Dirichlet series
| L(s) = 1 | − 6·2-s − 12·3-s + 15·4-s − 3·5-s + 72·6-s + 2·7-s − 15·8-s + 64·9-s + 18·10-s − 7·11-s − 180·12-s − 7·13-s − 12·14-s + 36·15-s − 17·16-s − 6·17-s − 384·18-s + 16·19-s − 45·20-s − 24·21-s + 42·22-s + 23-s + 180·24-s + 18·25-s + 42·26-s − 183·27-s + 30·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 6.92·3-s + 15/2·4-s − 1.34·5-s + 29.3·6-s + 0.755·7-s − 5.30·8-s + 64/3·9-s + 5.69·10-s − 2.11·11-s − 51.9·12-s − 1.94·13-s − 3.20·14-s + 9.29·15-s − 4.25·16-s − 1.45·17-s − 90.5·18-s + 3.67·19-s − 10.0·20-s − 5.23·21-s + 8.95·22-s + 0.208·23-s + 36.7·24-s + 18/5·25-s + 8.23·26-s − 35.2·27-s + 5.66·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.98710\times 10^{-10}\) |
| Root analytic conductor: | \(0.497530\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 31^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.129711493\times10^{-5}\) |
| \(L(\frac12)\) | \(\approx\) | \(9.129711493\times10^{-5}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 - 15 T + 158 T^{2} - 1635 T^{3} + 13788 T^{4} - 99390 T^{5} + 688351 T^{6} - 4312200 T^{7} + 24371915 T^{8} - 4312200 p T^{9} + 688351 p^{2} T^{10} - 99390 p^{3} T^{11} + 13788 p^{4} T^{12} - 1635 p^{5} T^{13} + 158 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) |
| good | 2 | \( 1 + 3 p T + 21 T^{2} + 51 T^{3} + 49 p T^{4} + 159 T^{5} + 231 T^{6} + 153 p T^{7} + 369 T^{8} + 399 T^{9} + 189 p T^{10} + 9 p^{5} T^{11} + 35 p T^{12} - 51 p^{3} T^{13} - 1395 T^{14} - 1479 p T^{15} - 4795 T^{16} - 1479 p^{2} T^{17} - 1395 p^{2} T^{18} - 51 p^{6} T^{19} + 35 p^{5} T^{20} + 9 p^{10} T^{21} + 189 p^{7} T^{22} + 399 p^{7} T^{23} + 369 p^{8} T^{24} + 153 p^{10} T^{25} + 231 p^{10} T^{26} + 159 p^{11} T^{27} + 49 p^{13} T^{28} + 51 p^{13} T^{29} + 21 p^{14} T^{30} + 3 p^{16} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + 4 p T + 80 T^{2} + 125 p T^{3} + 454 p T^{4} + 1346 p T^{5} + 10105 T^{6} + 7312 p T^{7} + 42530 T^{8} + 25505 p T^{9} + 134002 T^{10} + 79420 p T^{11} + 48404 p^{2} T^{12} + 268223 p T^{13} + 1467485 T^{14} + 872407 p T^{15} + 4573453 T^{16} + 872407 p^{2} T^{17} + 1467485 p^{2} T^{18} + 268223 p^{4} T^{19} + 48404 p^{6} T^{20} + 79420 p^{6} T^{21} + 134002 p^{6} T^{22} + 25505 p^{8} T^{23} + 42530 p^{8} T^{24} + 7312 p^{10} T^{25} + 10105 p^{10} T^{26} + 1346 p^{12} T^{27} + 454 p^{13} T^{28} + 125 p^{14} T^{29} + 80 p^{14} T^{30} + 4 p^{16} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 + 3 T - 9 T^{2} - 18 T^{3} + 61 T^{4} - 21 T^{5} - 291 T^{6} + 591 T^{7} + 172 T^{8} - 4164 T^{9} + 1989 T^{10} + 14124 T^{11} - 21959 T^{12} - 41793 T^{13} + 231951 T^{14} + 136788 T^{15} - 1318469 T^{16} + 136788 p T^{17} + 231951 p^{2} T^{18} - 41793 p^{3} T^{19} - 21959 p^{4} T^{20} + 14124 p^{5} T^{21} + 1989 p^{6} T^{22} - 4164 p^{7} T^{23} + 172 p^{8} T^{24} + 591 p^{9} T^{25} - 291 p^{10} T^{26} - 21 p^{11} T^{27} + 61 p^{12} T^{28} - 18 p^{13} T^{29} - 9 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 - 2 T + 8 T^{2} - 57 