Dirichlet series
| L(s) = 1 | + 9·2-s + 12·3-s + 18·4-s + 16·5-s + 108·6-s + 58·7-s − 61·8-s − 20·9-s + 144·10-s + 142·11-s + 216·12-s + 28·13-s + 522·14-s + 192·15-s − 249·16-s + 160·17-s − 180·18-s + 100·19-s + 288·20-s + 696·21-s + 1.27e3·22-s + 664·23-s − 732·24-s − 485·25-s + 252·26-s − 994·27-s + 1.04e3·28-s + ⋯ |
| L(s) = 1 | + 3.18·2-s + 2.30·3-s + 9/4·4-s + 1.43·5-s + 7.34·6-s + 3.13·7-s − 2.69·8-s − 0.740·9-s + 4.55·10-s + 3.89·11-s + 5.19·12-s + 0.597·13-s + 9.96·14-s + 3.30·15-s − 3.89·16-s + 2.28·17-s − 2.35·18-s + 1.20·19-s + 3.21·20-s + 7.23·21-s + 12.3·22-s + 6.01·23-s − 6.22·24-s − 3.87·25-s + 1.90·26-s − 7.08·27-s + 7.04·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(97^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(97^{13}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(97^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.06791\times 10^{9}\) |
| Root analytic conductor: | \(2.39231\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 97^{13} ,\ ( \ : [3/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(369.7238208\) |
| \(L(\frac12)\) | \(\approx\) | \(369.7238208\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 97 | \( ( 1 + p T )^{13} \) |
| good | 2 | \( 1 - 9 T + 63 T^{2} - 43 p^{3} T^{3} + 831 p T^{4} - 7051 T^{5} + 27655 T^{6} - 50321 p T^{7} + 349131 T^{8} - 577077 p T^{9} + 28813 p^{7} T^{10} - 1418819 p^{3} T^{11} + 132395 p^{8} T^{12} - 760555 p^{7} T^{13} + 132395 p^{11} T^{14} - 1418819 p^{9} T^{15} + 28813 p^{16} T^{16} - 577077 p^{13} T^{17} + 349131 p^{15} T^{18} - 50321 p^{19} T^{19} + 27655 p^{21} T^{20} - 7051 p^{24} T^{21} + 831 p^{28} T^{22} - 43 p^{33} T^{23} + 63 p^{33} T^{24} - 9 p^{36} T^{25} + p^{39} T^{26} \) |
| 3 | \( 1 - 4 p T + 164 T^{2} - 1214 T^{3} + 10237 T^{4} - 60158 T^{5} + 134243 p T^{6} - 2001028 T^{7} + 11751484 T^{8} - 53216266 T^{9} + 316658264 T^{10} - 1410124090 T^{11} + 8678930891 T^{12} - 37837586144 T^{13} + 8678930891 p^{3} T^{14} - 1410124090 p^{6} T^{15} + 316658264 p^{9} T^{16} - 53216266 p^{12} T^{17} + 11751484 p^{15} T^{18} - 2001028 p^{18} T^{19} + 134243 p^{22} T^{20} - 60158 p^{24} T^{21} + 10237 p^{27} T^{22} - 1214 p^{30} T^{23} + 164 p^{33} T^{24} - 4 p^{37} T^{25} + p^{39} T^{26} \) | |
| 5 | \( 1 - 16 T + 741 T^{2} - 7968 T^{3} + 241534 T^{4} - 1902302 T^{5} + 53769664 T^{6} - 347961884 T^{7} + 10134795222 T^{8} - 11780134766 p T^{9} + 1674970090193 T^{10} - 8872174107888 T^{11} + 9574083012037 p^{2} T^{12} - 1171585409282304 T^{13} + 9574083012037 p^{5} T^{14} - 8872174107888 p^{6} T^{15} + 1674970090193 p^{9} T^{16} - 11780134766 p^{13} T^{17} + 10134795222 p^{15} T^{18} - 347961884 p^{18} T^{19} + 53769664 p^{21} T^{20} - 1902302 p^{24} T^{21} + 241534 p^{27} T^{22} - 7968 p^{30} T^{23} + 741 p^{33} T^{24} - 16 p^{36} T^{25} + p^{39} T^{26} \) | |
