Dirichlet series
L(s) = 1 | + 5·2-s − 3-s + 30·4-s − 60·5-s − 5·6-s + 38·7-s + 95·8-s + 89·9-s − 300·10-s + 181·11-s − 30·12-s + 190·14-s + 60·15-s + 338·16-s + 55·17-s + 445·18-s + 161·19-s − 1.80e3·20-s − 38·21-s + 905·22-s + 204·23-s − 95·24-s + 982·25-s − 552·27-s + 1.14e3·28-s − 280·29-s + 300·30-s + ⋯ |
L(s) = 1 | + 1.76·2-s − 0.192·3-s + 15/4·4-s − 5.36·5-s − 0.340·6-s + 2.05·7-s + 4.19·8-s + 3.29·9-s − 9.48·10-s + 4.96·11-s − 0.721·12-s + 3.62·14-s + 1.03·15-s + 5.28·16-s + 0.784·17-s + 5.82·18-s + 1.94·19-s − 20.1·20-s − 0.394·21-s + 8.77·22-s + 1.84·23-s − 0.807·24-s + 7.85·25-s − 3.93·27-s + 7.69·28-s − 1.79·29-s + 1.82·30-s + ⋯ |
Functional equation
Invariants
Degree: | \(36\) |
Conductor: | \(13^{36}\) |
Sign: | $1$ |
Analytic conductor: | \(9.49620\times 10^{17}\) |
Root analytic conductor: | \(3.15774\) |
Motivic weight: | \(3\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((36,\ 13^{36} ,\ ( \ : [3/2]^{18} ),\ 1 )\) |
Particular Values
\(L(2)\) | \(\approx\) | \(33.13008848\) |
\(L(\frac12)\) | \(\approx\) | \(33.13008848\) |
\(L(\frac{5}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 - 5 T - 5 T^{2} + 5 p^{4} T^{3} - 113 T^{4} - 125 p^{2} T^{5} + 2421 T^{6} - 2867 T^{7} - 3717 p^{2} T^{8} + 26481 p T^{9} + 2165 p^{4} T^{10} - 183553 p T^{11} + 93085 T^{12} + 930683 p T^{13} - 451715 p^{2} T^{14} - 1271587 p^{4} T^{15} + 5046691 p^{4} T^{16} + 558401 p^{7} T^{17} - 13751043 p^{6} T^{18} + 558401 p^{10} T^{19} + 5046691 p^{10} T^{20} - 1271587 p^{13} T^{21} - 451715 p^{14} T^{22} + 930683 p^{16} T^{23} + 93085 p^{18} T^{24} - 183553 p^{22} T^{25} + 2165 p^{28} T^{26} + 26481 p^{28} T^{27} - 3717 p^{32} T^{28} - 2867 p^{33} T^{29} + 2421 p^{36} T^{30} - 125 p^{41} T^{31} - 113 p^{42} T^{32} + 5 p^{49} T^{33} - 5 p^{48} T^{34} - 5 p^{51} T^{35} + p^{54} T^{36} \) |
3 | \( 1 + T - 88 T^{2} + 125 p T^{3} + 409 p^{2} T^{4} - 34786 T^{5} - 3673 T^{6} + 1260532 T^{7} - 5105383 T^{8} - 579149 p^{3} T^{9} + 16635910 p^{2} T^{10} - 150954817 T^{11} - 465483760 T^{12} - 9971656481 T^{13} + 1849387190 T^{14} + 302947002901 p T^{15} - 495799032625 p^{2} T^{16} - 14501929124200 T^{17} + 207410196954085 T^{18} - 14501929124200 p^{3} T^{19} - 495799032625 p^{8} T^{20} + 302947002901 p^{10} T^{21} + 1849387190 p^{12} T^{22} - 9971656481 p^{15} T^{23} - 465483760 p^{18} T^{24} - 150954817 p^{21} T^{25} + 16635910 p^{26} T^{26} - 579149 p^{30} T^{27} - 5105383 p^{30} T^{28} + 1260532 p^{33} T^{29} - 3673 p^{36} T^{30} - 34786 p^{39} T^{31} + 409 p^{44} T^{32} + 125 p^{46} T^{33} - 88 p^{48} T^{34} + p^{51} T^{35} + p^{54} T^{36} \) | |
5 | \( ( 1 + 6 p T + 859 T^{2} + 17314 T^{3} + 340943 T^{4} + 5555612 T^{5} + 85166304 T^{6} + 1151915472 T^{7} + 14768999549 T^{8} + 170409744152 T^{9} + 14768999549 p^{3} T^{10} + 1151915472 p^{6} T^{11} + 85166304 p^{9} T^{12} + 5555612 p^{12} T^{13} + 340943 p^{15} T^{14} + 17314 p^{18} T^{15} + 859 p^{21} T^{16} + 6 p^{25} T^{17} + p^{27} T^{18} )^{2} \) | |
