Dirichlet series
| L(s) = 1 | + 16·2-s + 121·3-s − 179·4-s + 376·5-s + 1.93e3·6-s + 2.24e3·7-s − 3.62e3·8-s + 205·9-s + 6.01e3·10-s + 9.41e3·11-s − 2.16e4·12-s + 1.25e4·13-s + 3.58e4·14-s + 4.54e4·15-s + 3.68e4·16-s + 5.43e4·17-s + 3.28e3·18-s + 9.71e4·19-s − 6.73e4·20-s + 2.71e5·21-s + 1.50e5·22-s + 1.07e5·23-s − 4.38e5·24-s − 2.76e5·25-s + 2.00e5·26-s − 5.05e5·27-s − 4.01e5·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.58·3-s − 1.39·4-s + 1.34·5-s + 3.65·6-s + 2.47·7-s − 2.50·8-s + 0.0937·9-s + 1.90·10-s + 2.13·11-s − 3.61·12-s + 1.57·13-s + 3.49·14-s + 3.48·15-s + 2.24·16-s + 2.68·17-s + 0.132·18-s + 3.25·19-s − 1.88·20-s + 6.39·21-s + 3.01·22-s + 1.83·23-s − 6.47·24-s − 3.53·25-s + 2.23·26-s − 4.93·27-s − 3.45·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{11}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(37^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.91825\times 10^{11}\) |
| Root analytic conductor: | \(3.39974\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 37^{11} ,\ ( \ : [7/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(107.5561661\) |
| \(L(\frac12)\) | \(\approx\) | \(107.5561661\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 37 | \( ( 1 + p^{3} T )^{11} \) |
| good | 2 | \( 1 - p^{4} T + 435 T^{2} - 3101 p T^{3} + 41155 p T^{4} - 61947 p^{4} T^{5} + 619329 p^{4} T^{6} - 1701345 p^{6} T^{7} + 44628999 p^{5} T^{8} - 137393889 p^{7} T^{9} + 999800209 p^{8} T^{10} - 1383216179 p^{11} T^{11} + 999800209 p^{15} T^{12} - 137393889 p^{21} T^{13} + 44628999 p^{26} T^{14} - 1701345 p^{34} T^{15} + 619329 p^{39} T^{16} - 61947 p^{46} T^{17} + 41155 p^{50} T^{18} - 3101 p^{57} T^{19} + 435 p^{63} T^{20} - p^{74} T^{21} + p^{77} T^{22} \) |
| 3 | \( 1 - 121 T + 1604 p^{2} T^{2} - 1216897 T^{3} + 96651856 T^{4} - 2171733847 p T^{5} + 140323942618 p T^{6} - 2707642251991 p^{2} T^{7} + 17043330127628 p^{4} T^{8} - 885063079842607 p^{4} T^{9} + 15024211026032065 p^{5} T^{10} - 78883518894917834 p^{7} T^{11} + 15024211026032065 p^{12} T^{12} - 885063079842607 p^{18} T^{13} + 17043330127628 p^{25} T^{14} - 2707642251991 p^{30} T^{15} + 140323942618 p^{36} T^{16} - 2171733847 p^{43} T^{17} + 96651856 p^{49} T^{18} - 1216897 p^{56} T^{19} + 1604 p^{65} T^{20} - 121 p^{70} T^{21} + p^{77} T^{22} \) | |
| 5 | \( 1 - 376 T + 16714 p^{2} T^{2} - 157703612 T^{3} + 100355325303 T^{4} - 6837786745942 p T^{5} + 675626202952452 p^{2} T^{6} - 41471370494631724 p^{3} T^{7} + 3401696644633360711 p^{4} T^{8} - \)\(18\!\cdots\!06\)\( p^{5} T^{9} + \)\(13\!\cdots\!79\)\( p^{6} T^{10} - \)\(13\!\cdots\!84\)\( p^{8} T^{11} + \)\(13\!\cdots\!79\)\( p^{13} T^{12} - \)\(18\!\cdots\!06\)\( p^{19} T^{13} + 3401696644633360711 p^{25} T^{14} - 41471370494631724 p^{31} T^{15} + 675626202952452 p^{37} T^{16} - 6837786745942 p^{43} T^{17} + 100355325303 p^{49} T^{18} - 157703612 p^{56} T^{19} + 16714 p^{65} T^{20} - 376 p^{70} T^{21} + p^{77} T^{22} \) | |
