11-s + 4/13·13-s + 159·16-s + 2.11·17-s + 56.8·18-s − 0.842·19-s − 12.7·20-s + 14.5·22-s + 0.695·23-s + 0.959·25-s + 2.46·26-s − 0.689·29-s + 2.96·31-s + 222·32-s + 16.9·34-s + 227.·36-s + 0.864·37-s − 6.73·38-s − 33.5·40-s − 1.46·41-s + 1.20·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{16} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.49307\times 10^{10}\) |
| Root analytic conductor: | \(2.15224\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(11707.37793\) |
| \(L(\frac12)\) | \(\approx\) | \(11707.37793\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{8} \) |
| 5 | \( 1 + 2 T - 4 p T^{2} + 186 T^{3} + 416 T^{4} - 1426 p T^{5} + 796 p^{2} T^{6} + 534 p^{3} T^{7} - 1898 p^{4} T^{8} + 534 p^{5} T^{9} + 796 p^{6} T^{10} - 1426 p^{7} T^{11} + 416 p^{8} T^{12} + 186 p^{10} T^{13} - 4 p^{13} T^{14} + 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 36 T + 172 T^{2} + 16172 T^{3} - 283596 T^{4} - 185756 p T^{5} + 24100 p^{3} T^{6} + 28116 p^{3} T^{7} - 387834 p^{4} T^{8} + 28116 p^{5} T^{9} + 24100 p^{7} T^{10} - 185756 p^{7} T^{11} - 283596 p^{8} T^{12} + 16172 p^{10} T^{13} + 172 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| good | 3 | \( 1 - 64 T^{2} + 724 p T^{4} - 51484 T^{6} + 105244 p^{2} T^{8} - 14290828 T^{10} + 182579036 T^{12} - 2013679448 T^{14} + 19373200858 T^{16} - 2013679448 p^{4} T^{18} + 182579036 p^{8} T^{20} - 14290828 p^{12} T^{22} + 105244 p^{18} T^{24} - 51484 p^{20} T^{26} + 724 p^{25} T^{28} - 64 p^{28} T^{30} + p^{32} T^{32} \) |
| 7 | \( 1 - 460 T^{2} + 105724 T^{4} - 16106672 T^{6} + 1823541484 T^{8} - 163253596456 T^{10} + 12009563868380 T^{12} - 743951010154108 T^{14} + 39356069818599514 T^{16} - 743951010154108 p^{4} T^{18} + 12009563868380 p^{8} T^{20} - 163253596456 p^{12} T^{22} + 1823541484 p^{16} T^{24} - 16106672 p^{20} T^{26} + 105724 p^{24} T^{28} - 460 p^{28} T^{30} + p^{32} T^{32} \) | |
| 11 | \( 1 - 20 T + 200 T^{2} - 3952 T^{3} + 73120 T^{4} - 63592 p T^{5} + 7175392 T^{6} - 119376360 T^{7} + 1026010444 T^{8} - 5669689312 T^{9} + 99960928192 T^{10} - 1252820848744 T^{11} + 472762529416 p T^{12} - 59210038600304 T^{13} + 1515474373774624 T^{14} - 12720998019560652 T^{15} + 76154444922718394 T^{16} - 12720998019560652 p^{2} T^{17} + 1515474373774624 p^{4} T^{18} - 59210038600304 p^{6} T^{19} + 472762529416 p^{9} T^{20} - 1252820848744 p^{10} T^{21} + 99960928192 p^{12} T^{22} - 5669689312 p^{14} T^{23} + 1026010444 p^{16} T^{24} - 119376360 p^{18} T^{25} + 7175392 p^{20} T^{26} - 63592 p^{23} T^{27} + 73120 p^{24} T^{28} - 3952 p^{26} T^{29} + 200 p^{28} T^{30} - 20 p^{30} T^{31} + p^{32} T^{32} \) | |