| 7 | \( 1 - 58 T + 4021 T^{2} - 174124 T^{3} + 7296018 T^{4} - 251372344 T^{5} + 8115283258 T^{6} - 232542233702 T^{7} + 895789962124 p T^{8} - 154200540627048 T^{9} + 3591388798029449 T^{10} - 77298243442653560 T^{11} + 1579237580471034915 T^{12} - 30017149813122654208 T^{13} + 1579237580471034915 p^{3} T^{14} - 77298243442653560 p^{6} T^{15} + 3591388798029449 p^{9} T^{16} - 154200540627048 p^{12} T^{17} + 895789962124 p^{16} T^{18} - 232542233702 p^{18} T^{19} + 8115283258 p^{21} T^{20} - 251372344 p^{24} T^{21} + 7296018 p^{27} T^{22} - 174124 p^{30} T^{23} + 4021 p^{33} T^{24} - 58 p^{36} T^{25} + p^{39} T^{26} \) | |
| 11 | \( 1 - 142 T + 20539 T^{2} - 1875594 T^{3} + 163105376 T^{4} - 11272446918 T^{5} + 738299602198 T^{6} - 41610057865072 T^{7} + 2233755701614024 T^{8} - 107141890159795422 T^{9} + 4926685167994714671 T^{10} - \)\(20\!\cdots\!62\)\( T^{11} + \)\(83\!\cdots\!03\)\( T^{12} - \)\(30\!\cdots\!16\)\( T^{13} + \)\(83\!\cdots\!03\)\( p^{3} T^{14} - \)\(20\!\cdots\!62\)\( p^{6} T^{15} + 4926685167994714671 p^{9} T^{16} - 107141890159795422 p^{12} T^{17} + 2233755701614024 p^{15} T^{18} - 41610057865072 p^{18} T^{19} + 738299602198 p^{21} T^{20} - 11272446918 p^{24} T^{21} + 163105376 p^{27} T^{22} - 1875594 p^{30} T^{23} + 20539 p^{33} T^{24} - 142 p^{36} T^{25} + p^{39} T^{26} \) | |
| 13 | \( 1 - 28 T + 23555 T^{2} - 703062 T^{3} + 264193625 T^{4} - 8095700850 T^{5} + 1869898275438 T^{6} - 56851846509486 T^{7} + 9320905068229188 T^{8} - 272158860043919708 T^{9} + 34555015320223470824 T^{10} - \)\(93\!\cdots\!28\)\( T^{11} + \)\(98\!\cdots\!33\)\( T^{12} - \)\(23\!\cdots\!00\)\( T^{13} + \)\(98\!\cdots\!33\)\( p^{3} T^{14} - \)\(93\!\cdots\!28\)\( p^{6} T^{15} + 34555015320223470824 p^{9} T^{16} - 272158860043919708 p^{12} T^{17} + 9320905068229188 p^{15} T^{18} - 56851846509486 p^{18} T^{19} + 1869898275438 p^{21} T^{20} - 8095700850 p^{24} T^{21} + 264193625 p^{27} T^{22} - 703062 p^{30} T^{23} + 23555 p^{33} T^{24} - 28 p^{36} T^{25} + p^{39} T^{26} \) | |