| 7 | \( 1 - 2243 T + 5903743 T^{2} - 9579800520 T^{3} + 16875259675310 T^{4} - 3263171041614894 p T^{5} + 653475345443309432 p^{2} T^{6} - \)\(37\!\cdots\!94\)\( T^{7} + \)\(44\!\cdots\!87\)\( T^{8} - \)\(45\!\cdots\!99\)\( T^{9} + \)\(46\!\cdots\!47\)\( T^{10} - \)\(42\!\cdots\!36\)\( T^{11} + \)\(46\!\cdots\!47\)\( p^{7} T^{12} - \)\(45\!\cdots\!99\)\( p^{14} T^{13} + \)\(44\!\cdots\!87\)\( p^{21} T^{14} - \)\(37\!\cdots\!94\)\( p^{28} T^{15} + 653475345443309432 p^{37} T^{16} - 3263171041614894 p^{43} T^{17} + 16875259675310 p^{49} T^{18} - 9579800520 p^{56} T^{19} + 5903743 p^{63} T^{20} - 2243 p^{70} T^{21} + p^{77} T^{22} \) | |
| 11 | \( 1 - 9415 T + 144474560 T^{2} - 1125113346099 T^{3} + 9941605656077900 T^{4} - 65445486536604019475 T^{5} + \)\(43\!\cdots\!30\)\( T^{6} - \)\(22\!\cdots\!59\)\( p T^{7} + \)\(13\!\cdots\!28\)\( T^{8} - \)\(68\!\cdots\!13\)\( T^{9} + \)\(32\!\cdots\!11\)\( T^{10} - \)\(14\!\cdots\!18\)\( T^{11} + \)\(32\!\cdots\!11\)\( p^{7} T^{12} - \)\(68\!\cdots\!13\)\( p^{14} T^{13} + \)\(13\!\cdots\!28\)\( p^{21} T^{14} - \)\(22\!\cdots\!59\)\( p^{29} T^{15} + \)\(43\!\cdots\!30\)\( p^{35} T^{16} - 65445486536604019475 p^{42} T^{17} + 9941605656077900 p^{49} T^{18} - 1125113346099 p^{56} T^{19} + 144474560 p^{63} T^{20} - 9415 p^{70} T^{21} + p^{77} T^{22} \) | |
| 13 | \( 1 - 12512 T + 3320162 p^{2} T^{2} - 5902748640596 T^{3} + 144440686482649935 T^{4} - \)\(13\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!84\)\( T^{6} - \)\(18\!\cdots\!48\)\( T^{7} + \)\(25\!\cdots\!07\)\( T^{8} - \)\(17\!\cdots\!50\)\( T^{9} + \)\(21\!\cdots\!99\)\( T^{10} - \)\(13\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!99\)\( p^{7} T^{12} - \)\(17\!\cdots\!50\)\( p^{14} T^{13} + \)\(25\!\cdots\!07\)\( p^{21} T^{14} - \)\(18\!\cdots\!48\)\( p^{28} T^{15} + \)\(22\!\cdots\!84\)\( p^{35} T^{16} - \)\(13\!\cdots\!02\)\( p^{42} T^{17} + 144440686482649935 p^{49} T^{18} - 5902748640596 p^{56} T^{19} + 3320162 p^{65} T^{20} - 12512 p^{70} T^{21} + p^{77} T^{22} \) | |
| 17 | \( 1 - 54312 T + 4450770039 T^{2} - 179840065713312 T^{3} + 8497274565076480451 T^{4} - \)\(27\!\cdots\!56\)\( T^{5} + \)\(96\!\cdots\!09\)\( T^{6} - \)\(26\!\cdots\!92\)\( T^{7} + \)\(43\!\cdots\!90\)\( p T^{8} - \)\(17\!\cdots\!32\)\( T^{9} + \)\(40\!\cdots\!22\)\( T^{10} - \)\(82\!\cdots\!16\)\( T^{11} + \)\(40\!\cdots\!22\)\( p^{7} T^{12} - \)\(17\!\cdots\!32\)\( p^{14} T^{13} + \)\(43\!\cdots\!90\)\( p^{22} T^{14} - \)\(26\!\cdots\!92\)\( p^{28} T^{15} + \)\(96\!\cdots\!09\)\( p^{35} T^{16} - \)\(27\!\cdots\!56\)\( p^{42} T^{17} + 8497274565076480451 p^{49} T^{18} - 179840065713312 p^{56} T^{19} + 4450770039 p^{63} T^{20} - 54312 p^{70} T^{21} + p^{77} T^{22} \) | |