| 13 | \( 1 - 4 T + 8 T^{2} - 3104 T^{3} + 45192 T^{4} - 27192 T^{5} + 4564640 T^{6} - 196848012 T^{7} + 1440993908 T^{8} + 2826473516 T^{9} + 382266875472 T^{10} - 511636632888 p T^{11} + 5095399485464 T^{12} - 17077765820400 p T^{13} + 16937531414858920 T^{14} - 132400086161974620 T^{15} - 437803876354002330 T^{16} - 132400086161974620 p^{2} T^{17} + 16937531414858920 p^{4} T^{18} - 17077765820400 p^{7} T^{19} + 5095399485464 p^{8} T^{20} - 511636632888 p^{11} T^{21} + 382266875472 p^{12} T^{22} + 2826473516 p^{14} T^{23} + 1440993908 p^{16} T^{24} - 196848012 p^{18} T^{25} + 4564640 p^{20} T^{26} - 27192 p^{22} T^{27} + 45192 p^{24} T^{28} - 3104 p^{26} T^{29} + 8 p^{28} T^{30} - 4 p^{30} T^{31} + p^{32} T^{32} \) | |
| 19 | \( ( 1 + 8 T + 1452 T^{2} - 4248 T^{3} + 887188 T^{4} - 13183592 T^{5} + 342360068 T^{6} - 9587467880 T^{7} + 114896865926 T^{8} - 9587467880 p^{2} T^{9} + 342360068 p^{4} T^{10} - 13183592 p^{6} T^{11} + 887188 p^{8} T^{12} - 4248 p^{10} T^{13} + 1452 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 8 T + 3296 T^{2} - 21254 T^{3} + 5061938 T^{4} - 25817406 T^{5} + 4764786268 T^{6} - 19563297516 T^{7} + 3027263622904 T^{8} - 19563297516 p^{2} T^{9} + 4764786268 p^{4} T^{10} - 25817406 p^{6} T^{11} + 5061938 p^{8} T^{12} - 21254 p^{10} T^{13} + 3296 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 29 | \( 1 + 20 T + 200 T^{2} - 11276 T^{3} + 381272 T^{4} + 42983836 T^{5} + 846996408 T^{6} + 28221224252 T^{7} + 917025187356 T^{8} + 22445967260596 T^{9} + 548160847241640 T^{10} + 14114701559014420 T^{11} + 1277972793768068712 T^{12} + 1372669810226687948 p T^{13} + \)\(75\!\cdots\!00\)\( T^{14} + \)\(21\!\cdots\!04\)\( T^{15} + \)\(33\!\cdots\!74\)\( T^{16} + \)\(21\!\cdots\!04\)\( p^{2} T^{17} + \)\(75\!\cdots\!00\)\( p^{4} T^{18} + 1372669810226687948 p^{7} T^{19} + 1277972793768068712 p^{8} T^{20} + 14114701559014420 p^{10} T^{21} + 548160847241640 p^{12} T^{22} + 22445967260596 p^{14} T^{23} + 917025187356 p^{16} T^{24} + 28221224252 p^{18} T^{25} + 846996408 p^{20} T^{26} + 42983836 p^{22} T^{27} + 381272 p^{24} T^{28} - 11276 p^{26} T^{29} + 200 p^{28} T^{30} + 20 p^{30} T^{31} + p^{32} T^{32} \) | |
| 31 | \( 1 - 92 T + 4232 T^{2} - 123016 T^{3} + 3722360 T^{4} - 143310692 T^{5} + 4998024272 T^{6} - 109321116956 T^{7} + 3645924777340 T^{8} - 221310285054860 T^{9} + 10043770682441048 T^{10} - 289084626695696012 T^{11} + 8322735011005247744 T^{12} - \)\(30\!\cdots\!24\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} - \)\(22\!\cdots\!12\)\( T^{15} + \)\(50\!\cdots\!94\)\( T^{16} - \)\(22\!\cdots\!12\)\( p^{2} T^{17} + \)\(10\!\cdots\!20\)\( p^{4} T^{18} - \)\(30\!\cdots\!24\)\( p^{6} T^{19} + 8322735011005247744 p^{8} T^{20} - 289084626695696012 p^{10} T^{21} + 10043770682441048 p^{12} T^{22} - 221310285054860 p^{14} T^{23} + 3645924777340 p^{16} T^{24} - 109321116956 p^{18} T^{25} + 4998024272 p^{20} T^{26} - 143310692 p^{22} T^{27} + 3722360 p^{24} T^{28} - 123016 p^{26} T^{29} + 4232 p^{28} T^{30} - 92 p^{30} T^{31} + p^{32} T^{32} \) | |