| 17 | \( 1 - 160 T + 2434 p T^{2} - 5303192 T^{3} + 811235687 T^{4} - 88440804844 T^{5} + 10370431109440 T^{6} - 58319108639410 p T^{7} + 97880496322525915 T^{8} - 490717434979881140 p T^{9} + \)\(72\!\cdots\!02\)\( T^{10} - \)\(55\!\cdots\!70\)\( T^{11} + \)\(43\!\cdots\!45\)\( T^{12} - \)\(30\!\cdots\!52\)\( T^{13} + \)\(43\!\cdots\!45\)\( p^{3} T^{14} - \)\(55\!\cdots\!70\)\( p^{6} T^{15} + \)\(72\!\cdots\!02\)\( p^{9} T^{16} - 490717434979881140 p^{13} T^{17} + 97880496322525915 p^{15} T^{18} - 58319108639410 p^{19} T^{19} + 10370431109440 p^{21} T^{20} - 88440804844 p^{24} T^{21} + 811235687 p^{27} T^{22} - 5303192 p^{30} T^{23} + 2434 p^{34} T^{24} - 160 p^{36} T^{25} + p^{39} T^{26} \) | |
| 19 | \( 1 - 100 T + 50866 T^{2} - 4131368 T^{3} + 1211245390 T^{4} - 76674791120 T^{5} + 17679351970447 T^{6} - 792664798134256 T^{7} + 176633684171822415 T^{8} - 4366313798551995536 T^{9} + \)\(13\!\cdots\!74\)\( T^{10} - \)\(56\!\cdots\!72\)\( T^{11} + \)\(85\!\cdots\!61\)\( T^{12} + \)\(55\!\cdots\!08\)\( T^{13} + \)\(85\!\cdots\!61\)\( p^{3} T^{14} - \)\(56\!\cdots\!72\)\( p^{6} T^{15} + \)\(13\!\cdots\!74\)\( p^{9} T^{16} - 4366313798551995536 p^{12} T^{17} + 176633684171822415 p^{15} T^{18} - 792664798134256 p^{18} T^{19} + 17679351970447 p^{21} T^{20} - 76674791120 p^{24} T^{21} + 1211245390 p^{27} T^{22} - 4131368 p^{30} T^{23} + 50866 p^{33} T^{24} - 100 p^{36} T^{25} + p^{39} T^{26} \) | |
| 23 | \( 1 - 664 T + 298748 T^{2} - 99896100 T^{3} + 27758653711 T^{4} - 6597298187236 T^{5} + 1385656488450341 T^{6} - 11331847889311390 p T^{7} + 1935983998062443176 p T^{8} - \)\(69\!\cdots\!56\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} - \)\(13\!\cdots\!96\)\( T^{11} + \)\(16\!\cdots\!61\)\( T^{12} - \)\(18\!\cdots\!80\)\( T^{13} + \)\(16\!\cdots\!61\)\( p^{3} T^{14} - \)\(13\!\cdots\!96\)\( p^{6} T^{15} + \)\(10\!\cdots\!58\)\( p^{9} T^{16} - \)\(69\!\cdots\!56\)\( p^{12} T^{17} + 1935983998062443176 p^{16} T^{18} - 11331847889311390 p^{19} T^{19} + 1385656488450341 p^{21} T^{20} - 6597298187236 p^{24} T^{21} + 27758653711 p^{27} T^{22} - 99896100 p^{30} T^{23} + 298748 p^{33} T^{24} - 664 p^{36} T^{25} + p^{39} T^{26} \) | |
| 29 | \( 1 - 4 p T + 134301 T^{2} - 18095518 T^{3} + 9376438615 T^{4} - 1338395014144 T^{5} + 443603353379436 T^{6} - 63930732100167748 T^{7} + 15837098647867416596 T^{8} - \)\(22\!\cdots\!80\)\( T^{9} + \)\(46\!\cdots\!37\)\( T^{10} - \)\(65\!\cdots\!02\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} - \)\(16\!\cdots\!12\)\( T^{13} + \)\(11\!\cdots\!86\)\( p^{3} T^{14} - \)\(65\!\cdots\!02\)\( p^{6} T^{15} + \)\(46\!\cdots\!37\)\( p^{9} T^{16} - \)\(22\!\cdots\!80\)\( p^{12} T^{17} + 15837098647867416596 p^{15} T^{18} - 63930732100167748 p^{18} T^{19} + 443603353379436 p^{21} T^{20} - 1338395014144 p^{24} T^{21} + 9376438615 p^{27} T^{22} - 18095518 p^{30} T^{23} + 134301 p^{33} T^{24} - 4 p^{37} T^{25} + p^{39} T^{26} \) | |
