Dirichlet series
| L(s) = 1 | − 6·2-s − 9·3-s + 15·4-s + 6·5-s + 54·6-s + 11·7-s − 15·8-s + 49·9-s − 36·10-s − 14·11-s − 135·12-s + 13-s − 66·14-s − 54·15-s − 17·16-s + 3·17-s − 294·18-s + 13·19-s + 90·20-s − 99·21-s + 84·22-s − 23-s + 135·24-s − 9·25-s − 6·26-s − 207·27-s + 165·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 5.19·3-s + 15/2·4-s + 2.68·5-s + 22.0·6-s + 4.15·7-s − 5.30·8-s + 49/3·9-s − 11.3·10-s − 4.22·11-s − 38.9·12-s + 0.277·13-s − 17.6·14-s − 13.9·15-s − 4.25·16-s + 0.727·17-s − 69.2·18-s + 2.98·19-s + 20.1·20-s − 21.6·21-s + 17.9·22-s − 0.208·23-s + 27.5·24-s − 9/5·25-s − 1.17·26-s − 39.8·27-s + 31.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(31^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44546\times 10^{14}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 31^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2304071050\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2304071050\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| 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 | \( 1 + p^{2} T + 32 T^{2} + 2 p^{3} T^{3} + 13 p T^{4} + 16 p T^{5} + 343 T^{6} + 124 p^{2} T^{7} + 1883 T^{8} + 80 p^{3} T^{9} + 4252 T^{10} + 4175 p T^{11} + 8378 p T^{12} + 11231 p T^{13} + 49319 T^{14} + 39265 p T^{15} + 249187 T^{16} + 39265 p^{2} T^{17} + 49319 p^{2} T^{18} + 11231 p^{4} T^{19} + 8378 p^{5} T^{20} + 4175 p^{6} T^{21} + 4252 p^{6} T^{22} + 80 p^{10} T^{23} + 1883 p^{8} T^{24} + 124 p^{11} T^{25} + 343 p^{10} T^{26} + 16 p^{12} T^{27} + 13 p^{13} T^{28} + 2 p^{16} T^{29} + 32 p^{14} T^{30} + p^{17} T^{31} + p^{16} T^{32} \) | |
| 5 | \( ( 1 - 3 T + 18 T^{2} - 36 T^{3} + 31 p T^{4} - 261 T^{5} + 993 T^{6} - 1428 T^{7} + 5151 T^{8} - 1428 p T^{9} + 993 p^{2} T^{10} - 261 p^{3} T^{11} + 31 p^{5} T^{12} - 36 p^{5} T^{13} + 18 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - 11 T + 26 T^{2} + 114 T^{3} - 537 T^{4} - 24 T^{5} + 393 p T^{6} - 928 p T^{7} + 14554 T^{8} + 52936 T^{9} - 318562 T^{10} + 44822 T^{11} + 1295360 T^{12} - 425606 p T^{13} + 1626305 p T^{14} + 11006753 T^{15} - 152106685 T^{16} + 11006753 p T^{17} + 1626305 p^{3} T^{18} - 425606 p^{4} T^{19} + 1295360 p^{4} T^{20} + 44822 p^{5} T^{21} - 318562 p^{6} T^{22} + 52936 p^{7} T^{23} + 14554 p^{8} T^{24} - 928 p^{10} T^{25} + 393 p^{11} T^{26} - 24 p^{11} T^{27} - 537 p^{12} T^{28} + 114 p^{13} T^{29} + 26 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 14 T + 62 T^{2} - 74 T^{3} - 1753 T^{4} - 6678 T^{5} - 5744 T^{6} + 475 p^{2} T^{7} + 35550 p T^{8} + 1301937 T^{9} + 890594 T^{10} - 13041239 T^{11} - 6511992 p T^{12} - 200236159 T^{13} - 118286517 T^{14} + 2147418214 T^{15} + 11388348009 T^{16} + 2147418214 p T^{17} - 118286517 p^{2} T^{18} - 200236159 p^{3} T^{19} - 6511992 p^{5} T^{20} - 13041239 p^{5} T^{21} + 890594 p^{6} T^{22} + 1301937 p^{7} T^{23} + 35550 p^{9} T^{24} + 475 p^{11} T^{25} - 5744 p^{10} T^{26} - 6678 p^{11} T^{27} - 1753 p^{12} T^{28} - 74 p^{13} T^{29} + 62 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - T - 16 T^{2} - 3 p T^{3} + 420 T^{4} - 252 T^{5} - 3129 T^{6} - 8588 T^{7} + 85303 T^{8} - 84673 T^{9} - 