7 | \( 1 - 38 T - 620 T^{2} + 31800 T^{3} + 338763 T^{4} - 14217512 T^{5} - 3605440 p^{2} T^{6} + 3729180788 T^{7} + 64225457875 T^{8} + 150016924480 T^{9} - 22974006228702 T^{10} - 507437261666584 T^{11} + 5431353147153050 T^{12} + 242484969403614160 T^{13} + 719781609802199729 T^{14} - 74602539403577047146 T^{15} - \)\(12\!\cdots\!68\)\( T^{16} + \)\(97\!\cdots\!26\)\( T^{17} + \)\(63\!\cdots\!75\)\( T^{18} + \)\(97\!\cdots\!26\)\( p^{3} T^{19} - \)\(12\!\cdots\!68\)\( p^{6} T^{20} - 74602539403577047146 p^{9} T^{21} + 719781609802199729 p^{12} T^{22} + 242484969403614160 p^{15} T^{23} + 5431353147153050 p^{18} T^{24} - 507437261666584 p^{21} T^{25} - 22974006228702 p^{24} T^{26} + 150016924480 p^{27} T^{27} + 64225457875 p^{30} T^{28} + 3729180788 p^{33} T^{29} - 3605440 p^{38} T^{30} - 14217512 p^{39} T^{31} + 338763 p^{42} T^{32} + 31800 p^{45} T^{33} - 620 p^{48} T^{34} - 38 p^{51} T^{35} + p^{54} T^{36} \) | |
11 | \( 1 - 181 T + 9948 T^{2} + 55953 T^{3} - 17191591 T^{4} - 356652938 T^{5} + 46632465051 T^{6} + 464521680020 T^{7} - 61627390349683 T^{8} - 2679999554420613 T^{9} + 158448016353971594 T^{10} + 2648000354329436885 T^{11} - \)\(15\!\cdots\!64\)\( T^{12} - \)\(55\!\cdots\!27\)\( T^{13} + \)\(21\!\cdots\!10\)\( T^{14} + \)\(41\!\cdots\!53\)\( T^{15} - \)\(14\!\cdots\!85\)\( T^{16} - \)\(54\!\cdots\!32\)\( T^{17} + \)\(34\!\cdots\!89\)\( T^{18} - \)\(54\!\cdots\!32\)\( p^{3} T^{19} - \)\(14\!\cdots\!85\)\( p^{6} T^{20} + \)\(41\!\cdots\!53\)\( p^{9} T^{21} + \)\(21\!\cdots\!10\)\( p^{12} T^{22} - \)\(55\!\cdots\!27\)\( p^{15} T^{23} - \)\(15\!\cdots\!64\)\( p^{18} T^{24} + 2648000354329436885 p^{21} T^{25} + 158448016353971594 p^{24} T^{26} - 2679999554420613 p^{27} T^{27} - 61627390349683 p^{30} T^{28} + 464521680020 p^{33} T^{29} + 46632465051 p^{36} T^{30} - 356652938 p^{39} T^{31} - 17191591 p^{42} T^{32} + 55953 p^{45} T^{33} + 9948 p^{48} T^{34} - 181 p^{51} T^{35} + p^{54} T^{36} \) | |
17 | \( 1 - 55 T - 17756 T^{2} + 888255 T^{3} + 165130699 T^{4} - 8064583152 T^{5} - 927459314241 T^{6} + 51004731718318 T^{7} + 2890328590914139 T^{8} - 292844451004211711 T^{9} + 1632168354585153246 T^{10} + \)\(18\!\cdots\!77\)\( T^{11} - \)\(91\!\cdots\!26\)\( T^{12} - \)\(10\!\cdots\!73\)\( T^{13} + \)\(80\!\cdots\!58\)\( T^{14} + \)\(46\!\cdots\!03\)\( T^{15} - \)\(51\!\cdots\!07\)\( T^{16} - \)\(95\!\cdots\!58\)\( T^{17} + \)\(27\!\cdots\!67\)\( T^{18} - \)\(95\!\cdots\!58\)\( p^{3} T^{19} - \)\(51\!\cdots\!07\)\( p^{6} T^{20} + \)\(46\!\cdots\!03\)\( p^{9} T^{21} + \)\(80\!\cdots\!58\)\( p^{12} T^{22} - \)\(10\!\cdots\!73\)\( p^{15} T^{23} - \)\(91\!\cdots\!26\)\( p^{18} T^{24} + \)\(18\!\cdots\!77\)\( p^{21} T^{25} + 1632168354585153246 p^{24} T^{26} - 292844451004211711 p^{27} T^{27} + 2890328590914139 p^{30} T^{28} + 51004731718318 p^{33} T^{29} - 927459314241 p^{36} T^{30} - 8064583152 p^{39} T^{31} + 165130699 p^{42} T^{32} + 888255 p^{45} T^{33} - 17756 p^{48} T^{34} - 55 p^{51} T^{35} + p^{54} T^{36} \) | |