| 19 | \( 1 - 97192 T + 11197976549 T^{2} - 762950650644896 T^{3} + 52967051224472637283 T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!95\)\( T^{6} - \)\(62\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!18\)\( p T^{8} - \)\(94\!\cdots\!84\)\( T^{9} + \)\(32\!\cdots\!10\)\( T^{10} - \)\(10\!\cdots\!88\)\( T^{11} + \)\(32\!\cdots\!10\)\( p^{7} T^{12} - \)\(94\!\cdots\!84\)\( p^{14} T^{13} + \)\(13\!\cdots\!18\)\( p^{22} T^{14} - \)\(62\!\cdots\!16\)\( p^{28} T^{15} + \)\(14\!\cdots\!95\)\( p^{35} T^{16} - \)\(27\!\cdots\!08\)\( p^{42} T^{17} + 52967051224472637283 p^{49} T^{18} - 762950650644896 p^{56} T^{19} + 11197976549 p^{63} T^{20} - 97192 p^{70} T^{21} + p^{77} T^{22} \) | |
| 23 | \( 1 - 107342 T + 30790542994 T^{2} - 2571970084412050 T^{3} + \)\(41\!\cdots\!75\)\( T^{4} - \)\(28\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!28\)\( p T^{6} - \)\(19\!\cdots\!24\)\( T^{7} + \)\(19\!\cdots\!59\)\( T^{8} - \)\(93\!\cdots\!90\)\( T^{9} + \)\(81\!\cdots\!37\)\( T^{10} - \)\(35\!\cdots\!92\)\( T^{11} + \)\(81\!\cdots\!37\)\( p^{7} T^{12} - \)\(93\!\cdots\!90\)\( p^{14} T^{13} + \)\(19\!\cdots\!59\)\( p^{21} T^{14} - \)\(19\!\cdots\!24\)\( p^{28} T^{15} + \)\(14\!\cdots\!28\)\( p^{36} T^{16} - \)\(28\!\cdots\!44\)\( p^{42} T^{17} + \)\(41\!\cdots\!75\)\( p^{49} T^{18} - 2571970084412050 p^{56} T^{19} + 30790542994 p^{63} T^{20} - 107342 p^{70} T^{21} + p^{77} T^{22} \) | |
| 29 | \( 1 - 41748 T + 103324143986 T^{2} - 4754067213639048 T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(29\!\cdots\!70\)\( T^{5} + \)\(17\!\cdots\!32\)\( T^{6} - \)\(11\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!75\)\( T^{8} - \)\(32\!\cdots\!06\)\( T^{9} + \)\(32\!\cdots\!03\)\( p T^{10} - \)\(22\!\cdots\!56\)\( p T^{11} + \)\(32\!\cdots\!03\)\( p^{8} T^{12} - \)\(32\!\cdots\!06\)\( p^{14} T^{13} + \)\(45\!\cdots\!75\)\( p^{21} T^{14} - \)\(11\!\cdots\!36\)\( p^{28} T^{15} + \)\(17\!\cdots\!32\)\( p^{35} T^{16} - \)\(29\!\cdots\!70\)\( p^{42} T^{17} + \)\(53\!\cdots\!95\)\( p^{49} T^{18} - 4754067213639048 p^{56} T^{19} + 103324143986 p^{63} T^{20} - 41748 p^{70} T^{21} + p^{77} T^{22} \) | |
| 31 | \( 1 + 272248 T + 163831120198 T^{2} + 39452553775212912 T^{3} + \)\(13\!\cdots\!15\)\( T^{4} + \)\(28\!\cdots\!60\)\( T^{5} + \)\(77\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!55\)\( T^{8} + \)\(55\!\cdots\!36\)\( T^{9} + \)\(11\!\cdots\!05\)\( T^{10} + \)\(55\!\cdots\!92\)\( p T^{11} + \)\(11\!\cdots\!05\)\( p^{7} T^{12} + \)\(55\!\cdots\!36\)\( p^{14} T^{13} + \)\(33\!\cdots\!55\)\( p^{21} T^{14} + \)\(14\!\cdots\!40\)\( p^{28} T^{15} + \)\(77\!\cdots\!24\)\( p^{35} T^{16} + \)\(28\!\cdots\!60\)\( p^{42} T^{17} + \)\(13\!\cdots\!15\)\( p^{49} T^{18} + 39452553775212912 p^{56} T^{19} + 163831120198 p^{63} T^{20} + 272248 p^{70} T^{21} + p^{77} T^{22} \) | |