504652 T^{10} - 941345 T^{11} + 14420915 T^{12} - 20300131 T^{13} - 101800924 T^{14} - 65606699 T^{15} + 2882308673 T^{16} - 65606699 p T^{17} - 101800924 p^{2} T^{18} - 20300131 p^{3} T^{19} + 14420915 p^{4} T^{20} - 941345 p^{5} T^{21} - 504652 p^{6} T^{22} - 84673 p^{7} T^{23} + 85303 p^{8} T^{24} - 8588 p^{9} T^{25} - 3129 p^{10} T^{26} - 252 p^{11} T^{27} + 420 p^{12} T^{28} - 3 p^{14} T^{29} - 16 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 3 T - 27 T^{2} - 27 T^{3} + 569 T^{4} + 3894 T^{5} - 3228 T^{6} - 103758 T^{7} - 246447 T^{8} + 1152930 T^{9} + 7711143 T^{10} + 10883595 T^{11} - 96849386 T^{12} - 497768766 T^{13} - 239070624 T^{14} + 3806364165 T^{15} + 27543128747 T^{16} + 3806364165 p T^{17} - 239070624 p^{2} T^{18} - 497768766 p^{3} T^{19} - 96849386 p^{4} T^{20} + 10883595 p^{5} T^{21} + 7711143 p^{6} T^{22} + 1152930 p^{7} T^{23} - 246447 p^{8} T^{24} - 103758 p^{9} T^{25} - 3228 p^{10} T^{26} + 3894 p^{11} T^{27} + 569 p^{12} T^{28} - 27 p^{13} T^{29} - 27 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 13 T + 33 T^{2} + 312 T^{3} - 1908 T^{4} + 2731 T^{5} - 13411 T^{6} + 73020 T^{7} + 593325 T^{8} - 4640339 T^{9} + 6777681 T^{10} - 5319433 T^{11} + 105337353 T^{12} + 458849178 T^{13} - 5759091908 T^{14} + 6305034888 T^{15} + 53695883519 T^{16} + 6305034888 p T^{17} - 5759091908 p^{2} T^{18} + 458849178 p^{3} T^{19} + 105337353 p^{4} T^{20} - 5319433 p^{5} T^{21} + 6777681 p^{6} T^{22} - 4640339 p^{7} T^{23} + 593325 p^{8} T^{24} + 73020 p^{9} T^{25} - 13411 p^{10} T^{26} + 2731 p^{11} T^{27} - 1908 p^{12} T^{28} + 312 p^{13} T^{29} + 33 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( ( 1 + 8 T + 184 T^{2} + 1003 T^{3} + 14093 T^{4} + 50978 T^{5} + 642206 T^{6} + 1564129 T^{7} + 23810023 T^{8} + 1564129 p T^{9} + 642206 p^{2} T^{10} + 50978 p^{3} T^{11} + 14093 p^{4} T^{12} + 1003 p^{5} T^{13} + 184 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 16 T - 35 T^{2} + 2224 T^{3} - 10078 T^{4} - 97956 T^{5} + 955832 T^{6} + 404026 T^{7} - 33611466 T^{8} + 101421405 T^{9} + 594078289 T^{10} - 7155602297 T^{11} + 13674463334 T^{12} + 460384716635 T^{13} - 3662338572861 T^{14} - 11015607753041 T^{15} + 235150335460009 T^{16} - 11015607753041 p T^{17} - 3662338572861 p^{2} T^{18} + 460384716635 p^{3} T^{19} + 13674463334 p^{4} T^{20} - 7155602297 p^{5} T^{21} + 594078289 p^{6} T^{22} + 101421405 p^{7} T^{23} - 33611466 p^{8} T^{24} + 404026 p^{9} T^{25} + 955832 p^{10} T^{26} - 97956 p^{11} T^{27} - 10078 p^{12} T^{28} + 2224 p^{13} T^{29} - 35 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 14 T - 2 p T^{2} - 1614 T^{3} + 4335 T^{4} + 68958 T^{5} - 315804 T^{6} - 497433 T^{7} + 13768788 T^{8} - 148711803 T^{9} - 53612742 T^{10} + 12378130305 T^{11} - 20300504490 T^{12} - 493998074661 T^{13} + 1341726421551 T^{14} + 8204342474996 T^{15} - 63655947997847 T^{16} + 8204342474996 p T^{17} + 1341726421551 p^{2} T^{18} - 493998074661 p^{3} T^{19} - 20300504490 p^{4} T^{20} + 12378130305 p^{5} T^{21} - 53612742 p^{6} T^{22} - 148711803 p^{7} T^{23} + 13768788 p^{8} T^{24} - 497433 p^{9} T^{25} - 315804 p^{10} T^{26} + 68958 p^{11} T^{27} + 