Dirichlet series
| L(s) = 1 | + 2-s − 3·3-s + 15·5-s − 3·6-s − 7·7-s + 20·9-s + 15·10-s + 22·13-s − 7·14-s − 45·15-s − 15·17-s + 20·18-s + 32·19-s + 21·21-s + 5·23-s + 107·25-s + 22·26-s − 51·27-s − 6·29-s − 45·30-s + 4·31-s − 15·34-s − 105·35-s − 4·37-s + 32·38-s − 66·39-s − 35·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 6.70·5-s − 1.22·6-s − 2.64·7-s + 20/3·9-s + 4.74·10-s + 6.10·13-s − 1.87·14-s − 11.6·15-s − 3.63·17-s + 4.71·18-s + 7.34·19-s + 4.58·21-s + 1.04·23-s + 21.3·25-s + 4.31·26-s − 9.81·27-s − 1.11·29-s − 8.21·30-s + 0.718·31-s − 2.57·34-s − 17.7·35-s − 0.657·37-s + 5.19·38-s − 10.5·39-s − 5.46·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 103^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 103^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 103^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2872.82\) |
| Root analytic conductor: | \(1.28254\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 103^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(29.38842457\) |
| \(L(\frac12)\) | \(\approx\) | \(29.38842457\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 103 | \( 1 - 33 T + 477 T^{2} - 3127 T^{3} - 14245 T^{4} + 443327 T^{5} - 1073957 T^{6} - 63206645 T^{7} + 1021644751 T^{8} - 63206645 p T^{9} - 1073957 p^{2} T^{10} + 443327 p^{3} T^{11} - 14245 p^{4} T^{12} - 3127 p^{5} T^{13} + 477 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + p T - 11 T^{2} - 14 p T^{3} + 20 p T^{4} + 34 p^{2} T^{5} - 163 T^{6} - 1424 T^{7} + 25 T^{8} + 1534 p T^{9} + 599 p T^{10} - 10591 T^{11} - 9097 T^{12} + 1892 p^{2} T^{13} + 10094 p T^{14} - 4936 p T^{15} - 90917 T^{16} - 4936 p^{2} T^{17} + 10094 p^{3} T^{18} + 1892 p^{5} T^{19} - 9097 p^{4} T^{20} - 10591 p^{5} T^{21} + 599 p^{7} T^{22} + 1534 p^{8} T^{23} + 25 p^{8} T^{24} - 1424 p^{9} T^{25} - 163 p^{10} T^{26} + 34 p^{13} T^{27} + 20 p^{13} T^{28} - 14 p^{14} T^{29} - 11 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 - 3 p T + 118 T^{2} - 641 T^{3} + 2667 T^{4} - 8971 T^{5} + 5036 p T^{6} - 12067 p T^{7} + 126784 T^{8} - 48799 p T^{9} + 457084 T^{10} - 866408 T^{11} + 1585882 T^{12} - 2415849 T^{13} + 2079926 T^{14} + 2132004 T^{15} - 10984719 T^{16} + 2132004 p T^{17} + 2079926 p^{2} T^{18} - 2415849 p^{3} T^{19} + 1585882 p^{4} T^{20} - 866408 p^{5} T^{21} + 457084 p^{6} T^{22} - 48799 p^{8} T^{23} + 126784 p^{8} T^{24} - 12067 p^{10} T^{25} + 5036 p^{11} T^{26} - 8971 p^{11} T^{27} + 2667 p^{12} T^{28} - 641 p^{13} T^{29} + 118 p^{14} T^{30} - 3 p^{16} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + p T - 9 T^{2} - 163 T^{3} - 18 p T^{4} + 1636 T^{5} + 470 p T^{6} - 5575 T^{7} - 4122 p T^{8} - 50294 T^{9} + 104869 T^{10} + 798093 T^{11} + 224471 T^{12} - 5023913 T^{13} - 588310 p T^{14} + 13364329 T^{15} + 28601073 T^{16} + 13364329 p T^{17} - 588310 p^{3} T^{18} - 5023913 p^{3} T^{19} + 224471 p^{4} T^{20} + 798093 p^{5} T^{21} + 104869 p^{6} T^{22} - 50294 p^{7} T^{23} - 4122 p^{9} T^{24} - 5575 p^{9} T^{25} + 470 p^{11} T^{26} + 1636 p^{11} T^{27} - 18 p^{13} T^{28} - 163 p^{13} T^{29} - 9 