T^{3} + 180 T^{4} - 447 T^{5} + 291 p T^{6} - 5539 T^{7} + 16183 T^{8} - 49082 T^{9} + 136106 T^{10} - 412273 T^{11} + 1249403 T^{12} - 421202 p T^{13} + 10225949 T^{14} - 27209176 T^{15} + 63276497 T^{16} - 27209176 p T^{17} + 10225949 p^{2} T^{18} - 421202 p^{4} T^{19} + 1249403 p^{4} T^{20} - 412273 p^{5} T^{21} + 136106 p^{6} T^{22} - 49082 p^{7} T^{23} + 16183 p^{8} T^{24} - 5539 p^{9} T^{25} + 291 p^{11} T^{26} - 447 p^{11} T^{27} + 180 p^{12} T^{28} - 57 p^{13} T^{29} + 8 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 7 T + 35 T^{2} + 158 T^{3} + 743 T^{4} + 2658 T^{5} + 9655 T^{6} + 30605 T^{7} + 105459 T^{8} + 298602 T^{9} + 903884 T^{10} + 2105465 T^{11} + 6983649 T^{12} + 12627823 T^{13} + 32620956 T^{14} + 51556382 T^{15} + 388064517 T^{16} + 51556382 p T^{17} + 32620956 p^{2} T^{18} + 12627823 p^{3} T^{19} + 6983649 p^{4} T^{20} + 2105465 p^{5} T^{21} + 903884 p^{6} T^{22} + 298602 p^{7} T^{23} + 105459 p^{8} T^{24} + 30605 p^{9} T^{25} + 9655 p^{10} T^{26} + 2658 p^{11} T^{27} + 743 p^{12} T^{28} + 158 p^{13} T^{29} + 35 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 7 T + 44 T^{2} + 249 T^{3} + 75 p T^{4} + 3375 T^{5} + 9585 T^{6} + 9146 T^{7} - 2963 p T^{8} - 357284 T^{9} - 1668070 T^{10} - 3915103 T^{11} - 4062397 T^{12} + 34368784 T^{13} + 322154357 T^{14} + 123798758 p T^{15} + 6221461841 T^{16} + 123798758 p^{2} T^{17} + 322154357 p^{2} T^{18} + 34368784 p^{3} T^{19} - 4062397 p^{4} T^{20} - 3915103 p^{5} T^{21} - 1668070 p^{6} T^{22} - 357284 p^{7} T^{23} - 2963 p^{9} T^{24} + 9146 p^{9} T^{25} + 9585 p^{10} T^{26} + 3375 p^{11} T^{27} + 75 p^{13} T^{28} + 249 p^{13} T^{29} + 44 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 6 T + 75 T^{2} + 495 T^{3} + 3677 T^{4} + 23094 T^{5} + 144360 T^{6} + 817152 T^{7} + 4567650 T^{8} + 23850960 T^{9} + 122226918 T^{10} + 594432255 T^{11} + 2843619706 T^{12} + 12936362562 T^{13} + 57908087850 T^{14} + 248770132872 T^{15} + 1043483763143 T^{16} + 248770132872 p T^{17} + 57908087850 p^{2} T^{18} + 12936362562 p^{3} T^{19} + 2843619706 p^{4} T^{20} + 594432255 p^{5} T^{21} + 122226918 p^{6} T^{22} + 23850960 p^{7} T^{23} + 4567650 p^{8} T^{24} + 817152 p^{9} T^{25} + 144360 p^{10} T^{26} + 23094 p^{11} T^{27} + 3677 p^{12} T^{28} + 495 p^{13} T^{29} + 75 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 16 T + 195 T^{2} - 1746 T^{3} + 13353 T^{4} - 89969 T^{5} + 29375 p T^{6} - 3294825 T^{7} + 18744924 T^{8} - 103602551 T^{9} + 554225766 T^{10} - 2858170600 T^{11} + 14222538534 T^{12} - 68363619099 T^{13} + 318565822489 T^{14} - 1445186621304 T^{15} + 6375621526007 T^{16} - 1445186621304 p T^{17} + 318565822489 p^{2} T^{18} - 68363619099 p^{3} T^{19} + 14222538534 p^{4} T^{20} - 2858170600 p^{5} T^{21} + 554225766 p^{6} T^{22} - 103602551 p^{7} T^{23} + 18744924 p^{8} T^{24} - 3294825 p^{9} T^{25} + 29375 p^{11} T^{26} - 89969 p^{11} T^{27} + 13353 p^{12} T^{28} - 1746 p^{13} T^{29} + 195 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - T - 32 T^{2} + 