| 41 | \( 1 - 525465 T + 610767101122 T^{2} - 218617383936935055 T^{3} + \)\(19\!\cdots\!64\)\( T^{4} - \)\(73\!\cdots\!19\)\( T^{5} + \)\(50\!\cdots\!50\)\( T^{6} - \)\(20\!\cdots\!97\)\( T^{7} + \)\(99\!\cdots\!54\)\( T^{8} - \)\(45\!\cdots\!87\)\( T^{9} + \)\(19\!\cdots\!13\)\( T^{10} - \)\(10\!\cdots\!62\)\( T^{11} + \)\(19\!\cdots\!13\)\( p^{7} T^{12} - \)\(45\!\cdots\!87\)\( p^{14} T^{13} + \)\(99\!\cdots\!54\)\( p^{21} T^{14} - \)\(20\!\cdots\!97\)\( p^{28} T^{15} + \)\(50\!\cdots\!50\)\( p^{35} T^{16} - \)\(73\!\cdots\!19\)\( p^{42} T^{17} + \)\(19\!\cdots\!64\)\( p^{49} T^{18} - 218617383936935055 p^{56} T^{19} + 610767101122 p^{63} T^{20} - 525465 p^{70} T^{21} + p^{77} T^{22} \) | |
| 43 | \( 1 + 32002 p T + 2216697310361 T^{2} + 53149759616269996 p T^{3} + \)\(22\!\cdots\!15\)\( T^{4} + \)\(18\!\cdots\!34\)\( T^{5} + \)\(14\!\cdots\!67\)\( T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(69\!\cdots\!30\)\( T^{8} + \)\(41\!\cdots\!92\)\( T^{9} + \)\(24\!\cdots\!70\)\( T^{10} + \)\(12\!\cdots\!88\)\( T^{11} + \)\(24\!\cdots\!70\)\( p^{7} T^{12} + \)\(41\!\cdots\!92\)\( p^{14} T^{13} + \)\(69\!\cdots\!30\)\( p^{21} T^{14} + \)\(10\!\cdots\!12\)\( p^{28} T^{15} + \)\(14\!\cdots\!67\)\( p^{35} T^{16} + \)\(18\!\cdots\!34\)\( p^{42} T^{17} + \)\(22\!\cdots\!15\)\( p^{49} T^{18} + 53149759616269996 p^{57} T^{19} + 2216697310361 p^{63} T^{20} + 32002 p^{71} T^{21} + p^{77} T^{22} \) | |
| 47 | \( 1 - 2269179 T + 5150633824335 T^{2} - 7578137983354454664 T^{3} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(12\!\cdots\!82\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(12\!\cdots\!82\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} - \)\(92\!\cdots\!55\)\( T^{9} + \)\(74\!\cdots\!59\)\( T^{10} - \)\(53\!\cdots\!20\)\( T^{11} + \)\(74\!\cdots\!59\)\( p^{7} T^{12} - \)\(92\!\cdots\!55\)\( p^{14} T^{13} + \)\(11\!\cdots\!03\)\( p^{21} T^{14} - \)\(12\!\cdots\!82\)\( p^{28} T^{15} + \)\(13\!\cdots\!04\)\( p^{35} T^{16} - \)\(12\!\cdots\!82\)\( p^{42} T^{17} + \)\(10\!\cdots\!58\)\( p^{49} T^{18} - 7578137983354454664 p^{56} T^{19} + 5150633824335 p^{63} T^{20} - 2269179 p^{70} T^{21} + p^{77} T^{22} \) | |