19 | \( 1 - 161 T - 24518 T^{2} + 5314715 T^{3} + 290439061 T^{4} - 93407858712 T^{5} - 1946784476247 T^{6} + 1160126327903368 T^{7} + 1988048802051383 T^{8} - 11349795969532892501 T^{9} + \)\(16\!\cdots\!48\)\( T^{10} + \)\(90\!\cdots\!77\)\( T^{11} - \)\(28\!\cdots\!02\)\( T^{12} - \)\(58\!\cdots\!33\)\( T^{13} + \)\(32\!\cdots\!58\)\( T^{14} + \)\(28\!\cdots\!63\)\( T^{15} - \)\(29\!\cdots\!39\)\( T^{16} - \)\(68\!\cdots\!00\)\( T^{17} + \)\(21\!\cdots\!57\)\( T^{18} - \)\(68\!\cdots\!00\)\( p^{3} T^{19} - \)\(29\!\cdots\!39\)\( p^{6} T^{20} + \)\(28\!\cdots\!63\)\( p^{9} T^{21} + \)\(32\!\cdots\!58\)\( p^{12} T^{22} - \)\(58\!\cdots\!33\)\( p^{15} T^{23} - \)\(28\!\cdots\!02\)\( p^{18} T^{24} + \)\(90\!\cdots\!77\)\( p^{21} T^{25} + \)\(16\!\cdots\!48\)\( p^{24} T^{26} - 11349795969532892501 p^{27} T^{27} + 1988048802051383 p^{30} T^{28} + 1160126327903368 p^{33} T^{29} - 1946784476247 p^{36} T^{30} - 93407858712 p^{39} T^{31} + 290439061 p^{42} T^{32} + 5314715 p^{45} T^{33} - 24518 p^{48} T^{34} - 161 p^{51} T^{35} + p^{54} T^{36} \) | |
23 | \( 1 - 204 T - 38756 T^{2} + 12501872 T^{3} + 306851848 T^{4} - 345839924612 T^{5} + 12154121680850 T^{6} + 5865186708315724 T^{7} - 437719529286705000 T^{8} - 67115888508638332640 T^{9} + \)\(76\!\cdots\!96\)\( T^{10} + \)\(49\!\cdots\!92\)\( T^{11} - \)\(83\!\cdots\!65\)\( T^{12} - \)\(70\!\cdots\!08\)\( p T^{13} + \)\(52\!\cdots\!36\)\( T^{14} - \)\(56\!\cdots\!32\)\( T^{15} - \)\(45\!\cdots\!60\)\( T^{16} + \)\(55\!\cdots\!56\)\( T^{17} - \)\(21\!\cdots\!72\)\( T^{18} + \)\(55\!\cdots\!56\)\( p^{3} T^{19} - \)\(45\!\cdots\!60\)\( p^{6} T^{20} - \)\(56\!\cdots\!32\)\( p^{9} T^{21} + \)\(52\!\cdots\!36\)\( p^{12} T^{22} - \)\(70\!\cdots\!08\)\( p^{16} T^{23} - \)\(83\!\cdots\!65\)\( p^{18} T^{24} + \)\(49\!\cdots\!92\)\( p^{21} T^{25} + \)\(76\!\cdots\!96\)\( p^{24} T^{26} - 67115888508638332640 p^{27} T^{27} - 437719529286705000 p^{30} T^{28} + 5865186708315724 p^{33} T^{29} + 12154121680850 p^{36} T^{30} - 345839924612 p^{39} T^{31} + 306851848 p^{42} T^{32} + 12501872 p^{45} T^{33} - 38756 p^{48} T^{34} - 204 p^{51} T^{35} + p^{54} T^{36} \) | |
29 | \( 1 + 280 T - 92566 T^{2} - 36256000 T^{3} + 3348031575 T^{4} + 2249910525728 T^{5} - 21002259657414 T^{6} - 86687731820936266 T^{7} - 2749201725668908063 T^{8} + \)\(22\!\cdots\!52\)\( T^{9} + \)\(95\!\cdots\!52\)\( T^{10} - \)\(38\!\cdots\!54\)\( T^{11} + \)\(33\!\cdots\!64\)\( T^{12} + \)\(14\!\cdots\!92\)\( p T^{13} - \)\(13\!\cdots\!41\)\( T^{14} - \)\(23\!\cdots\!82\)\( T^{15} + \)\(63\!\cdots\!42\)\( T^{16} + \)\(17\!\cdots\!00\)\( T^{17} - \)\(18\!\cdots\!83\)\( T^{18} + \)\(17\!\cdots\!00\)\( p^{3} T^{19} + \)\(63\!\cdots\!42\)\( p^{6} T^{20} - \)\(23\!\cdots\!82\)\( p^{9} T^{21} - \)\(13\!\cdots\!41\)\( p^{12} T^{22} + \)\(14\!\cdots\!92\)\( p^{16} T^{23} + \)\(33\!\cdots\!64\)\( p^{18} T^{24} - \)\(38\!\cdots\!54\)\( p^{21} T^{25} + \)\(95\!\cdots\!52\)\( p^{24} T^{26} + \)\(22\!\cdots\!52\)\( p^{27} T^{27} - 2749201725668908063 p^{30} T^{28} - 86687731820936266 p^{33} T^{29} - 21002259657414 p^{36} T^{30} + 2249910525728 p^{39} T^{31} + 3348031575 p^{42} T^{32} - 36256000 p^{45} T^{33} - 92566 p^{48} T^{34} + 280 p^{51} T^{35} + p^{54} T^{36} \) | |