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 6 T^{2} + 85 T^{3} + 87 T^{4} + 765 T^{5} + 4228 T^{6} + 1020 p T^{7} + 37897 T^{8} + 191335 T^{9} + 715452 T^{10} + 1425008 T^{11} + 8918582 T^{12} + 26321831 T^{13} + 5817477 p T^{14} + 316195954 T^{15} + 879333975 T^{16} + 316195954 p T^{17} + 5817477 p^{3} T^{18} + 26321831 p^{3} T^{19} + 8918582 p^{4} T^{20} + 1425008 p^{5} T^{21} + 715452 p^{6} T^{22} + 191335 p^{7} T^{23} + 37897 p^{8} T^{24} + 1020 p^{10} T^{25} + 4228 p^{10} T^{26} + 765 p^{11} T^{27} + 87 p^{12} T^{28} + 85 p^{13} T^{29} + 6 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 22 T + 250 T^{2} - 1916 T^{3} + 10801 T^{4} - 46624 T^{5} + 160945 T^{6} - 499955 T^{7} + 1820054 T^{8} - 8773130 T^{9} + 44317595 T^{10} - 196972636 T^{11} + 754959845 T^{12} - 2621956101 T^{13} + 8883332179 T^{14} - 30921722940 T^{15} + 110627646165 T^{16} - 30921722940 p T^{17} + 8883332179 p^{2} T^{18} - 2621956101 p^{3} T^{19} + 754959845 p^{4} T^{20} - 196972636 p^{5} T^{21} + 44317595 p^{6} T^{22} - 8773130 p^{7} T^{23} + 1820054 p^{8} T^{24} - 499955 p^{9} T^{25} + 160945 p^{10} T^{26} - 46624 p^{11} T^{27} + 10801 p^{12} T^{28} - 1916 p^{13} T^{29} + 250 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 15 T + 140 T^{2} + 1080 T^{3} + 6799 T^{4} + 36297 T^{5} + 173549 T^{6} + 733422 T^{7} + 2651134 T^{8} + 8315412 T^{9} + 20456444 T^{10} + 20478396 T^{11} - 124966304 T^{12} - 1126349295 T^{13} - 6720590641 T^{14} - 31919777295 T^{15} - 132603796388 T^{16} - 31919777295 p T^{17} - 6720590641 p^{2} T^{18} - 1126349295 p^{3} T^{19} - 124966304 p^{4} T^{20} + 20478396 p^{5} T^{21} + 20456444 p^{6} T^{22} + 8315412 p^{7} T^{23} + 2651134 p^{8} T^{24} + 733422 p^{9} T^{25} + 173549 p^{10} T^{26} + 36297 p^{11} T^{27} + 6799 p^{12} T^{28} + 1080 p^{13} T^{29} + 140 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 32 T + 512 T^{2} - 5593 T^{3} + 47426 T^{4} - 331865 T^{5} + 1984487 T^{6} - 10431829 T^{7} + 49501942 T^{8} - 217160790 T^{9} + 895110161 T^{10} - 3464167656 T^{11} + 12270215906 T^{12} - 37567754428 T^{13} + 89457177189 T^{14} - 134520575867 T^{15} + 162295399129 T^{16} - 134520575867 p T^{17} + 89457177189 p^{2} T^{18} - 37567754428 p^{3} T^{19} + 12270215906 p^{4} T^{20} - 3464167656 p^{5} T^{21} + 895110161 p^{6} T^{22} - 217160790 p^{7} T^{23} + 49501942 p^{8} T^{24} - 10431829 p^{9} T^{25} + 1984487 p^{10} T^{26} - 331865 p^{11} T^{27} + 47426 p^{12} T^{28} - 5593 p^{13} T^{29} + 512 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 5 T + 19 T^{2} - 303 T^{3} + 1928 T^{4} - 7856 T^{5} + 39153 T^{6} - 237624 T^{7} + 1183926 T^{8} - 2473725 T^{9} - 350201 T^{10} + 15371769 T^{11} - 252153795 T^{12} + 3451604990 T^{13} - 23484066937 T^{14} + 110030383364 T^{15} - 504338756287 T^{16} + 110030383364 p T^{17} - 23484066937 p^{2} T^{18} + 3451604990 p^{3} T^{19} - 252153795 p^{4} T^{20} + 15371769 p^{5} T^{21} - 350201 p^{6} T^{22} - 2473725 p^{7} T^{23} + 1183926 p^{8} T^{24} - 237624 p^{9} T^{25} + 39153 p^{10} T^{26} - 7856 p^{11} T^{27} + 1928 p^{12} T^{28} - 303 p^{13} T^{29} + 19 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 