| 53 | \( 1 + 346415 T + 5990263327153 T^{2} + 2004936166133257456 T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(43\!\cdots\!02\)\( T^{5} + \)\(39\!\cdots\!96\)\( T^{6} + \)\(42\!\cdots\!82\)\( T^{7} + \)\(63\!\cdots\!39\)\( T^{8} - \)\(35\!\cdots\!05\)\( T^{9} + \)\(84\!\cdots\!17\)\( T^{10} - \)\(36\!\cdots\!32\)\( T^{11} + \)\(84\!\cdots\!17\)\( p^{7} T^{12} - \)\(35\!\cdots\!05\)\( p^{14} T^{13} + \)\(63\!\cdots\!39\)\( p^{21} T^{14} + \)\(42\!\cdots\!82\)\( p^{28} T^{15} + \)\(39\!\cdots\!96\)\( p^{35} T^{16} + \)\(43\!\cdots\!02\)\( p^{42} T^{17} + \)\(18\!\cdots\!54\)\( p^{49} T^{18} + 2004936166133257456 p^{56} T^{19} + 5990263327153 p^{63} T^{20} + 346415 p^{70} T^{21} + p^{77} T^{22} \) | |
| 59 | \( 1 - 4598828 T + 30382309329961 T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!47\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{5} + \)\(29\!\cdots\!35\)\( T^{6} - \)\(65\!\cdots\!16\)\( T^{7} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(27\!\cdots\!36\)\( T^{9} + \)\(51\!\cdots\!66\)\( T^{10} - \)\(81\!\cdots\!68\)\( T^{11} + \)\(51\!\cdots\!66\)\( p^{7} T^{12} - \)\(27\!\cdots\!36\)\( p^{14} T^{13} + \)\(14\!\cdots\!14\)\( p^{21} T^{14} - \)\(65\!\cdots\!16\)\( p^{28} T^{15} + \)\(29\!\cdots\!35\)\( p^{35} T^{16} - \)\(10\!\cdots\!68\)\( p^{42} T^{17} + \)\(39\!\cdots\!47\)\( p^{49} T^{18} - \)\(10\!\cdots\!68\)\( p^{56} T^{19} + 30382309329961 p^{63} T^{20} - 4598828 p^{70} T^{21} + p^{77} T^{22} \) | |
| 61 | \( 1 - 6208418 T + 40200898514190 T^{2} - \)\(15\!\cdots\!82\)\( T^{3} + \)\(57\!\cdots\!95\)\( T^{4} - \)\(15\!\cdots\!18\)\( T^{5} + \)\(42\!\cdots\!44\)\( T^{6} - \)\(92\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!47\)\( T^{8} - \)\(35\!\cdots\!16\)\( T^{9} + \)\(67\!\cdots\!83\)\( T^{10} - \)\(11\!\cdots\!72\)\( T^{11} + \)\(67\!\cdots\!83\)\( p^{7} T^{12} - \)\(35\!\cdots\!16\)\( p^{14} T^{13} + \)\(19\!\cdots\!47\)\( p^{21} T^{14} - \)\(92\!\cdots\!60\)\( p^{28} T^{15} + \)\(42\!\cdots\!44\)\( p^{35} T^{16} - \)\(15\!\cdots\!18\)\( p^{42} T^{17} + \)\(57\!\cdots\!95\)\( p^{49} T^{18} - \)\(15\!\cdots\!82\)\( p^{56} T^{19} + 40200898514190 p^{63} T^{20} - 6208418 p^{70} T^{21} + p^{77} T^{22} \) | |
| 67 | \( 1 - 2199016 T + 33018631376050 T^{2} - 67968839492168482000 T^{3} + \)\(57\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(69\!\cdots\!52\)\( T^{6} - \)\(11\!\cdots\!68\)\( T^{7} + \)\(64\!\cdots\!35\)\( T^{8} - \)\(95\!\cdots\!08\)\( T^{9} + \)\(47\!\cdots\!01\)\( T^{10} - \)\(63\!\cdots\!52\)\( T^{11} + \)\(47\!\cdots\!01\)\( p^{7} T^{12} - \)\(95\!\cdots\!08\)\( p^{14} T^{13} + \)\(64\!\cdots\!35\)\( p^{21} T^{14} - \)\(11\!\cdots\!68\)\( p^{28} T^{15} + \)\(69\!\cdots\!52\)\( p^{35} T^{16} - \)\(10\!\cdots\!04\)\( p^{42} T^{17} + \)\(57\!\cdots\!39\)\( p^{49} T^{18} - 67968839492168482000 p^{56} T^{19} + 33018631376050 p^{63} T^{20} - 2199016 p^{70} T^{21} + p^{77} T^{22} \) | |