| 37 | \( ( 1 - 16 T + 6268 T^{2} - 92168 T^{3} + 21317396 T^{4} - 274774248 T^{5} + 47901314228 T^{6} - 541622434784 T^{7} + 76961899778534 T^{8} - 541622434784 p^{2} T^{9} + 47901314228 p^{4} T^{10} - 274774248 p^{6} T^{11} + 21317396 p^{8} T^{12} - 92168 p^{10} T^{13} + 6268 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 + 60 T + 1800 T^{2} - 92412 T^{3} - 12360304 T^{4} - 562449084 T^{5} - 7228408968 T^{6} + 553208308028 T^{7} + 38894379624428 T^{8} + 928006774023012 T^{9} + 3376590721340040 T^{10} + 353810612510504604 T^{11} + 91986744581976593008 T^{12} + \)\(43\!\cdots\!16\)\( T^{13} + \)\(18\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!28\)\( T^{15} - \)\(67\!\cdots\!22\)\( T^{16} - \)\(10\!\cdots\!28\)\( p^{2} T^{17} + \)\(18\!\cdots\!60\)\( p^{4} T^{18} + \)\(43\!\cdots\!16\)\( p^{6} T^{19} + 91986744581976593008 p^{8} T^{20} + 353810612510504604 p^{10} T^{21} + 3376590721340040 p^{12} T^{22} + 928006774023012 p^{14} T^{23} + 38894379624428 p^{16} T^{24} + 553208308028 p^{18} T^{25} - 7228408968 p^{20} T^{26} - 562449084 p^{22} T^{27} - 12360304 p^{24} T^{28} - 92412 p^{26} T^{29} + 1800 p^{28} T^{30} + 60 p^{30} T^{31} + p^{32} T^{32} \) | |
| 43 | \( 1 - 52 T + 1352 T^{2} - 305288 T^{3} + 19561776 T^{4} - 386317168 T^{5} + 40241353056 T^{6} - 2952988759484 T^{7} + 52957870885284 T^{8} - 2938723220359172 T^{9} + 270154741154507024 T^{10} - 5030829863590288000 T^{11} + \)\(18\!\cdots\!24\)\( T^{12} - \)\(20\!\cdots\!04\)\( T^{13} + \)\(42\!\cdots\!56\)\( T^{14} - \)\(16\!\cdots\!56\)\( T^{15} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(16\!\cdots\!56\)\( p^{2} T^{17} + \)\(42\!\cdots\!56\)\( p^{4} T^{18} - \)\(20\!\cdots\!04\)\( p^{6} T^{19} + \)\(18\!\cdots\!24\)\( p^{8} T^{20} - 5030829863590288000 p^{10} T^{21} + 270154741154507024 p^{12} T^{22} - 2938723220359172 p^{14} T^{23} + 52957870885284 p^{16} T^{24} - 2952988759484 p^{18} T^{25} + 40241353056 p^{20} T^{26} - 386317168 p^{22} T^{27} + 19561776 p^{24} T^{28} - 305288 p^{26} T^{29} + 1352 p^{28} T^{30} - 52 p^{30} T^{31} + p^{32} T^{32} \) | |
| 47 | \( 1 - 112 T + 6272 T^{2} - 534532 T^{3} + 38144184 T^{4} - 1398250204 T^{5} + 60225930312 T^{6} - 2989504227096 T^{7} + 9951057112628 T^{8} + 2524029703390568 T^{9} - 66835834371151960 T^{10} + 12245621388480813756 T^{11} - \)\(32\!\cdots\!92\)\( T^{12} - \)\(20\!\cdots\!36\)\( T^{13} + \)\(61\!\cdots\!04\)\( T^{14} - \)\(65\!\cdots\!24\)\( T^{15} + \)\(63\!\cdots\!62\)\( T^{16} - \)\(65\!\cdots\!24\)\( p^{2} T^{17} + \)\(61\!\cdots\!04\)\( p^{4} T^{18} - \)\(20\!\cdots\!36\)\( p^{6} T^{19} - \)\(32\!\cdots\!92\)\( p^{8} T^{20} + 12245621388480813756 p^{10} T^{21} - 66835834371151960 p^{12} T^{22} + 2524029703390568 p^{14} T^{23} + 9951057112628 p^{16} T^{24} - 2989504227096 p^{18} T^{25} + 60225930312 p^{20} T^{26} - 1398250204 p^{22} T^{27} + 38144184 p^{24} T^{28} - 534532 p^{26} T^{29} + 6272 p^{28} T^{30} - 112 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 - 48 T + 1152 T^{2} + 79672 T^{3} - 4630664 T^{4} + 122607896 T^{5} + 2623159712 T^{6} + 428141841824 T^{7} - 93112612753476 T^{8} + 5706345118867200 T^{9} - 112864599666437344 T^{10} - 3033242998287223064 T^{11} + \)\(35\!