265 T^{3} + 1220 T^{4} - 3468 T^{5} - 10927 T^{6} + 173296 T^{7} + 1397175 T^{8} - 394035 T^{9} - 17319068 T^{10} + 116114803 T^{11} + 1091799191 T^{12} + 1263133445 T^{13} - 14131047870 T^{14} + 39184802221 T^{15} + 769047681799 T^{16} + 39184802221 p T^{17} - 14131047870 p^{2} T^{18} + 1263133445 p^{3} T^{19} + 1091799191 p^{4} T^{20} + 116114803 p^{5} T^{21} - 17319068 p^{6} T^{22} - 394035 p^{7} T^{23} + 1397175 p^{8} T^{24} + 173296 p^{9} T^{25} - 10927 p^{10} T^{26} - 3468 p^{11} T^{27} + 1220 p^{12} T^{28} + 265 p^{13} T^{29} - 32 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 14 T + 65 T^{2} + 271 T^{3} + 4124 T^{4} + 28761 T^{5} + 57004 T^{6} + 203557 T^{7} + 4414002 T^{8} + 20510481 T^{9} - 1562410 T^{10} + 162325096 T^{11} + 3747178047 T^{12} + 10722269693 T^{13} - 22040075979 T^{14} + 162043565485 T^{15} + 2719402913313 T^{16} + 162043565485 p T^{17} - 22040075979 p^{2} T^{18} + 10722269693 p^{3} T^{19} + 3747178047 p^{4} T^{20} + 162325096 p^{5} T^{21} - 1562410 p^{6} T^{22} + 20510481 p^{7} T^{23} + 4414002 p^{8} T^{24} + 203557 p^{9} T^{25} + 57004 p^{10} T^{26} + 28761 p^{11} T^{27} + 4124 p^{12} T^{28} + 271 p^{13} T^{29} + 65 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 8 T - 120 T^{2} - 534 T^{3} + 11739 T^{4} + 10042 T^{5} - 710515 T^{6} + 2077152 T^{7} + 30531852 T^{8} - 207297908 T^{9} - 600747795 T^{10} + 11697189185 T^{11} - 16371022929 T^{12} - 395056153140 T^{13} + 2176783134259 T^{14} + 6190914463491 T^{15} - 102798721686295 T^{16} + 6190914463491 p T^{17} + 2176783134259 p^{2} T^{18} - 395056153140 p^{3} T^{19} - 16371022929 p^{4} T^{20} + 11697189185 p^{5} T^{21} - 600747795 p^{6} T^{22} - 207297908 p^{7} T^{23} + 30531852 p^{8} T^{24} + 2077152 p^{9} T^{25} - 710515 p^{10} T^{26} + 10042 p^{11} T^{27} + 11739 p^{12} T^{28} - 534 p^{13} T^{29} - 120 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 8 T + 106 T^{2} + 976 T^{3} + 7985 T^{4} + 55050 T^{5} + 396119 T^{6} + 1966432 T^{7} + 11020956 T^{8} + 52272537 T^{9} + 2453888 p T^{10} + 252719935 T^{11} - 1900853239 T^{12} - 50449760350 T^{13} - 359337416133 T^{14} - 1943490188168 T^{15} - 20983245640805 T^{16} - 1943490188168 p T^{17} - 359337416133 p^{2} T^{18} - 50449760350 p^{3} T^{19} - 1900853239 p^{4} T^{20} + 252719935 p^{5} T^{21} + 2453888 p^{7} T^{22} + 52272537 p^{7} T^{23} + 11020956 p^{8} T^{24} + 1966432 p^{9} T^{25} + 396119 p^{10} T^{26} + 55050 p^{11} T^{27} + 7985 p^{12} T^{28} + 976 p^{13} T^{29} + 106 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 23 T + 289 T^{2} - 72 p T^{3} + 26895 T^{4} - 183450 T^{5} + 1103565 T^{6} - 5143929 T^{7} + 16278591 T^{8} - 38987724 T^{9} - 224772150 T^{10} + 3232536297 T^{11} - 26862162747 T^{12} + 175142751 p^{2} T^{13} - 2730755315628 T^{14} + 21633024576604 T^{15} - 167060950817939 T^{16} + 21633024576604 p T^{17} - 2730755315628 p^{2} T^{18} + 175142751 p^{5} T^{19} - 26862162747 p^{4} T^{20} + 3232536297 p^{5} T^{21} - 224772150 p^{6} T^{22} - 38987724 p^{7} T^{23} + 