4335 p^{12} T^{28} - 1614 p^{13} T^{29} - 2 p^{15} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 + 3 T - 191 T^{2} - 858 T^{3} + 17818 T^{4} + 123267 T^{5} - 1187686 T^{6} - 10991538 T^{7} + 57748754 T^{8} + 695788332 T^{9} - 1934957318 T^{10} - 33662879799 T^{11} + 67962099005 T^{12} + 1151132657706 T^{13} - 3251337842320 T^{14} - 19351107451191 T^{15} + 155798026196935 T^{16} - 19351107451191 p T^{17} - 3251337842320 p^{2} T^{18} + 1151132657706 p^{3} T^{19} + 67962099005 p^{4} T^{20} - 33662879799 p^{5} T^{21} - 1934957318 p^{6} T^{22} + 695788332 p^{7} T^{23} + 57748754 p^{8} T^{24} - 10991538 p^{9} T^{25} - 1187686 p^{10} T^{26} + 123267 p^{11} T^{27} + 17818 p^{12} T^{28} - 858 p^{13} T^{29} - 191 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 7 T - 206 T^{2} + 1705 T^{3} + 18410 T^{4} - 178716 T^{5} - 1103533 T^{6} + 12158116 T^{7} + 58853415 T^{8} - 766335945 T^{9} - 1599172616 T^{10} + 42242182912 T^{11} - 96377490364 T^{12} - 1358509997230 T^{13} + 10474043014890 T^{14} + 16320769807219 T^{15} - 574764125356553 T^{16} + 16320769807219 p T^{17} + 10474043014890 p^{2} T^{18} - 1358509997230 p^{3} T^{19} - 96377490364 p^{4} T^{20} + 42242182912 p^{5} T^{21} - 1599172616 p^{6} T^{22} - 766335945 p^{7} T^{23} + 58853415 p^{8} T^{24} + 12158116 p^{9} T^{25} - 1103533 p^{10} T^{26} - 178716 p^{11} T^{27} + 18410 p^{12} T^{28} + 1705 p^{13} T^{29} - 206 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( ( 1 + 13 T + 427 T^{2} + 4781 T^{3} + 85148 T^{4} + 816694 T^{5} + 10335824 T^{6} + 84073193 T^{7} + 837177553 T^{8} + 84073193 p T^{9} + 10335824 p^{2} T^{10} + 816694 p^{3} T^{11} + 85148 p^{4} T^{12} + 4781 p^{5} T^{13} + 427 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 17 T - 119 T^{2} - 2795 T^{3} + 15695 T^{4} + 325176 T^{5} - 1168108 T^{6} - 23627039 T^{7} + 70260315 T^{8} + 1317744285 T^{9} - 2740448420 T^{10} - 47751224735 T^{11} + 302787955055 T^{12} + 1831459149335 T^{13} - 31662351108435 T^{14} + 1406741703730 T^{15} + 3353975253559105 T^{16} + 1406741703730 p T^{17} - 31662351108435 p^{2} T^{18} + 1831459149335 p^{3} T^{19} + 302787955055 p^{4} T^{20} - 47751224735 p^{5} T^{21} - 2740448420 p^{6} T^{22} + 1317744285 p^{7} T^{23} + 70260315 p^{8} T^{24} - 23627039 p^{9} T^{25} - 1168108 p^{10} T^{26} + 325176 p^{11} T^{27} + 15695 p^{12} T^{28} - 2795 p^{13} T^{29} - 119 p^{14} T^{30} + 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 11 T - 76 T^{2} - 2346 T^{3} - 10680 T^{4} + 143727 T^{5} + 2061846 T^{6} + 6298318 T^{7} - 101497067 T^{8} - 1126414867 T^{9} - 6096347392 T^{10} - 9455759000 T^{11} + 526181419955 T^{12} + 9385347807791 T^{13} + 48727700967671 T^{14} - 439951746648806 T^{15} - 7445129257432987 T^{16} - 439951746648806 p T^{17} + 48727700967671 p^{2} T^{18} + 9385347807791 p^{3} T^{19} + 526181419955 p^{4} T^{20} - 9455759000 p^{5} T^{21} - 6096347392 p^{6} T^{22} - 1126414867 p^{7} T^{23} - 101497067 p^{8} T^{24} + 6298318 p^{9} T^{25} + 2061846 p^{10} T^{26} + 143727 p^{11} T^{27} - 10680 p^{12} T^{28} - 2346 p^{13} T^{29} - 76 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 6 T - 215 T^{2} + 1736 