31 | \( ( 1 + 706 T + 416348 T^{2} + 167129464 T^{3} + 1911356387 p T^{4} + 17113845091756 T^{5} + 4471497849821796 T^{6} + 1005049239010198696 T^{7} + \)\(20\!\cdots\!78\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(20\!\cdots\!78\)\( p^{3} T^{10} + 1005049239010198696 p^{6} T^{11} + 4471497849821796 p^{9} T^{12} + 17113845091756 p^{12} T^{13} + 1911356387 p^{16} T^{14} + 167129464 p^{18} T^{15} + 416348 p^{21} T^{16} + 706 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
37 | \( 1 - 298 T - 134975 T^{2} - 1052114 T^{3} + 22249721318 T^{4} + 2437271850390 T^{5} - 1200901989788393 T^{6} - 475005049275098254 T^{7} + 12505203597200012412 T^{8} + \)\(81\!\cdots\!66\)\( p T^{9} + \)\(55\!\cdots\!29\)\( T^{10} - \)\(85\!\cdots\!22\)\( T^{11} - \)\(41\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!10\)\( T^{13} + \)\(14\!\cdots\!71\)\( T^{14} + \)\(36\!\cdots\!58\)\( T^{15} + \)\(24\!\cdots\!64\)\( T^{16} - \)\(30\!\cdots\!62\)\( p T^{17} - \)\(28\!\cdots\!89\)\( T^{18} - \)\(30\!\cdots\!62\)\( p^{4} T^{19} + \)\(24\!\cdots\!64\)\( p^{6} T^{20} + \)\(36\!\cdots\!58\)\( p^{9} T^{21} + \)\(14\!\cdots\!71\)\( p^{12} T^{22} - \)\(39\!\cdots\!10\)\( p^{15} T^{23} - \)\(41\!\cdots\!26\)\( p^{18} T^{24} - \)\(85\!\cdots\!22\)\( p^{21} T^{25} + \)\(55\!\cdots\!29\)\( p^{24} T^{26} + \)\(81\!\cdots\!66\)\( p^{28} T^{27} + 12505203597200012412 p^{30} T^{28} - 475005049275098254 p^{33} T^{29} - 1200901989788393 p^{36} T^{30} + 2437271850390 p^{39} T^{31} + 22249721318 p^{42} T^{32} - 1052114 p^{45} T^{33} - 134975 p^{48} T^{34} - 298 p^{51} T^{35} + p^{54} T^{36} \) | |
41 | \( 1 - 1201 T + 327977 T^{2} + 99605028 T^{3} - 16852540214 T^{4} - 30459566181494 T^{5} + 5014151643050972 T^{6} + 3184573020734915249 T^{7} - 71030389086130070892 T^{8} - \)\(42\!\cdots\!10\)\( T^{9} - \)\(18\!\cdots\!87\)\( T^{10} + \)\(36\!\cdots\!98\)\( T^{11} + \)\(59\!\cdots\!03\)\( T^{12} - \)\(27\!\cdots\!41\)\( T^{13} - \)\(77\!\cdots\!14\)\( T^{14} + \)\(14\!\cdots\!54\)\( T^{15} + \)\(74\!\cdots\!30\)\( T^{16} - \)\(36\!\cdots\!71\)\( T^{17} - \)\(57\!\cdots\!52\)\( T^{18} - \)\(36\!\cdots\!71\)\( p^{3} T^{19} + \)\(74\!\cdots\!30\)\( p^{6} T^{20} + \)\(14\!\cdots\!54\)\( p^{9} T^{21} - \)\(77\!\cdots\!14\)\( p^{12} T^{22} - \)\(27\!\cdots\!41\)\( p^{15} T^{23} + \)\(59\!\cdots\!03\)\( p^{18} T^{24} + \)\(36\!\cdots\!98\)\( p^{21} T^{25} - \)\(18\!\cdots\!87\)\( p^{24} T^{26} - \)\(42\!\cdots\!10\)\( p^{27} T^{27} - 71030389086130070892 p^{30} T^{28} + 3184573020734915249 p^{33} T^{29} + 5014151643050972 p^{36} T^{30} - 30459566181494 p^{39} T^{31} - 16852540214 p^{42} T^{32} + 99605028 p^{45} T^{33} + 327977 p^{48} T^{34} - 1201 p^{51} T^{35} + p^{54} T^{36} \) | |