16278591 p^{8} T^{24} - 5143929 p^{9} T^{25} + 1103565 p^{10} T^{26} - 183450 p^{11} T^{27} + 26895 p^{12} T^{28} - 72 p^{14} T^{29} + 289 p^{14} T^{30} - 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 14 T - 64 T^{2} + 2051 T^{3} - 6367 T^{4} - 96576 T^{5} + 874231 T^{6} - 748939 T^{7} - 39432966 T^{8} + 323481909 T^{9} + 441998288 T^{10} - 20651272567 T^{11} + 54844789185 T^{12} + 746473355167 T^{13} - 4553042500275 T^{14} - 12229830802363 T^{15} + 232672311744285 T^{16} - 12229830802363 p T^{17} - 4553042500275 p^{2} T^{18} + 746473355167 p^{3} T^{19} + 54844789185 p^{4} T^{20} - 20651272567 p^{5} T^{21} + 441998288 p^{6} T^{22} + 323481909 p^{7} T^{23} - 39432966 p^{8} T^{24} - 748939 p^{9} T^{25} + 874231 p^{10} T^{26} - 96576 p^{11} T^{27} - 6367 p^{12} T^{28} + 2051 p^{13} T^{29} - 64 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 6 T + 37 T^{2} + 126 T^{3} + 1690 T^{4} - 1806 T^{5} - 120187 T^{6} + 3508437 T^{7} - 15460642 T^{8} + 56872014 T^{9} + 164905174 T^{10} + 7036349469 T^{11} - 9006554212 T^{12} - 185117474262 T^{13} + 4685233247216 T^{14} - 8469368678412 T^{15} + 48444353804527 T^{16} - 8469368678412 p T^{17} + 4685233247216 p^{2} T^{18} - 185117474262 p^{3} T^{19} - 9006554212 p^{4} T^{20} + 7036349469 p^{5} T^{21} + 164905174 p^{6} T^{22} + 56872014 p^{7} T^{23} - 15460642 p^{8} T^{24} + 3508437 p^{9} T^{25} - 120187 p^{10} T^{26} - 1806 p^{11} T^{27} + 1690 p^{12} T^{28} + 126 p^{13} T^{29} + 37 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 4 T + 157 T^{2} - 458 T^{3} + 14849 T^{4} - 42150 T^{5} + 924686 T^{6} - 2509544 T^{7} + 45834753 T^{8} - 195555639 T^{9} + 1870878718 T^{10} - 18828370418 T^{11} + 109287552140 T^{12} - 1909212618757 T^{13} + 8459795171586 T^{14} - 146067983115188 T^{15} + 587919268827301 T^{16} - 146067983115188 p T^{17} + 8459795171586 p^{2} T^{18} - 1909212618757 p^{3} T^{19} + 109287552140 p^{4} T^{20} - 18828370418 p^{5} T^{21} + 1870878718 p^{6} T^{22} - 195555639 p^{7} T^{23} + 45834753 p^{8} T^{24} - 2509544 p^{9} T^{25} + 924686 p^{10} T^{26} - 42150 p^{11} T^{27} + 14849 p^{12} T^{28} - 458 p^{13} T^{29} + 157 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 30 T + 776 T^{2} + 13665 T^{3} + 208005 T^{4} + 2593710 T^{5} + 28390399 T^{6} + 268109040 T^{7} + 2240276459 T^{8} + 268109040 p T^{9} + 28390399 p^{2} T^{10} + 2593710 p^{3} T^{11} + 208005 p^{4} T^{12} + 13665 p^{5} T^{13} + 776 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 13 T - 258 T^{2} + 4011 T^{3} + 35028 T^{4} - 644333 T^{5} - 51427 p T^{6} + 71925183 T^{7} + 280896507 T^{8} - 6225465362 T^{9} - 19715222826 T^{10} + 429031194656 T^{11} + 1171728043770 T^{12} - 21978520425855 T^{13} - 61484782388300 T^{14} + 555554083027185 T^{15} + 3540421883673161 T^{16} + 555554083027185 p T^{17} - 61484782388300 p^{2} T^{18} - 21978520425855 p^{3} T^{19} + 1171728043770 p^{4} T^{20} + 429031194656 p^{5} T^{21} - 19715222826 p^{6} T^{22} - 6225465362 p^{7} T^{23} + 280896507 p^{8} T^{24} + 71925183 