\cdots\!48\)\( T^{12} - \)\(88\!\cdots\!44\)\( T^{13} + \)\(28\!\cdots\!92\)\( T^{14} - \)\(75\!\cdots\!76\)\( T^{15} + \)\(49\!\cdots\!58\)\( T^{16} - \)\(75\!\cdots\!76\)\( p^{2} T^{17} + \)\(28\!\cdots\!92\)\( p^{4} T^{18} - \)\(88\!\cdots\!44\)\( p^{6} T^{19} + \)\(35\!\cdots\!48\)\( p^{8} T^{20} - 3033242998287223064 p^{10} T^{21} - 112864599666437344 p^{12} T^{22} + 5706345118867200 p^{14} T^{23} - 93112612753476 p^{16} T^{24} + 428141841824 p^{18} T^{25} + 2623159712 p^{20} T^{26} + 122607896 p^{22} T^{27} - 4630664 p^{24} T^{28} + 79672 p^{26} T^{29} + 1152 p^{28} T^{30} - 48 p^{30} T^{31} + p^{32} T^{32} \) | |
| 59 | \( ( 1 + 12728 T^{2} - 68240 T^{3} + 92964036 T^{4} - 550102480 T^{5} + 483774146344 T^{6} - 2328307337984 T^{7} + 1922657466781670 T^{8} - 2328307337984 p^{2} T^{9} + 483774146344 p^{4} T^{10} - 550102480 p^{6} T^{11} + 92964036 p^{8} T^{12} - 68240 p^{10} T^{13} + 12728 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 61 | \( 1 - 76 T + 2888 T^{2} - 312228 T^{3} - 15268960 T^{4} + 2401837580 T^{5} - 89699737608 T^{6} + 9310828679524 T^{7} - 239682635216788 T^{8} - 17160830177028164 T^{9} + 604115301812885256 T^{10} - 57028761906956517676 T^{11} + \)\(27\!\cdots\!44\)\( T^{12} - \)\(18\!\cdots\!76\)\( T^{13} + \)\(11\!\cdots\!56\)\( T^{14} - \)\(18\!\cdots\!48\)\( T^{15} + \)\(24\!\cdots\!30\)\( T^{16} - \)\(18\!\cdots\!48\)\( p^{2} T^{17} + \)\(11\!\cdots\!56\)\( p^{4} T^{18} - \)\(18\!\cdots\!76\)\( p^{6} T^{19} + \)\(27\!\cdots\!44\)\( p^{8} T^{20} - 57028761906956517676 p^{10} T^{21} + 604115301812885256 p^{12} T^{22} - 17160830177028164 p^{14} T^{23} - 239682635216788 p^{16} T^{24} + 9310828679524 p^{18} T^{25} - 89699737608 p^{20} T^{26} + 2401837580 p^{22} T^{27} - 15268960 p^{24} T^{28} - 312228 p^{26} T^{29} + 2888 p^{28} T^{30} - 76 p^{30} T^{31} + p^{32} T^{32} \) | |
| 67 | \( 1 - 116 T + 6728 T^{2} + 402688 T^{3} - 94964104 T^{4} + 7339129384 T^{5} - 131341704160 T^{6} - 27374600847212 T^{7} + 2604819995212180 T^{8} - 79368028619094740 T^{9} - 4883298069776025968 T^{10} + \)\(45\!\cdots\!28\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} - \)\(22\!\cdots\!08\)\( T^{13} + \)\(10\!\cdots\!68\)\( T^{14} + \)\(74\!\cdots\!64\)\( T^{15} - \)\(92\!\cdots\!66\)\( T^{16} + \)\(74\!\cdots\!64\)\( p^{2} T^{17} + \)\(10\!\cdots\!68\)\( p^{4} T^{18} - \)\(22\!\cdots\!08\)\( p^{6} T^{19} + \)\(22\!\cdots\!88\)\( p^{8} T^{20} + \)\(45\!\cdots\!28\)\( p^{10} T^{21} - 4883298069776025968 p^{12} T^{22} - 79368028619094740 p^{14} T^{23} + 2604819995212180 p^{16} T^{24} - 27374600847212 p^{18} T^{25} - 131341704160 p^{20} T^{26} + 7339129384 p^{22} T^{27} - 94964104 p^{24} T^{28} + 402688 p^{26} T^{29} + 6728 p^{28} T^{30} - 116 p^{30} T^{31} + p^{32} T^{32} \) | |