6 T + 2 p T^{2} + 361 T^{3} + 722 T^{4} - 5474 T^{5} - 53459 T^{6} - 684962 T^{7} - 2594838 T^{8} - 1797301 T^{9} + 1127138 p T^{10} + 578627096 T^{11} + 4001468842 T^{12} + 5209182019 T^{13} - 11252582713 T^{14} - 451259344585 T^{15} - 4133778677677 T^{16} - 451259344585 p T^{17} - 11252582713 p^{2} T^{18} + 5209182019 p^{3} T^{19} + 4001468842 p^{4} T^{20} + 578627096 p^{5} T^{21} + 1127138 p^{7} T^{22} - 1797301 p^{7} T^{23} - 2594838 p^{8} T^{24} - 684962 p^{9} T^{25} - 53459 p^{10} T^{26} - 5474 p^{11} T^{27} + 722 p^{12} T^{28} + 361 p^{13} T^{29} + 2 p^{15} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 4 T - 15 T^{2} + 14 T^{3} + 69 T^{4} + 1211 T^{5} + 7569 T^{6} - 25912 T^{7} + 310125 T^{8} - 5173360 T^{9} + 5213018 T^{10} + 76126674 T^{11} - 189324142 T^{12} + 2131274987 T^{13} + 7808856496 T^{14} - 157157719638 T^{15} + 445865474398 T^{16} - 157157719638 p T^{17} + 7808856496 p^{2} T^{18} + 2131274987 p^{3} T^{19} - 189324142 p^{4} T^{20} + 76126674 p^{5} T^{21} + 5213018 p^{6} T^{22} - 5173360 p^{7} T^{23} + 310125 p^{8} T^{24} - 25912 p^{9} T^{25} + 7569 p^{10} T^{26} + 1211 p^{11} T^{27} + 69 p^{12} T^{28} + 14 p^{13} T^{29} - 15 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 4 T - 72 T^{2} - 912 T^{3} + 733 T^{4} + 42184 T^{5} + 197766 T^{6} - 720410 T^{7} - 7258747 T^{8} - 24965440 T^{9} + 195625616 T^{10} + 1773982684 T^{11} + 1206960438 T^{12} - 117667702970 T^{13} - 555732180324 T^{14} + 2149629861902 T^{15} + 38004539275723 T^{16} + 2149629861902 p T^{17} - 555732180324 p^{2} T^{18} - 117667702970 p^{3} T^{19} + 1206960438 p^{4} T^{20} + 1773982684 p^{5} T^{21} + 195625616 p^{6} T^{22} - 24965440 p^{7} T^{23} - 7258747 p^{8} T^{24} - 720410 p^{9} T^{25} + 197766 p^{10} T^{26} + 42184 p^{11} T^{27} + 733 p^{12} T^{28} - 912 p^{13} T^{29} - 72 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 35 T + 589 T^{2} + 6532 T^{3} + 1420 p T^{4} + 499478 T^{5} + 4362066 T^{6} + 35865932 T^{7} + 265002713 T^{8} + 1845330908 T^{9} + 13033078777 T^{10} + 92725285336 T^{11} + 624142288553 T^{12} + 3938855251947 T^{13} + 24881315313666 T^{14} + 164066521400542 T^{15} + 1075697545914419 T^{16} + 164066521400542 p T^{17} + 24881315313666 p^{2} T^{18} + 3938855251947 p^{3} T^{19} + 624142288553 p^{4} T^{20} + 92725285336 p^{5} T^{21} + 13033078777 p^{6} T^{22} + 1845330908 p^{7} T^{23} + 265002713 p^{8} T^{24} + 35865932 p^{9} T^{25} + 4362066 p^{10} T^{26} + 499478 p^{11} T^{27} + 1420 p^{13} T^{28} + 6532 p^{13} T^{29} + 589 p^{14} T^{30} + 35 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + T - 127 T^{2} - 680 T^{3} + 7450 T^{4} + 76742 T^{5} - 204525 T^{6} - 4907696 T^{7} - 1709317 T^{8} + 224574096 T^{9} + 681063647 T^{10} - 8233660439 T^{11} - 49719835829 T^{12} + 221228184094 T^{13} + 2564148245170 T^{14} - 3350279872030 T^{15} - 112076879824273 T^{16} - 3350279872030 p T^{17} + 2564148245170 p^{2} T^{18} + 221228184094 p^{3} T^{19} - 49719835829 p^{4} T^{20} - 8233660439 p^{5} T^{21} + 681063647 p^{6} T^{22} + 224574096 p^{7} T^{23} - 1709317 p^{8} T^{24} - 4907696 