p^{9} T^{25} - 51427 p^{11} T^{26} - 644333 p^{11} T^{27} + 35028 p^{12} T^{28} + 4011 p^{13} T^{29} - 258 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 14 T + 196 T^{2} + 2695 T^{3} + 17090 T^{4} + 118017 T^{5} + 577928 T^{6} - 4343303 T^{7} - 27945240 T^{8} - 279502470 T^{9} - 2826368429 T^{10} + 131657464 T^{11} - 99138557149 T^{12} - 2207242010815 T^{13} - 23038847142600 T^{14} - 294198893013362 T^{15} - 2969589635047583 T^{16} - 294198893013362 p T^{17} - 23038847142600 p^{2} T^{18} - 2207242010815 p^{3} T^{19} - 99138557149 p^{4} T^{20} + 131657464 p^{5} T^{21} - 2826368429 p^{6} T^{22} - 279502470 p^{7} T^{23} - 27945240 p^{8} T^{24} - 4343303 p^{9} T^{25} + 577928 p^{10} T^{26} + 118017 p^{11} T^{27} + 17090 p^{12} T^{28} + 2695 p^{13} T^{29} + 196 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 2 T + 14 T^{2} + 1041 T^{3} - 15525 T^{4} + 39720 T^{5} + 424590 T^{6} - 12956116 T^{7} + 131189071 T^{8} - 324722351 T^{9} - 2585580235 T^{10} + 84857314523 T^{11} - 864267778837 T^{12} + 4587359474806 T^{13} + 4622090434727 T^{14} - 447090776699989 T^{15} + 5618971555814261 T^{16} - 447090776699989 p T^{17} + 4622090434727 p^{2} T^{18} + 4587359474806 p^{3} T^{19} - 864267778837 p^{4} T^{20} + 84857314523 p^{5} T^{21} - 2585580235 p^{6} T^{22} - 324722351 p^{7} T^{23} + 131189071 p^{8} T^{24} - 12956116 p^{9} T^{25} + 424590 p^{10} T^{26} + 39720 p^{11} T^{27} - 15525 p^{12} T^{28} + 1041 p^{13} T^{29} + 14 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 18 T + 226 T^{2} - 2498 T^{3} + 40839 T^{4} - 469468 T^{5} + 5768280 T^{6} - 65435894 T^{7} + 780288208 T^{8} - 7561516414 T^{9} + 85206384568 T^{10} - 852745741894 T^{11} + 8569871201298 T^{12} - 77514808977636 T^{13} + 796329583287446 T^{14} - 7002575994516930 T^{15} + 64612388642450549 T^{16} - 7002575994516930 p T^{17} + 796329583287446 p^{2} T^{18} - 77514808977636 p^{3} T^{19} + 8569871201298 p^{4} T^{20} - 852745741894 p^{5} T^{21} + 85206384568 p^{6} T^{22} - 7561516414 p^{7} T^{23} + 780288208 p^{8} T^{24} - 65435894 p^{9} T^{25} + 5768280 p^{10} T^{26} - 469468 p^{11} T^{27} + 40839 p^{12} T^{28} - 2498 p^{13} T^{29} + 226 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 16 T + 185 T^{2} + 3773 T^{3} + 53729 T^{4} + 585909 T^{5} + 6686260 T^{6} + 69214526 T^{7} + 675636522 T^{8} + 5645389119 T^{9} + 32133713345 T^{10} + 172336220420 T^{11} + 356263146291 T^{12} - 19484713577420 T^{13} - 322899642624249 T^{14} - 3433895006273167 T^{15} - 33440240762794785 T^{16} - 3433895006273167 p T^{17} - 322899642624249 p^{2} T^{18} - 19484713577420 p^{3} T^{19} + 356263146291 p^{4} T^{20} + 172336220420 p^{5} T^{21} + 32133713345 p^{6} T^{22} + 5645389119 p^{7} T^{23} + 675636522 p^{8} T^{24} + 69214526 p^{9} T^{25} + 6686260 p^{10} T^{26} + 585909 p^{11} T^{27} + 53729 p^{12} T^{28} + 3773 p^{13} T^{29} + 185 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - T - 247 T^{2} + 1489 T^{3} + 31955 T^{4} - 480540 T^{5} - 2146982 T^{6} + 82322077 T^{7} - 115178013 