| 31 | \( 1 + 254 T + 228915 T^{2} + 46871674 T^{3} + 23868702463 T^{4} + 3923689107906 T^{5} + 1504863173401632 T^{6} + 192025055645599158 T^{7} + 64487840930203417566 T^{8} + \)\(19\!\cdots\!18\)\( p T^{9} + \)\(20\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!48\)\( T^{11} + \)\(18\!\cdots\!15\)\( p T^{12} + \)\(25\!\cdots\!24\)\( T^{13} + \)\(18\!\cdots\!15\)\( p^{4} T^{14} + \)\(12\!\cdots\!48\)\( p^{6} T^{15} + \)\(20\!\cdots\!32\)\( p^{9} T^{16} + \)\(19\!\cdots\!18\)\( p^{13} T^{17} + 64487840930203417566 p^{15} T^{18} + 192025055645599158 p^{18} T^{19} + 1504863173401632 p^{21} T^{20} + 3923689107906 p^{24} T^{21} + 23868702463 p^{27} T^{22} + 46871674 p^{30} T^{23} + 228915 p^{33} T^{24} + 254 p^{36} T^{25} + p^{39} T^{26} \) | |
| 37 | \( 1 - 50 T + 452332 T^{2} - 15884900 T^{3} + 98380661157 T^{4} - 2207100961910 T^{5} + 13747984044659501 T^{6} - 4393183116828048 p T^{7} + \)\(13\!\cdots\!90\)\( T^{8} - \)\(51\!\cdots\!34\)\( T^{9} + \)\(10\!\cdots\!56\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{11} + \)\(67\!\cdots\!95\)\( T^{12} + \)\(17\!\cdots\!36\)\( T^{13} + \)\(67\!\cdots\!95\)\( p^{3} T^{14} + \)\(13\!\cdots\!40\)\( p^{6} T^{15} + \)\(10\!\cdots\!56\)\( p^{9} T^{16} - \)\(51\!\cdots\!34\)\( p^{12} T^{17} + \)\(13\!\cdots\!90\)\( p^{15} T^{18} - 4393183116828048 p^{19} T^{19} + 13747984044659501 p^{21} T^{20} - 2207100961910 p^{24} T^{21} + 98380661157 p^{27} T^{22} - 15884900 p^{30} T^{23} + 452332 p^{33} T^{24} - 50 p^{36} T^{25} + p^{39} T^{26} \) | |
| 41 | \( 1 + 104 T + 6834 p T^{2} - 25798036 T^{3} + 33219408895 T^{4} - 8480861231480 T^{5} + 3891781104931456 T^{6} - 814202128445118790 T^{7} + \)\(41\!\cdots\!99\)\( T^{8} - \)\(64\!\cdots\!04\)\( T^{9} + \)\(31\!\cdots\!98\)\( T^{10} - \)\(52\!\cdots\!58\)\( T^{11} + \)\(20\!\cdots\!97\)\( T^{12} - \)\(35\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!97\)\( p^{3} T^{14} - \)\(52\!\cdots\!58\)\( p^{6} T^{15} + \)\(31\!\cdots\!98\)\( p^{9} T^{16} - \)\(64\!\cdots\!04\)\( p^{12} T^{17} + \)\(41\!\cdots\!99\)\( p^{15} T^{18} - 814202128445118790 p^{18} T^{19} + 3891781104931456 p^{21} T^{20} - 8480861231480 p^{24} T^{21} + 33219408895 p^{27} T^{22} - 25798036 p^{30} T^{23} + 6834 p^{34} T^{24} + 104 p^{36} T^{25} + p^{39} T^{26} \) | |