43 | \( 1 - 533 T - 278395 T^{2} + 104585676 T^{3} + 80771741583 T^{4} - 12223492764483 T^{5} - 14508095787212709 T^{6} + 16443906934953719 T^{7} + \)\(18\!\cdots\!35\)\( T^{8} + \)\(19\!\cdots\!73\)\( T^{9} - \)\(17\!\cdots\!88\)\( T^{10} - \)\(32\!\cdots\!31\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} + \)\(31\!\cdots\!66\)\( T^{13} - \)\(57\!\cdots\!29\)\( T^{14} - \)\(19\!\cdots\!12\)\( T^{15} + \)\(23\!\cdots\!69\)\( T^{16} + \)\(58\!\cdots\!89\)\( T^{17} - \)\(13\!\cdots\!03\)\( T^{18} + \)\(58\!\cdots\!89\)\( p^{3} T^{19} + \)\(23\!\cdots\!69\)\( p^{6} T^{20} - \)\(19\!\cdots\!12\)\( p^{9} T^{21} - \)\(57\!\cdots\!29\)\( p^{12} T^{22} + \)\(31\!\cdots\!66\)\( p^{15} T^{23} + \)\(11\!\cdots\!86\)\( p^{18} T^{24} - \)\(32\!\cdots\!31\)\( p^{21} T^{25} - \)\(17\!\cdots\!88\)\( p^{24} T^{26} + \)\(19\!\cdots\!73\)\( p^{27} T^{27} + \)\(18\!\cdots\!35\)\( p^{30} T^{28} + 16443906934953719 p^{33} T^{29} - 14508095787212709 p^{36} T^{30} - 12223492764483 p^{39} T^{31} + 80771741583 p^{42} T^{32} + 104585676 p^{45} T^{33} - 278395 p^{48} T^{34} - 533 p^{51} T^{35} + p^{54} T^{36} \) | |
47 | \( ( 1 + 956 T + 1071019 T^{2} + 666300230 T^{3} + 434228172393 T^{4} + 202420083652096 T^{5} + 97312087273615934 T^{6} + 36574221973341845292 T^{7} + \)\(14\!\cdots\!23\)\( T^{8} + \)\(44\!\cdots\!76\)\( T^{9} + \)\(14\!\cdots\!23\)\( p^{3} T^{10} + 36574221973341845292 p^{6} T^{11} + 97312087273615934 p^{9} T^{12} + 202420083652096 p^{12} T^{13} + 434228172393 p^{15} T^{14} + 666300230 p^{18} T^{15} + 1071019 p^{21} T^{16} + 956 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
53 | \( ( 1 + 278 T + 773518 T^{2} + 207620330 T^{3} + 319892405009 T^{4} + 77727314367624 T^{5} + 87908254531904768 T^{6} + 19135768589714679110 T^{7} + \)\(17\!\cdots\!16\)\( T^{8} + \)\(33\!\cdots\!56\)\( T^{9} + \)\(17\!\cdots\!16\)\( p^{3} T^{10} + 19135768589714679110 p^{6} T^{11} + 87908254531904768 p^{9} T^{12} + 77727314367624 p^{12} T^{13} + 319892405009 p^{15} T^{14} + 207620330 p^{18} T^{15} + 773518 p^{21} T^{16} + 278 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
59 | \( 1 - 1377 T - 205215 T^{2} + 887158676 T^{3} + 134978179534 T^{4} - 489716129743400 T^{5} - 29769900343747010 T^{6} + \)\(17\!\cdots\!79\)\( T^{7} + \)\(97\!\cdots\!20\)\( T^{8} - \)\(51\!\cdots\!90\)\( T^{9} - \)\(39\!\cdots\!73\)\( T^{10} + \)\(11\!\cdots\!40\)\( T^{11} - \)\(76\!\cdots\!85\)\( T^{12} - \)\(21\!\cdots\!63\)\( T^{13} + \)\(49\!\cdots\!88\)\( T^{14} + \)\(28\!\cdots\!18\)\( T^{15} - \)\(16\!\cdots\!02\)\( T^{16} - \)\(21\!\cdots\!93\)\( T^{17} + \)\(40\!\cdots\!56\)\( T^{18} - \)\(21\!\cdots\!93\)\( p^{3} T^{19} - \)\(16\!\cdots\!02\)\( p^{6} T^{20} + \)\(28\!\cdots\!18\)\( p^{9} T^{21} + \)\(49\!\cdots\!88\)\( p^{12} T^{22} - \)\(21\!\cdots\!63\)\( p^{15} T^{23} - \)\(76\!\cdots\!85\)\( p^{18} T^{24} + \)\(11\!\cdots\!40\)\( p^{21} T^{25} - \)\(39\!\cdots\!73\)\( p^{24} T^{26} - \)\(51\!\cdots\!90\)\( p^{27} T^{27} + \)\(97\!\cdots\!20\)\( p^{30} T^{28} + \)\(17\!\cdots\!79\)\( p^{33} T^{29} - 29769900343747010 p^{36} T^{30} - 489716129743400 p^{39} T^{31} + 134978179534 p^{42} T^{32} + 887158676 p^{45} T^{33} - 205215 p^{48} T^{34} - 1377 p^{51} T^{35} + p^{54} T^{36} \) | |