| 71 | \( 1 - 4653285 T + 52819226480179 T^{2} - \)\(20\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} - \)\(50\!\cdots\!62\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} - \)\(86\!\cdots\!62\)\( T^{7} + \)\(38\!\cdots\!87\)\( T^{8} - \)\(11\!\cdots\!65\)\( T^{9} + \)\(43\!\cdots\!79\)\( T^{10} - \)\(11\!\cdots\!72\)\( T^{11} + \)\(43\!\cdots\!79\)\( p^{7} T^{12} - \)\(11\!\cdots\!65\)\( p^{14} T^{13} + \)\(38\!\cdots\!87\)\( p^{21} T^{14} - \)\(86\!\cdots\!62\)\( p^{28} T^{15} + \)\(26\!\cdots\!32\)\( p^{35} T^{16} - \)\(50\!\cdots\!62\)\( p^{42} T^{17} + \)\(14\!\cdots\!06\)\( p^{49} T^{18} - \)\(20\!\cdots\!64\)\( p^{56} T^{19} + 52819226480179 p^{63} T^{20} - 4653285 p^{70} T^{21} + p^{77} T^{22} \) | |
| 73 | \( 1 + 1080699 T + 41160308967742 T^{2} + 51618872539420533029 T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(10\!\cdots\!93\)\( T^{5} + \)\(19\!\cdots\!98\)\( T^{6} + \)\(16\!\cdots\!43\)\( T^{7} + \)\(29\!\cdots\!94\)\( T^{8} + \)\(20\!\cdots\!09\)\( T^{9} + \)\(37\!\cdots\!57\)\( T^{10} + \)\(24\!\cdots\!06\)\( T^{11} + \)\(37\!\cdots\!57\)\( p^{7} T^{12} + \)\(20\!\cdots\!09\)\( p^{14} T^{13} + \)\(29\!\cdots\!94\)\( p^{21} T^{14} + \)\(16\!\cdots\!43\)\( p^{28} T^{15} + \)\(19\!\cdots\!98\)\( p^{35} T^{16} + \)\(10\!\cdots\!93\)\( p^{42} T^{17} + \)\(10\!\cdots\!56\)\( p^{49} T^{18} + 51618872539420533029 p^{56} T^{19} + 41160308967742 p^{63} T^{20} + 1080699 p^{70} T^{21} + p^{77} T^{22} \) | |
| 79 | \( 1 + 1336084 T + 120037204532306 T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(69\!\cdots\!31\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} + \)\(45\!\cdots\!32\)\( T^{7} + \)\(74\!\cdots\!19\)\( T^{8} + \)\(13\!\cdots\!16\)\( T^{9} + \)\(16\!\cdots\!61\)\( T^{10} + \)\(29\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!61\)\( p^{7} T^{12} + \)\(13\!\cdots\!16\)\( p^{14} T^{13} + \)\(74\!\cdots\!19\)\( p^{21} T^{14} + \)\(45\!\cdots\!32\)\( p^{28} T^{15} + \)\(26\!\cdots\!32\)\( p^{35} T^{16} + \)\(11\!\cdots\!96\)\( p^{42} T^{17} + \)\(69\!\cdots\!31\)\( p^{49} T^{18} + \)\(17\!\cdots\!00\)\( p^{56} T^{19} + 120037204532306 p^{63} T^{20} + 1336084 p^{70} T^{21} + p^{77} T^{22} \) | |
| 83 | \( 1 - 28551309 T + 532716594842759 T^{2} - \)\(73\!\cdots\!96\)\( T^{3} + \)\(83\!\cdots\!86\)\( T^{4} - \)\(79\!\cdots\!54\)\( T^{5} + \)\(67\!\cdots\!96\)\( T^{6} - \)\(51\!\cdots\!22\)\( T^{7} + \)\(35\!\cdots\!91\)\( T^{8} - \)\(22\!\cdots\!61\)\( T^{9} + \)\(12\!\cdots\!59\)\( T^{10} - \)\(69\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!59\)\( p^{7} T^{12} - \)\(22\!\cdots\!61\)\( p^{14} T^{13} + \)\(35\!\cdots\!91\)\( p^{21} T^{14} - \)\(51\!\cdots\!22\)\( p^{28} T^{15} + \)\(67\!\cdots\!96\)\( p^{35} T^{16} - \)\(79\!\cdots\!54\)\( p^{42} T^{17} + \)\(83\!\cdots\!86\)\( p^{49} T^{18} - \)\(73\!\cdots\!96\)\( p^{56} T^{19} + 532716594842759 p^{63} T^{20} - 28551309 p^{70} T^{21} + p^{77} T^{22} \) | |