| 43 | \( 1 + 92 T + 320775 T^{2} + 71878134 T^{3} + 58805333254 T^{4} + 414100176024 p T^{5} + 9184267310483402 T^{6} + 63537414139702832 p T^{7} + \)\(12\!\cdots\!36\)\( T^{8} + \)\(33\!\cdots\!70\)\( T^{9} + \)\(13\!\cdots\!23\)\( T^{10} + \)\(34\!\cdots\!38\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} + \)\(29\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!89\)\( p^{3} T^{14} + \)\(34\!\cdots\!38\)\( p^{6} T^{15} + \)\(13\!\cdots\!23\)\( p^{9} T^{16} + \)\(33\!\cdots\!70\)\( p^{12} T^{17} + \)\(12\!\cdots\!36\)\( p^{15} T^{18} + 63537414139702832 p^{19} T^{19} + 9184267310483402 p^{21} T^{20} + 414100176024 p^{25} T^{21} + 58805333254 p^{27} T^{22} + 71878134 p^{30} T^{23} + 320775 p^{33} T^{24} + 92 p^{36} T^{25} + p^{39} T^{26} \) | |
| 47 | \( 1 - 1176 T + 1416692 T^{2} - 1074897588 T^{3} + 784125786858 T^{4} - 455205188641160 T^{5} + 252918148189841749 T^{6} - \)\(12\!\cdots\!04\)\( T^{7} + \)\(56\!\cdots\!73\)\( T^{8} - \)\(23\!\cdots\!44\)\( T^{9} + \)\(92\!\cdots\!14\)\( T^{10} - \)\(34\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!59\)\( T^{12} - \)\(39\!\cdots\!76\)\( T^{13} + \)\(12\!\cdots\!59\)\( p^{3} T^{14} - \)\(34\!\cdots\!88\)\( p^{6} T^{15} + \)\(92\!\cdots\!14\)\( p^{9} T^{16} - \)\(23\!\cdots\!44\)\( p^{12} T^{17} + \)\(56\!\cdots\!73\)\( p^{15} T^{18} - \)\(12\!\cdots\!04\)\( p^{18} T^{19} + 252918148189841749 p^{21} T^{20} - 455205188641160 p^{24} T^{21} + 784125786858 p^{27} T^{22} - 1074897588 p^{30} T^{23} + 1416692 p^{33} T^{24} - 1176 p^{36} T^{25} + p^{39} T^{26} \) | |
| 53 | \( 1 - 898 T + 1545331 T^{2} - 1108761758 T^{3} + 1097334430285 T^{4} - 656591462470676 T^{5} + 482190091953636102 T^{6} - \)\(24\!\cdots\!08\)\( T^{7} + \)\(14\!\cdots\!68\)\( T^{8} - \)\(67\!\cdots\!42\)\( T^{9} + \)\(34\!\cdots\!12\)\( T^{10} - \)\(26\!\cdots\!02\)\( p T^{11} + \)\(63\!\cdots\!05\)\( T^{12} - \)\(23\!\cdots\!00\)\( T^{13} + \)\(63\!\cdots\!05\)\( p^{3} T^{14} - \)\(26\!\cdots\!02\)\( p^{7} T^{15} + \)\(34\!\cdots\!12\)\( p^{9} T^{16} - \)\(67\!\cdots\!42\)\( p^{12} T^{17} + \)\(14\!\cdots\!68\)\( p^{15} T^{18} - \)\(24\!\cdots\!08\)\( p^{18} T^{19} + 482190091953636102 p^{21} T^{20} - 656591462470676 p^{24} T^{21} + 1097334430285 p^{27} T^{22} - 1108761758 p^{30} T^{23} + 1545331 p^{33} T^{24} - 898 p^{36} T^{25} + p^{39} T^{26} \) | |
| 59 | \( 1 - 660 T + 1514714 T^{2} - 1012517050 T^{3} + 1132006280689 T^{4} - 727288235169450 T^{5} + 561809202257972114 T^{6} - \)\(33\!\cdots\!46\)\( T^{7} + \)\(20\!\cdots\!81\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{9} + \)\(61\!\cdots\!96\)\( T^{10} - \)\(30\!\cdots\!40\)\( T^{11} + \)\(15\!\cdots\!29\)\( T^{12} - \)\(68\!\cdots\!12\)\( T^{13} + \)\(15\!