61 | \( 1 - 136 T - 893327 T^{2} - 5194684 T^{3} + 362395503618 T^{4} + 40471015312996 T^{5} - 74056679658938765 T^{6} - 7091377177374787044 T^{7} + \)\(11\!\cdots\!28\)\( p T^{8} - \)\(53\!\cdots\!80\)\( T^{9} - \)\(19\!\cdots\!55\)\( T^{10} + \)\(26\!\cdots\!64\)\( T^{11} + \)\(13\!\cdots\!90\)\( T^{12} - \)\(35\!\cdots\!60\)\( T^{13} - \)\(33\!\cdots\!61\)\( T^{14} - \)\(41\!\cdots\!28\)\( T^{15} + \)\(88\!\cdots\!44\)\( T^{16} + \)\(10\!\cdots\!12\)\( T^{17} + \)\(85\!\cdots\!35\)\( T^{18} + \)\(10\!\cdots\!12\)\( p^{3} T^{19} + \)\(88\!\cdots\!44\)\( p^{6} T^{20} - \)\(41\!\cdots\!28\)\( p^{9} T^{21} - \)\(33\!\cdots\!61\)\( p^{12} T^{22} - \)\(35\!\cdots\!60\)\( p^{15} T^{23} + \)\(13\!\cdots\!90\)\( p^{18} T^{24} + \)\(26\!\cdots\!64\)\( p^{21} T^{25} - \)\(19\!\cdots\!55\)\( p^{24} T^{26} - \)\(53\!\cdots\!80\)\( p^{27} T^{27} + \)\(11\!\cdots\!28\)\( p^{31} T^{28} - 7091377177374787044 p^{33} T^{29} - 74056679658938765 p^{36} T^{30} + 40471015312996 p^{39} T^{31} + 362395503618 p^{42} T^{32} - 5194684 p^{45} T^{33} - 893327 p^{48} T^{34} - 136 p^{51} T^{35} + p^{54} T^{36} \) | |
67 | \( 1 + 931 T - 698771 T^{2} - 1240782812 T^{3} - 153832776213 T^{4} + 585043557396701 T^{5} + 331913903304290227 T^{6} - 65236424473939264533 T^{7} - \)\(13\!\cdots\!89\)\( T^{8} - \)\(42\!\cdots\!47\)\( T^{9} + \)\(15\!\cdots\!08\)\( T^{10} + \)\(17\!\cdots\!97\)\( T^{11} + \)\(45\!\cdots\!74\)\( T^{12} - \)\(25\!\cdots\!58\)\( T^{13} - \)\(20\!\cdots\!93\)\( T^{14} + \)\(11\!\cdots\!08\)\( T^{15} + \)\(54\!\cdots\!09\)\( T^{16} - \)\(16\!\cdots\!91\)\( T^{17} - \)\(15\!\cdots\!79\)\( T^{18} - \)\(16\!\cdots\!91\)\( p^{3} T^{19} + \)\(54\!\cdots\!09\)\( p^{6} T^{20} + \)\(11\!\cdots\!08\)\( p^{9} T^{21} - \)\(20\!\cdots\!93\)\( p^{12} T^{22} - \)\(25\!\cdots\!58\)\( p^{15} T^{23} + \)\(45\!\cdots\!74\)\( p^{18} T^{24} + \)\(17\!\cdots\!97\)\( p^{21} T^{25} + \)\(15\!\cdots\!08\)\( p^{24} T^{26} - \)\(42\!\cdots\!47\)\( p^{27} T^{27} - \)\(13\!\cdots\!89\)\( p^{30} T^{28} - 65236424473939264533 p^{33} T^{29} + 331913903304290227 p^{36} T^{30} + 585043557396701 p^{39} T^{31} - 153832776213 p^{42} T^{32} - 1240782812 p^{45} T^{33} - 698771 p^{48} T^{34} + 931 p^{51} T^{35} + p^{54} T^{36} \) | |
71 | \( 1 - 2046 T + 656630 T^{2} + 1598035416 T^{3} - 1514168802767 T^{4} + 55208225170502 T^{5} + 459186690642448730 T^{6} - \)\(26\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!81\)\( T^{8} - \)\(95\!\cdots\!90\)\( T^{9} - \)\(10\!\cdots\!46\)\( T^{10} + \)\(13\!\cdots\!38\)\( T^{11} - \)\(20\!\cdots\!64\)\( T^{12} - \)\(27\!\cdots\!82\)\( T^{13} + \)\(15\!\cdots\!51\)\( T^{14} - \)\(10\!\cdots\!48\)\( T^{15} + \)\(56\!\cdots\!20\)\( T^{16} + \)\(33\!\cdots\!20\)\( T^{17} - \)\(51\!\cdots\!17\)\( T^{18} + \)\(33\!\cdots\!20\)\( p^{3} T^{19} + \)\(56\!\cdots\!20\)\( p^{6} T^{20} - \)\(10\!\cdots\!48\)\( p^{9} T^{21} + \)\(15\!\cdots\!51\)\( p^{12} T^{22} - \)\(27\!\cdots\!82\)\( p^{15} T^{23} - \)\(20\!\cdots\!64\)\( p^{18} T^{24} + \)\(13\!\cdots\!38\)\( p^{21} T^{25} - \)\(10\!\cdots\!46\)\( p^{24} T^{26} - \)\(95\!\cdots\!90\)\( p^{27} T^{27} + \)\(20\!\cdots\!81\)\( p^{30} T^{28} - \)\(26\!\cdots\!92\)\( p^{33} T^{29} + 459186690642448730 p^{36} T^{30} + 55208225170502 p^{39} T^{31} - 1514168802767 p^{42} T^{32} + 1598035416 p^{45} T^{33} + 656630 p^{48} T^{34} - 2046 p^{51} T^{35} + p^{54} T^{36} \) | |