| 71 | \( 1 + 268 T + 35912 T^{2} + 4319988 T^{3} + 479745256 T^{4} + 40103061348 T^{5} + 2850156967864 T^{6} + 190707939102184 T^{7} + 9137987357730124 T^{8} + 297235752303772976 T^{9} + 12130268605399933640 T^{10} + \)\(64\!\cdots\!52\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(24\!\cdots\!20\)\( T^{13} + \)\(25\!\cdots\!96\)\( T^{14} + \)\(23\!\cdots\!40\)\( T^{15} + \)\(19\!\cdots\!22\)\( T^{16} + \)\(23\!\cdots\!40\)\( p^{2} T^{17} + \)\(25\!\cdots\!96\)\( p^{4} T^{18} + \)\(24\!\cdots\!20\)\( p^{6} T^{19} + \)\(14\!\cdots\!20\)\( p^{8} T^{20} + \)\(64\!\cdots\!52\)\( p^{10} T^{21} + 12130268605399933640 p^{12} T^{22} + 297235752303772976 p^{14} T^{23} + 9137987357730124 p^{16} T^{24} + 190707939102184 p^{18} T^{25} + 2850156967864 p^{20} T^{26} + 40103061348 p^{22} T^{27} + 479745256 p^{24} T^{28} + 4319988 p^{26} T^{29} + 35912 p^{28} T^{30} + 268 p^{30} T^{31} + p^{32} T^{32} \) | |
| 73 | \( 1 - 54188 T^{2} + 1467335352 T^{4} - 26281780618564 T^{6} + 347985145456223324 T^{8} - \)\(36\!\cdots\!56\)\( T^{10} + \)\(30\!\cdots\!88\)\( T^{12} - \)\(21\!\cdots\!96\)\( T^{14} + \)\(12\!\cdots\!98\)\( T^{16} - \)\(21\!\cdots\!96\)\( p^{4} T^{18} + \)\(30\!\cdots\!88\)\( p^{8} T^{20} - \)\(36\!\cdots\!56\)\( p^{12} T^{22} + 347985145456223324 p^{16} T^{24} - 26281780618564 p^{20} T^{26} + 1467335352 p^{24} T^{28} - 54188 p^{28} T^{30} + p^{32} T^{32} \) | |
| 79 | \( 1 + 88 T + 3872 T^{2} + 1788340 T^{3} + 135244744 T^{4} - 3599400064 T^{5} + 758665123400 T^{6} + 55432482891976 T^{7} - 12605750004834148 T^{8} - 317828135324554920 T^{9} + 23845084579593540384 T^{10} - \)\(89\!\cdots\!08\)\( T^{11} - \)\(24\!\cdots\!40\)\( T^{12} + \)\(46\!\cdots\!48\)\( T^{13} - \)\(20\!\cdots\!44\)\( T^{14} - \)\(33\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!70\)\( T^{16} - \)\(33\!\cdots\!44\)\( p^{2} T^{17} - \)\(20\!\cdots\!44\)\( p^{4} T^{18} + \)\(46\!\cdots\!48\)\( p^{6} T^{19} - \)\(24\!\cdots\!40\)\( p^{8} T^{20} - \)\(89\!\cdots\!08\)\( p^{10} T^{21} + 23845084579593540384 p^{12} T^{22} - 317828135324554920 p^{14} T^{23} - 12605750004834148 p^{16} T^{24} + 55432482891976 p^{18} T^{25} + 758665123400 p^{20} T^{26} - 3599400064 p^{22} T^{27} + 135244744 p^{24} T^{28} + 1788340 p^{26} T^{29} + 3872 p^{28} T^{30} + 88 p^{30} T^{31} + p^{32} T^{32} \) | |
| 83 | \( 1 + 160 T + 12800 T^{2} + 966044 T^{3} + 91405312 T^{4} + 10797089204 T^{5} + 1024166784008 T^{6} + 99030253388360 T^{7} + 11013952786074244 T^{8} + 1019893995403131944 T^{9} + 82546308914334461992 T^{10} + \)\(75\!\cdots\!72\)\( T^{11} + \)\(77\!\cdots\!36\)\( T^{12} + \)\(66\!\cdots\!84\)\( T^{13} + \)\(50\!\cdots\!28\)\( T^{14} + \)\(42\!\cdots\!04\)\( T^{15} + \)\(34\!\cdots\!18\)\( T^{16} + \)\(42\!\cdots\!04\)\( p^{2} T^{17} + \)\(50\!\cdots\!28\)\( p^{4} T^{18} + \)\(66\!\cdots\!84\)\( p^{6} T^{19} + \)\(77\!\cdots\!36\)\( p^{8} T^{20} + \)\(75\!\cdots\!72\)\( p^{10} T^{21} + 82546308914334461992 p^{12} T^{22} + 1019893995403131944 p^{14} T^{23} + 11013952786074244 p^{16} T^{24} + 99030253388360 p^{18} T^{25} + 1024166784008 p^{20} T^{26} + 10797089204 p^{22} T^{27} + 91405312 p^{24} T^{28} + 966044 p^{26} T^{29} + 12800 p^{28} T^{30} + 160 p^{30} T^{31} + p^{32} T^{32} \) | |