p^{9} T^{25} - 204525 p^{10} T^{26} + 76742 p^{11} T^{27} + 7450 p^{12} T^{28} - 680 p^{13} T^{29} - 127 p^{14} T^{30} + p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 9 T + 149 T^{2} + 1171 T^{3} + 13225 T^{4} + 84379 T^{5} + 865967 T^{6} + 5151189 T^{7} + 44959697 T^{8} + 5151189 p T^{9} + 865967 p^{2} T^{10} + 84379 p^{3} T^{11} + 13225 p^{4} T^{12} + 1171 p^{5} T^{13} + 149 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 84 T + 3195 T^{2} + 71318 T^{3} + 987597 T^{4} + 7403614 T^{5} - 8534333 T^{6} - 986137986 T^{7} - 13094149717 T^{8} - 85605080666 T^{9} - 16765678573 T^{10} + 6222412979816 T^{11} + 74079519169899 T^{12} + 429279830377094 T^{13} + 90428329500217 T^{14} - 24438328184182140 T^{15} - 257792890443810747 T^{16} - 24438328184182140 p T^{17} + 90428329500217 p^{2} T^{18} + 429279830377094 p^{3} T^{19} + 74079519169899 p^{4} T^{20} + 6222412979816 p^{5} T^{21} - 16765678573 p^{6} T^{22} - 85605080666 p^{7} T^{23} - 13094149717 p^{8} T^{24} - 986137986 p^{9} T^{25} - 8534333 p^{10} T^{26} + 7403614 p^{11} T^{27} + 987597 p^{12} T^{28} + 71318 p^{13} T^{29} + 3195 p^{14} T^{30} + 84 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 13 T - 26 T^{2} + 1853 T^{3} - 18985 T^{4} + 97154 T^{5} + 565707 T^{6} - 18287240 T^{7} + 170927944 T^{8} - 530958488 T^{9} - 4723586773 T^{10} + 90774827801 T^{11} - 823207167752 T^{12} + 3662496230598 T^{13} + 11145502540400 T^{14} - 310732062438119 T^{15} + 2934000378783960 T^{16} - 310732062438119 p T^{17} + 11145502540400 p^{2} T^{18} + 3662496230598 p^{3} T^{19} - 823207167752 p^{4} T^{20} + 90774827801 p^{5} T^{21} - 4723586773 p^{6} T^{22} - 530958488 p^{7} T^{23} + 170927944 p^{8} T^{24} - 18287240 p^{9} T^{25} + 565707 p^{10} T^{26} + 97154 p^{11} T^{27} - 18985 p^{12} T^{28} + 1853 p^{13} T^{29} - 26 p^{14} T^{30} - 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 28 T + 281 T^{2} + 164 T^{3} - 33531 T^{4} + 331518 T^{5} - 451801 T^{6} - 19037446 T^{7} + 177097427 T^{8} - 44173754 T^{9} - 11811120885 T^{10} + 92777604356 T^{11} + 69245155993 T^{12} - 6211427565204 T^{13} + 37826309233993 T^{14} + 106116542078544 T^{15} - 2581455236133347 T^{16} + 106116542078544 p T^{17} + 37826309233993 p^{2} T^{18} - 6211427565204 p^{3} T^{19} + 69245155993 p^{4} T^{20} + 92777604356 p^{5} T^{21} - 11811120885 p^{6} T^{22} - 44173754 p^{7} T^{23} + 177097427 p^{8} T^{24} - 19037446 p^{9} T^{25} - 451801 p^{10} T^{26} + 331518 p^{11} T^{27} - 33531 p^{12} T^{28} + 164 p^{13} T^{29} + 281 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 18 T + 155 T^{2} + 683 T^{3} - 8478 T^{4} - 167204 T^{5} - 994844 T^{6} + 2145183 T^{7} + 1595909 p T^{8} + 1159017198 T^{9} + 5218535377 T^{10} - 21329810589 T^{11} - 483684087266 T^{12} - 3251534129123 T^{13} - 3560128054969 T^{14} + 200527084436588 T^{15} + 2573287149835487 T^{16} + 200527084436588 p T^{17} - 3560128054969 p^{2} T^{18} - 3251534129123 p^{3} T^{19} - 483684087266 p^{4} T^{20} - 21329810589 p^{5} T^{21} + 5218535377 p^{6} T^{22} + 1159017198 p^{7} T^{23} + 1595909 p^{9} T^{24} + 2145183 p^{9} T^{25} - 994844 