73 | \( ( 1 - 45 T + 1891466 T^{2} + 33830186 T^{3} + 1674355114259 T^{4} + 33519963542817 T^{5} + 983076362313015860 T^{6} - 18994886524736826951 T^{7} + \)\(45\!\cdots\!72\)\( T^{8} - \)\(18\!\cdots\!55\)\( T^{9} + \)\(45\!\cdots\!72\)\( p^{3} T^{10} - 18994886524736826951 p^{6} T^{11} + 983076362313015860 p^{9} T^{12} + 33519963542817 p^{12} T^{13} + 1674355114259 p^{15} T^{14} + 33830186 p^{18} T^{15} + 1891466 p^{21} T^{16} - 45 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
79 | \( ( 1 - 412 T + 3404882 T^{2} - 1342603600 T^{3} + 5422429448055 T^{4} - 2004760282589858 T^{5} + 5353334618487262810 T^{6} - \)\(18\!\cdots\!82\)\( T^{7} + \)\(36\!\cdots\!76\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{9} + \)\(36\!\cdots\!76\)\( p^{3} T^{10} - \)\(18\!\cdots\!82\)\( p^{6} T^{11} + 5353334618487262810 p^{9} T^{12} - 2004760282589858 p^{12} T^{13} + 5422429448055 p^{15} T^{14} - 1342603600 p^{18} T^{15} + 3404882 p^{21} T^{16} - 412 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
83 | \( ( 1 + 3709 T + 9253460 T^{2} + 15842855210 T^{3} + 21895132541837 T^{4} + 24308978625331839 T^{5} + 23407062219850258008 T^{6} + \)\(19\!\cdots\!91\)\( T^{7} + \)\(15\!\cdots\!88\)\( T^{8} + \)\(11\!\cdots\!75\)\( T^{9} + \)\(15\!\cdots\!88\)\( p^{3} T^{10} + \)\(19\!\cdots\!91\)\( p^{6} T^{11} + 23407062219850258008 p^{9} T^{12} + 24308978625331839 p^{12} T^{13} + 21895132541837 p^{15} T^{14} + 15842855210 p^{18} T^{15} + 9253460 p^{21} T^{16} + 3709 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
89 | \( 1 - 1663 T - 3307889 T^{2} + 4389505430 T^{3} + 9575717323487 T^{4} - 7610410478848107 T^{5} - 18811569012189027633 T^{6} + \)\(86\!\cdots\!97\)\( T^{7} + \)\(27\!\cdots\!53\)\( T^{8} - \)\(66\!\cdots\!83\)\( T^{9} - \)\(32\!\cdots\!12\)\( T^{10} + \)\(44\!\cdots\!23\)\( T^{11} + \)\(32\!\cdots\!72\)\( T^{12} - \)\(27\!\cdots\!12\)\( T^{13} - \)\(29\!\cdots\!15\)\( T^{14} + \)\(16\!\cdots\!42\)\( T^{15} + \)\(24\!\cdots\!35\)\( T^{16} - \)\(53\!\cdots\!23\)\( T^{17} - \)\(18\!\cdots\!77\)\( T^{18} - \)\(53\!\cdots\!23\)\( p^{3} T^{19} + \)\(24\!\cdots\!35\)\( p^{6} T^{20} + \)\(16\!\cdots\!42\)\( p^{9} T^{21} - \)\(29\!\cdots\!15\)\( p^{12} T^{22} - \)\(27\!\cdots\!12\)\( p^{15} T^{23} + \)\(32\!\cdots\!72\)\( p^{18} T^{24} + \)\(44\!\cdots\!23\)\( p^{21} T^{25} - \)\(32\!\cdots\!12\)\( p^{24} T^{26} - \)\(66\!\cdots\!83\)\( p^{27} T^{27} + \)\(27\!\cdots\!53\)\( p^{30} T^{28} + \)\(86\!\cdots\!97\)\( p^{33} T^{29} - 18811569012189027633 p^{36} T^{30} - 7610410478848107 p^{39} T^{31} + 9575717323487 p^{42} T^{32} + 4389505430 p^{45} T^{33} - 3307889 p^{48} T^{34} - 1663 p^{51} T^{35} + p^{54} T^{36} \) | |