\cdots\!29\)\( p^{3} T^{14} - \)\(30\!\cdots\!40\)\( p^{6} T^{15} + \)\(61\!\cdots\!96\)\( p^{9} T^{16} - \)\(11\!\cdots\!72\)\( p^{12} T^{17} + \)\(20\!\cdots\!81\)\( p^{15} T^{18} - \)\(33\!\cdots\!46\)\( p^{18} T^{19} + 561809202257972114 p^{21} T^{20} - 727288235169450 p^{24} T^{21} + 1132006280689 p^{27} T^{22} - 1012517050 p^{30} T^{23} + 1514714 p^{33} T^{24} - 660 p^{36} T^{25} + p^{39} T^{26} \) | |
| 61 | \( 1 + 748 T + 1601642 T^{2} + 1214492360 T^{3} + 1408964944095 T^{4} + 994999453227992 T^{5} + 850010892669524044 T^{6} + \)\(54\!\cdots\!78\)\( T^{7} + \)\(38\!\cdots\!99\)\( T^{8} + \)\(22\!\cdots\!92\)\( T^{9} + \)\(13\!\cdots\!82\)\( T^{10} + \)\(71\!\cdots\!22\)\( T^{11} + \)\(37\!\cdots\!13\)\( T^{12} + \)\(18\!\cdots\!96\)\( T^{13} + \)\(37\!\cdots\!13\)\( p^{3} T^{14} + \)\(71\!\cdots\!22\)\( p^{6} T^{15} + \)\(13\!\cdots\!82\)\( p^{9} T^{16} + \)\(22\!\cdots\!92\)\( p^{12} T^{17} + \)\(38\!\cdots\!99\)\( p^{15} T^{18} + \)\(54\!\cdots\!78\)\( p^{18} T^{19} + 850010892669524044 p^{21} T^{20} + 994999453227992 p^{24} T^{21} + 1408964944095 p^{27} T^{22} + 1214492360 p^{30} T^{23} + 1601642 p^{33} T^{24} + 748 p^{36} T^{25} + p^{39} T^{26} \) | |
| 67 | \( 1 + 82 T + 2031448 T^{2} + 68073298 T^{3} + 2077100941893 T^{4} - 634386521082 p T^{5} + 1434634607485680550 T^{6} - 91445931657965689070 T^{7} + \)\(75\!\cdots\!91\)\( T^{8} - \)\(66\!\cdots\!46\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} - \)\(30\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!17\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{13} + \)\(11\!\cdots\!17\)\( p^{3} T^{14} - \)\(30\!\cdots\!56\)\( p^{6} T^{15} + \)\(32\!\cdots\!04\)\( p^{9} T^{16} - \)\(66\!\cdots\!46\)\( p^{12} T^{17} + \)\(75\!\cdots\!91\)\( p^{15} T^{18} - 91445931657965689070 p^{18} T^{19} + 1434634607485680550 p^{21} T^{20} - 634386521082 p^{25} T^{21} + 2077100941893 p^{27} T^{22} + 68073298 p^{30} T^{23} + 2031448 p^{33} T^{24} + 82 p^{36} T^{25} + p^{39} T^{26} \) | |
| 71 | \( 1 - 2536 T + 5482712 T^{2} - 7998547756 T^{3} + 10437007995681 T^{4} - 11241347886179286 T^{5} + 11202545988851603638 T^{6} - \)\(99\!\cdots\!14\)\( T^{7} + \)\(82\!\cdots\!03\)\( T^{8} - \)\(63\!\cdots\!28\)\( T^{9} + \)\(46\!\cdots\!76\)\( T^{10} - \)\(31\!\cdots\!50\)\( T^{11} + \)\(20\!\cdots\!37\)\( T^{12} - \)\(12\!\cdots\!56\)\( T^{13} + \)\(20\!\cdots\!37\)\( p^{3} T^{14} - \)\(31\!\cdots\!50\)\( p^{6} T^{15} + \)\(46\!\cdots\!76\)\( p^{9} T^{16} - \)\(63\!\cdots\!28\)\( p^{12} T^{17} + \)\(82\!\cdots\!03\)\( p^{15} T^{18} - \)\(99\!\cdots\!14\)\( p^{18} T^{19} + 11202545988851603638 p^{21} T^{20} - 11241347886179286 p^{24} T^{21} + 10437007995681 p^{27} T^{22} - 7998547756 p^{30} T^{23} + 5482712 p^{33} T^{24} - 2536 p^{36} T^{25} + p^{39} T^{26} \) | |