p^{10} T^{26} - 167204 p^{11} T^{27} - 8478 p^{12} T^{28} + 683 p^{13} T^{29} + 155 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 23 T + 220 T^{2} - 449 T^{3} - 34090 T^{4} - 364999 T^{5} - 814152 T^{6} + 19561693 T^{7} + 225697454 T^{8} + 509550035 T^{9} - 7917096820 T^{10} - 57639342185 T^{11} + 386167253722 T^{12} + 7440621920527 T^{13} + 25977750674452 T^{14} - 444417338982905 T^{15} - 6370702157321805 T^{16} - 444417338982905 p T^{17} + 25977750674452 p^{2} T^{18} + 7440621920527 p^{3} T^{19} + 386167253722 p^{4} T^{20} - 57639342185 p^{5} T^{21} - 7917096820 p^{6} T^{22} + 509550035 p^{7} T^{23} + 225697454 p^{8} T^{24} + 19561693 p^{9} T^{25} - 814152 p^{10} T^{26} - 364999 p^{11} T^{27} - 34090 p^{12} T^{28} - 449 p^{13} T^{29} + 220 p^{14} T^{30} + 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 16 T - 106 T^{2} + 6094 T^{3} - 40330 T^{4} - 712635 T^{5} + 13473011 T^{6} - 10357087 T^{7} - 1640920660 T^{8} + 13944306038 T^{9} + 80455707864 T^{10} - 1929275315231 T^{11} + 5030120487237 T^{12} + 141404880719781 T^{13} - 1233313785252553 T^{14} - 4221232183272506 T^{15} + 117933107276555421 T^{16} - 4221232183272506 p T^{17} - 1233313785252553 p^{2} T^{18} + 141404880719781 p^{3} T^{19} + 5030120487237 p^{4} T^{20} - 1929275315231 p^{5} T^{21} + 80455707864 p^{6} T^{22} + 13944306038 p^{7} T^{23} - 1640920660 p^{8} T^{24} - 10357087 p^{9} T^{25} + 13473011 p^{10} T^{26} - 712635 p^{11} T^{27} - 40330 p^{12} T^{28} + 6094 p^{13} T^{29} - 106 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 44 T + 871 T^{2} - 10912 T^{3} + 113751 T^{4} - 15675 p T^{5} + 13569959 T^{6} - 134449391 T^{7} + 1215866832 T^{8} - 128184670 p T^{9} + 76421180550 T^{10} - 562996243704 T^{11} + 4219304760219 T^{12} - 26043525360660 T^{13} + 98341571758323 T^{14} - 182714534584662 T^{15} + 458261031346661 T^{16} - 182714534584662 p T^{17} + 98341571758323 p^{2} T^{18} - 26043525360660 p^{3} T^{19} + 4219304760219 p^{4} T^{20} - 562996243704 p^{5} T^{21} + 76421180550 p^{6} T^{22} - 128184670 p^{8} T^{23} + 1215866832 p^{8} T^{24} - 134449391 p^{9} T^{25} + 13569959 p^{10} T^{26} - 15675 p^{12} T^{27} + 113751 p^{12} T^{28} - 10912 p^{13} T^{29} + 871 p^{14} T^{30} - 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 77 T + 3092 T^{2} + 84830 T^{3} + 1776224 T^{4} + 30083283 T^{5} + 426190773 T^{6} + 5137243957 T^{7} + 52800397936 T^{8} + 454064153113 T^{9} + 3051785902857 T^{10} + 11753362610349 T^{11} - 65302827833202 T^{12} - 2093440136067901 T^{13} - 29870058742991287 T^{14} - 333636536542220472 T^{15} - 3228465305302170549 T^{16} - 333636536542220472 p T^{17} - 29870058742991287 p^{2} T^{18} - 2093440136067901 p^{3} T^{19} - 65302827833202 p^{4} T^{20} + 11753362610349 p^{5} T^{21} + 3051785902857 p^{6} T^{22} + 454064153113 p^{7} T^{23} + 52800397936 p^{8} T^{24} + 5137243957 p^{9} T^{25} + 426190773 p^{10} T^{26} + 30083283 p^{11} T^{27} + 1776224 p^{12} T^{28} + 84830 p^{13} T^{29} + 3092 p^{14} T^{30} + 77 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 16 T - 20 T^{2} - 1642 T^{3} + 9457 T^{4} + 193155 T^{5} - 1794626 