97 | \( 1 + 1087 T - 3953317 T^{2} - 6381339578 T^{3} + 6059982354107 T^{4} + 15831982212735351 T^{5} - 2301343398914614305 T^{6} - \)\(22\!\cdots\!61\)\( T^{7} - \)\(58\!\cdots\!35\)\( T^{8} + \)\(18\!\cdots\!55\)\( T^{9} + \)\(10\!\cdots\!44\)\( T^{10} - \)\(70\!\cdots\!35\)\( T^{11} - \)\(48\!\cdots\!52\)\( T^{12} - \)\(39\!\cdots\!56\)\( T^{13} - \)\(73\!\cdots\!59\)\( T^{14} + \)\(77\!\cdots\!50\)\( T^{15} + \)\(18\!\cdots\!55\)\( T^{16} - \)\(35\!\cdots\!69\)\( T^{17} - \)\(22\!\cdots\!69\)\( T^{18} - \)\(35\!\cdots\!69\)\( p^{3} T^{19} + \)\(18\!\cdots\!55\)\( p^{6} T^{20} + \)\(77\!\cdots\!50\)\( p^{9} T^{21} - \)\(73\!\cdots\!59\)\( p^{12} T^{22} - \)\(39\!\cdots\!56\)\( p^{15} T^{23} - \)\(48\!\cdots\!52\)\( p^{18} T^{24} - \)\(70\!\cdots\!35\)\( p^{21} T^{25} + \)\(10\!\cdots\!44\)\( p^{24} T^{26} + \)\(18\!\cdots\!55\)\( p^{27} T^{27} - \)\(58\!\cdots\!35\)\( p^{30} T^{28} - \)\(22\!\cdots\!61\)\( p^{33} T^{29} - 2301343398914614305 p^{36} T^{30} + 15831982212735351 p^{39} T^{31} + 6059982354107 p^{42} T^{32} - 6381339578 p^{45} T^{33} - 3953317 p^{48} T^{34} + 1087 p^{51} T^{35} + p^{54} T^{36} \) | |
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Imaginary part of the first few zeros on the critical line
−3.11588539398628207947916980311, −2.74896085645125159519046523020, −2.73534072107228434748806407223, −2.72269578192888692448763484002, −2.40153642769208601016800746175, −2.32689491322484646394107319854, −2.26994150005055281159950239397, −2.20082067939617071712804319998, −2.04600153381917965620708304898, −1.99703260440959398295008495831, −1.97629176241786259979680704586, −1.59203785230204965261393429457, −1.58493853555484201297146093501, −1.57840848239689795647703803351, −1.52719958905533159657651639032, −1.46605091526454200811447183631, −1.23177532812376166568771791980, −1.20759246794816029454293042948, −1.12551374477489632988270786602, −1.00158372965701569796191508454, −0.57477301280130435426359043692, −0.52243126721513021669868818410, −0.36227896339874846672695283354, −0.24597451070040081740506028349, −0.17931160111080829701213135013, 0.17931160111080829701213135013, 0.24597451070040081740506028349, 0.36227896339874846672695283354, 0.52243126721513021669868818410, 0.57477301280130435426359043692, 1.00158372965701569796191508454, 1.12551374477489632988270786602, 1.20759246794816029454293042948, 1.23177532812376166568771791980, 1.46605091526454200811447183631, 1.52719958905533159657651639032, 1.57840848239689795647703803351, 1.58493853555484201297146093501, 1.59203785230204965261393429457, 1.97629176241786259979680704586, 1.99703260440959398295008495831, 2.04600153381917965620708304898, 2.20082067939617071712804319998, 2.26994150005055281159950239397, 2.32689491322484646394107319854, 2.40153642769208601016800746175, 2.72269578192888692448763484002, 2.73534072107228434748806407223, 2.74896085645125159519046523020, 3.11588539398628207947916980311