| 73 | \( 1 - 994 T + 2801017 T^{2} - 2506624140 T^{3} + 3932943664956 T^{4} - 3213745602504326 T^{5} + 3759108749468414084 T^{6} - \)\(27\!\cdots\!24\)\( T^{7} + \)\(27\!\cdots\!44\)\( T^{8} - \)\(18\!\cdots\!26\)\( T^{9} + \)\(15\!\cdots\!25\)\( T^{10} - \)\(97\!\cdots\!34\)\( T^{11} + \)\(73\!\cdots\!17\)\( T^{12} - \)\(41\!\cdots\!48\)\( T^{13} + \)\(73\!\cdots\!17\)\( p^{3} T^{14} - \)\(97\!\cdots\!34\)\( p^{6} T^{15} + \)\(15\!\cdots\!25\)\( p^{9} T^{16} - \)\(18\!\cdots\!26\)\( p^{12} T^{17} + \)\(27\!\cdots\!44\)\( p^{15} T^{18} - \)\(27\!\cdots\!24\)\( p^{18} T^{19} + 3759108749468414084 p^{21} T^{20} - 3213745602504326 p^{24} T^{21} + 3932943664956 p^{27} T^{22} - 2506624140 p^{30} T^{23} + 2801017 p^{33} T^{24} - 994 p^{36} T^{25} + p^{39} T^{26} \) | |
| 79 | \( 1 - 2072 T + 5463115 T^{2} - 7723446292 T^{3} + 12163683962897 T^{4} - 13405980734003370 T^{5} + 16059514676065356782 T^{6} - \)\(18\!\cdots\!44\)\( p T^{7} + \)\(14\!\cdots\!42\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(78\!\cdots\!56\)\( T^{11} + \)\(63\!\cdots\!77\)\( T^{12} - \)\(42\!\cdots\!28\)\( T^{13} + \)\(63\!\cdots\!77\)\( p^{3} T^{14} - \)\(78\!\cdots\!56\)\( p^{6} T^{15} + \)\(10\!\cdots\!24\)\( p^{9} T^{16} - \)\(12\!\cdots\!44\)\( p^{12} T^{17} + \)\(14\!\cdots\!42\)\( p^{15} T^{18} - \)\(18\!\cdots\!44\)\( p^{19} T^{19} + 16059514676065356782 p^{21} T^{20} - 13405980734003370 p^{24} T^{21} + 12163683962897 p^{27} T^{22} - 7723446292 p^{30} T^{23} + 5463115 p^{33} T^{24} - 2072 p^{36} T^{25} + p^{39} T^{26} \) | |
| 83 | \( 1 - 2724 T + 7210489 T^{2} - 12230091100 T^{3} + 19157472651624 T^{4} - 23881766961480464 T^{5} + 27404327846525438912 T^{6} - \)\(26\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!01\)\( T^{8} - \)\(19\!\cdots\!12\)\( T^{9} + \)\(15\!\cdots\!21\)\( T^{10} - \)\(10\!\cdots\!56\)\( T^{11} + \)\(77\!\cdots\!84\)\( T^{12} - \)\(68\!\cdots\!76\)\( p T^{13} + \)\(77\!\cdots\!84\)\( p^{3} T^{14} - \)\(10\!\cdots\!56\)\( p^{6} T^{15} + \)\(15\!\cdots\!21\)\( p^{9} T^{16} - \)\(19\!\cdots\!12\)\( p^{12} T^{17} + \)\(24\!\cdots\!01\)\( p^{15} T^{18} - \)\(26\!\cdots\!48\)\( p^{18} T^{19} + 27404327846525438912 p^{21} T^{20} - 23881766961480464 p^{24} T^{21} + 19157472651624 p^{27} T^{22} - 12230091100 p^{30} T^{23} + 7210489 p^{33} T^{24} - 2724 p^{36} T^{25} + p^{39} T^{26} \) | |