T^{8} - 9374121609 T^{9} + 52814885666 T^{10} + 761747156077 T^{11} - 8384072473029 T^{12} - 44513145443047 T^{13} + 944971959403383 T^{14} + 1347434269510390 T^{15} - 88986302178103107 T^{16} + 1347434269510390 p T^{17} + 944971959403383 p^{2} T^{18} - 44513145443047 p^{3} T^{19} - 8384072473029 p^{4} T^{20} + 761747156077 p^{5} T^{21} + 52814885666 p^{6} T^{22} - 9374121609 p^{7} T^{23} - 115178013 p^{8} T^{24} + 82322077 p^{9} T^{25} - 2146982 p^{10} T^{26} - 480540 p^{11} T^{27} + 31955 p^{12} T^{28} + 1489 p^{13} T^{29} - 247 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 3 T - 107 T^{2} - 512 T^{3} + 11949 T^{4} + 162824 T^{5} - 68553 T^{6} - 17532563 T^{7} - 79037522 T^{8} + 475597250 T^{9} + 13552416748 T^{10} + 101483946140 T^{11} - 861406877856 T^{12} - 8094085378701 T^{13} - 80335748639044 T^{14} + 503130707568075 T^{15} + 13011656869749317 T^{16} + 503130707568075 p T^{17} - 80335748639044 p^{2} T^{18} - 8094085378701 p^{3} T^{19} - 861406877856 p^{4} T^{20} + 101483946140 p^{5} T^{21} + 13552416748 p^{6} T^{22} + 475597250 p^{7} T^{23} - 79037522 p^{8} T^{24} - 17532563 p^{9} T^{25} - 68553 p^{10} T^{26} + 162824 p^{11} T^{27} + 11949 p^{12} T^{28} - 512 p^{13} T^{29} - 107 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−6.08672357786122193649285099056, −5.98121444488268926644646859397, −5.82483504900735932203209099317, −5.56940224862133110592340098817, −5.36466432647098909702482918333, −5.32364987605313280391396869349, −5.24863161269803209441645806917, −5.22379061394187740150474457877, −5.22044426614243068574469174912, −5.05556253852467870297968779161, −4.87636712298002155565865665085, −4.87106169569145169939996010039, −4.82977166522661922656485005360, −4.47635727147183126633039920895, −4.39803154546424988575029186627, −4.34779856238566896989447204291, −4.02894472949931429174626488087, −4.01676555322226381565934810542, −3.53382860905873253360409113508, −3.14476156855467306061134251707, −3.10508788532872281543473427084, −2.76877362851943597799652067756, −2.63129835239756389634579536525, −2.40987483438552121348599809955, −1.25116026881693198379894299837, 1.25116026881693198379894299837, 2.40987483438552121348599809955, 2.63129835239756389634579536525, 2.76877362851943597799652067756, 3.10508788532872281543473427084, 3.14476156855467306061134251707, 3.53382860905873253360409113508, 4.01676555322226381565934810542, 4.02894472949931429174626488087, 4.34779856238566896989447204291, 4.39803154546424988575029186627, 4.47635727147183126633039920895, 4.82977166522661922656485005360, 4.87106169569145169939996010039, 4.87636712298002155565865665085, 5.05556253852467870297968779161, 5.22044426614243068574469174912, 5.22379061394187740150474457877, 5.24863161269803209441645806917, 5.32364987605313280391396869349, 5.36466432647098909702482918333, 5.56940224862133110592340098817, 5.82483504900735932203209099317, 5.98121444488268926644646859397, 6.08672357786122193649285099056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.