| 89 | \( 1 - 17920 T^{2} + 205239416 T^{4} - 2354093939072 T^{6} + 21146411199253476 T^{8} - 1992141789238244864 p T^{10} + \)\(18\!\cdots\!60\)\( T^{12} - \)\(17\!\cdots\!12\)\( T^{14} + \)\(14\!\cdots\!74\)\( T^{16} - \)\(17\!\cdots\!12\)\( p^{4} T^{18} + \)\(18\!\cdots\!60\)\( p^{8} T^{20} - 1992141789238244864 p^{13} T^{22} + 21146411199253476 p^{16} T^{24} - 2354093939072 p^{20} T^{26} + 205239416 p^{24} T^{28} - 17920 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 + 128 T + 28208 T^{2} + 3014056 T^{3} + 335990484 T^{4} + 40045013704 T^{5} + 4295669035216 T^{6} + 528615460779952 T^{7} + 52517321897130022 T^{8} + 528615460779952 p^{2} T^{9} + 4295669035216 p^{4} T^{10} + 40045013704 p^{6} T^{11} + 335990484 p^{8} T^{12} + 3014056 p^{10} T^{13} + 28208 p^{12} T^{14} + 128 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−3.50439488693495904403914909130, −3.44825090300576335233463314274, −3.42337865064008222240146419154, −3.40913800704142848666165017374, −3.38581524152712890294043842049, −3.16110002782571534923857027539, −2.94880086866907613925519104755, −2.88892454026098854118206131433, −2.79258490529831477453000616945, −2.71360592477924927678841806385, −2.49780135710891894653454747753, −2.43009352634332355120174610333, −2.38552461877825514548535001408, −2.38057882670026160489589641077, −2.02977233374994006764972391864, −1.98910585855986735012097194192, −1.90413839060300939082430140976, −1.56318804307880192987446912566, −1.30909432313704401840509901406, −1.30363403160145588884809582460, −1.27157153168207066276521703287, −1.20278577270036213786613897128, −0.800028732126912042557773046193, −0.75156254991360563761084919174, −0.71453145452543205256582821012, 0.71453145452543205256582821012, 0.75156254991360563761084919174, 0.800028732126912042557773046193, 1.20278577270036213786613897128, 1.27157153168207066276521703287, 1.30363403160145588884809582460, 1.30909432313704401840509901406, 1.56318804307880192987446912566, 1.90413839060300939082430140976, 1.98910585855986735012097194192, 2.02977233374994006764972391864, 2.38057882670026160489589641077, 2.38552461877825514548535001408, 2.43009352634332355120174610333, 2.49780135710891894653454747753, 2.71360592477924927678841806385, 2.79258490529831477453000616945, 2.88892454026098854118206131433, 2.94880086866907613925519104755, 3.16110002782571534923857027539, 3.38581524152712890294043842049, 3.40913800704142848666165017374, 3.42337865064008222240146419154, 3.44825090300576335233463314274, 3.50439488693495904403914909130
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.