| 89 | \( 1 - 1756 T + 7082478 T^{2} - 10328865210 T^{3} + 23330915443445 T^{4} - 29058609317949882 T^{5} + 48055609852059761835 T^{6} - \)\(52\!\cdots\!10\)\( T^{7} + \)\(70\!\cdots\!58\)\( T^{8} - \)\(67\!\cdots\!12\)\( T^{9} + \)\(77\!\cdots\!40\)\( T^{10} - \)\(67\!\cdots\!98\)\( T^{11} + \)\(68\!\cdots\!35\)\( T^{12} - \)\(52\!\cdots\!36\)\( T^{13} + \)\(68\!\cdots\!35\)\( p^{3} T^{14} - \)\(67\!\cdots\!98\)\( p^{6} T^{15} + \)\(77\!\cdots\!40\)\( p^{9} T^{16} - \)\(67\!\cdots\!12\)\( p^{12} T^{17} + \)\(70\!\cdots\!58\)\( p^{15} T^{18} - \)\(52\!\cdots\!10\)\( p^{18} T^{19} + 48055609852059761835 p^{21} T^{20} - 29058609317949882 p^{24} T^{21} + 23330915443445 p^{27} T^{22} - 10328865210 p^{30} T^{23} + 7082478 p^{33} T^{24} - 1756 p^{36} T^{25} + p^{39} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.13845937502476368015353108199, −4.02117145695810468569571797454, −3.86384963196572147779223677337, −3.82932213156270685492067820474, −3.70383390008827450068409300703, −3.68434343385559595599833756862, −3.65259475825099925833625066918, −3.56074493837258258759306387655, −3.37428817474785125071650511244, −3.24600753361693468864037727474, −3.14249727313849898898880136348, −2.76089977322000244205168738123, −2.69948999999887572817687011662, −2.58960634048273139930644364653, −2.33522756679068813051543614727, −2.19867452544458524327732986985, −2.10085434100086360670396491004, −1.78741894662634403657861151581, −1.66697207027680963639256884403, −1.38249876317833720931603470266, −1.30605988712263902858740297994, −1.21056432765674554401984197931, −0.952392242139031222634633194133, −0.66053323362024886833120364308, −0.38414590477859108297990410584, 0.38414590477859108297990410584, 0.66053323362024886833120364308, 0.952392242139031222634633194133, 1.21056432765674554401984197931, 1.30605988712263902858740297994, 1.38249876317833720931603470266, 1.66697207027680963639256884403, 1.78741894662634403657861151581, 2.10085434100086360670396491004, 2.19867452544458524327732986985, 2.33522756679068813051543614727, 2.58960634048273139930644364653, 2.69948999999887572817687011662, 2.76089977322000244205168738123, 3.14249727313849898898880136348, 3.24600753361693468864037727474, 3.37428817474785125071650511244, 3.56074493837258258759306387655, 3.65259475825099925833625066918, 3.68434343385559595599833756862, 3.70383390008827450068409300703, 3.82932213156270685492067820474, 3.86384963196572147779223677337, 4.02117145695810468569571797454, 4.13845937502476368015353108199
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.