T^{6} - 30569706 T^{7} + 18590026 T^{8} + 175345379 T^{9} - 23619855468 T^{10} - 79556825427 T^{11} + 1733457687483 T^{12} + 4036034814267 T^{13} - 105579770686569 T^{14} + 1095881013493578 T^{15} + 26878746846636025 T^{16} + 1095881013493578 p T^{17} - 105579770686569 p^{2} T^{18} + 4036034814267 p^{3} T^{19} + 1733457687483 p^{4} T^{20} - 79556825427 p^{5} T^{21} - 23619855468 p^{6} T^{22} + 175345379 p^{7} T^{23} + 18590026 p^{8} T^{24} - 30569706 p^{9} T^{25} - 1794626 p^{10} T^{26} + 193155 p^{11} T^{27} + 9457 p^{12} T^{28} - 1642 p^{13} T^{29} - 20 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 37 T + 745 T^{2} - 14303 T^{3} + 240400 T^{4} - 3359613 T^{5} + 44642319 T^{6} - 536318359 T^{7} + 5688100399 T^{8} - 56496891693 T^{9} + 480833082101 T^{10} - 3360045400314 T^{11} + 17393893160696 T^{12} + 21620599627342 T^{13} - 1811219071075594 T^{14} + 28218462475443897 T^{15} - 325673143439168125 T^{16} + 28218462475443897 p T^{17} - 1811219071075594 p^{2} T^{18} + 21620599627342 p^{3} T^{19} + 17393893160696 p^{4} T^{20} - 3360045400314 p^{5} T^{21} + 480833082101 p^{6} T^{22} - 56496891693 p^{7} T^{23} + 5688100399 p^{8} T^{24} - 536318359 p^{9} T^{25} + 44642319 p^{10} T^{26} - 3359613 p^{11} T^{27} + 240400 p^{12} T^{28} - 14303 p^{13} T^{29} + 745 p^{14} T^{30} - 37 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.70905544305991475649534055605, −3.69881845012748606323125431808, −3.34692052483285354014985015176, −3.23616369719941133972797307241, −3.16276899856516673587452044523, −3.11368417805801665079805716597, −3.03879198921173265102232502544, −3.00050235695388890063391728025, −2.97588682718025107515481415864, −2.96681112044429059995141866664, −2.86100172633390022888198885047, −2.25561330750345766521231049016, −2.22355059352775939776497992497, −2.08658289065163563628890805888, −2.03668180502688499557994463004, −1.91385482057419596534160341676, −1.79432516168716377649169040625, −1.67326333009570895175626889581, −1.64899530209295907980032085362, −1.49737499751005859730496577367, −1.29714615608471330059273648666, −1.25687050077995496612816671853, −1.25255020945776168972022457255, −1.11543561919698907315530600348, −0.878867919413828010660034334204, 0.878867919413828010660034334204, 1.11543561919698907315530600348, 1.25255020945776168972022457255, 1.25687050077995496612816671853, 1.29714615608471330059273648666, 1.49737499751005859730496577367, 1.64899530209295907980032085362, 1.67326333009570895175626889581, 1.79432516168716377649169040625, 1.91385482057419596534160341676, 2.03668180502688499557994463004, 2.08658289065163563628890805888, 2.22355059352775939776497992497, 2.25561330750345766521231049016, 2.86100172633390022888198885047, 2.96681112044429059995141866664, 2.97588682718025107515481415864, 3.00050235695388890063391728025, 3.03879198921173265102232502544, 3.11368417805801665079805716597, 3.16276899856516673587452044523, 3.23616369719941133972797307241, 3.34692052483285354014985015176, 3.69881845012748606323125431808, 3.70905544305991475649534055605
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.