T^{3} + 5334 T^{4} - 59234 T^{5} + 2801049 T^{6} - 29429038 T^{7} - 282023768 T^{8} + 3581875846 T^{9} - 5240495750 T^{10} + 75456406486 T^{11} + 1949641431702 T^{12} - 40751817048912 T^{13} - 22921475297614 T^{14} + 2013690079818150 T^{15} - 8011738335973837 T^{16} + 2013690079818150 p T^{17} - 22921475297614 p^{2} T^{18} - 40751817048912 p^{3} T^{19} + 1949641431702 p^{4} T^{20} + 75456406486 p^{5} T^{21} - 5240495750 p^{6} T^{22} + 3581875846 p^{7} T^{23} - 282023768 p^{8} T^{24} - 29429038 p^{9} T^{25} + 2801049 p^{10} T^{26} - 59234 p^{11} T^{27} + 5334 p^{12} T^{28} + 1736 p^{13} T^{29} - 215 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 28 T + 338 T^{2} - 4325 T^{3} + 820 p T^{4} - 768234 T^{5} + 6814942 T^{6} - 67486877 T^{7} + 625968765 T^{8} - 3558241110 T^{9} + 152025367 p T^{10} + 277703167 T^{11} - 2422831994577 T^{12} + 43230248588510 T^{13} - 468242061240150 T^{14} + 5132360572102819 T^{15} - 52527308502383847 T^{16} + 5132360572102819 p T^{17} - 468242061240150 p^{2} T^{18} + 43230248588510 p^{3} T^{19} - 2422831994577 p^{4} T^{20} + 277703167 p^{5} T^{21} + 152025367 p^{7} T^{22} - 3558241110 p^{7} T^{23} + 625968765 p^{8} T^{24} - 67486877 p^{9} T^{25} + 6814942 p^{10} T^{26} - 768234 p^{11} T^{27} + 820 p^{13} T^{28} - 4325 p^{13} T^{29} + 338 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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
−2.43686632768528485869138414973, −2.40643211683382267972212478578, −2.38384424112189464471056980019, −2.35929568720351234994290427836, −2.09322915353415608686873382225, −2.02581832494605686947444601214, −2.00161922712900307986070639205, −1.87628297791212400545230891004, −1.77976984417893830437866039451, −1.63430825489653836925300723276, −1.58300215510574198860500880239, −1.55411652555355709711813733558, −1.50182692631369926074183869676, −1.47251525682502830878403546538, −1.42427688248598703675580251293, −1.19077355757281115036006334698, −1.01526227929697719084339838003, −0.882722229527658776944497640420, −0.831055298037688781363658655473, −0.77322516631649267501931091960, −0.75671749346869862050392434269, −0.64826411980691904582851545592, −0.39696874425221610798430876891, −0.37094832384168314283848817885, −0.14969268218130349460330442062, 0.14969268218130349460330442062, 0.37094832384168314283848817885, 0.39696874425221610798430876891, 0.64826411980691904582851545592, 0.75671749346869862050392434269, 0.77322516631649267501931091960, 0.831055298037688781363658655473, 0.882722229527658776944497640420, 1.01526227929697719084339838003, 1.19077355757281115036006334698, 1.42427688248598703675580251293, 1.47251525682502830878403546538, 1.50182692631369926074183869676, 1.55411652555355709711813733558, 1.58300215510574198860500880239, 1.63430825489653836925300723276, 1.77976984417893830437866039451, 1.87628297791212400545230891004, 2.00161922712900307986070639205, 2.02581832494605686947444601214, 2.09322915353415608686873382225, 2.35929568720351234994290427836, 2.38384424112189464471056980019, 2.40643211683382267972212478578, 2.43686632768528485869138414973
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.