| 89 | \( 1 + 8994788 T + 377146170606231 T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(64\!\cdots\!19\)\( T^{4} + \)\(37\!\cdots\!96\)\( T^{5} + \)\(67\!\cdots\!17\)\( T^{6} + \)\(32\!\cdots\!92\)\( T^{7} + \)\(50\!\cdots\!50\)\( T^{8} + \)\(20\!\cdots\!64\)\( T^{9} + \)\(28\!\cdots\!10\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(28\!\cdots\!10\)\( p^{7} T^{12} + \)\(20\!\cdots\!64\)\( p^{14} T^{13} + \)\(50\!\cdots\!50\)\( p^{21} T^{14} + \)\(32\!\cdots\!92\)\( p^{28} T^{15} + \)\(67\!\cdots\!17\)\( p^{35} T^{16} + \)\(37\!\cdots\!96\)\( p^{42} T^{17} + \)\(64\!\cdots\!19\)\( p^{49} T^{18} + \)\(26\!\cdots\!60\)\( p^{56} T^{19} + 377146170606231 p^{63} T^{20} + 8994788 p^{70} T^{21} + p^{77} T^{22} \) | |
| 97 | \( 1 + 3968264 T + 309871406086295 T^{2} + \)\(29\!\cdots\!84\)\( T^{3} + \)\(69\!\cdots\!07\)\( T^{4} + \)\(69\!\cdots\!32\)\( p T^{5} + \)\(11\!\cdots\!93\)\( T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!02\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{9} + \)\(15\!\cdots\!54\)\( T^{10} + \)\(11\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!54\)\( p^{7} T^{12} + \)\(12\!\cdots\!12\)\( p^{14} T^{13} + \)\(15\!\cdots\!02\)\( p^{21} T^{14} + \)\(10\!\cdots\!12\)\( p^{28} T^{15} + \)\(11\!\cdots\!93\)\( p^{35} T^{16} + \)\(69\!\cdots\!32\)\( p^{43} T^{17} + \)\(69\!\cdots\!07\)\( p^{49} T^{18} + \)\(29\!\cdots\!84\)\( p^{56} T^{19} + 309871406086295 p^{63} T^{20} + 3968264 p^{70} T^{21} + p^{77} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.10647504780188132743307517862, −4.69006233743045664076635235280, −4.61903572998725857207479342440, −4.20827723092694388620753588808, −3.96475093978772444334821328697, −3.84552991494495652333892676751, −3.71812872251981761579952682874, −3.61797055365329499953029951058, −3.52108124210253391847801265695, −3.49739914711234879654348896596, −3.38441135977466307347827911875, −3.00820504982403643225610108791, −2.85598991471211841011020740978, −2.70446975359335587073669934009, −2.31241121641938364952898666074, −2.14923808116560336202324711305, −1.94400106817590540922421432321, −1.75921362187966111280090576783, −1.49108618240634153573658746524, −1.42298504082016234708605965643, −1.06741578527658326494846276300, −0.939844912586268810222203608743, −0.876447989551867241953528249837, −0.61541509549283714489822765426, −0.21130406350845247512526609604, 0.21130406350845247512526609604, 0.61541509549283714489822765426, 0.876447989551867241953528249837, 0.939844912586268810222203608743, 1.06741578527658326494846276300, 1.42298504082016234708605965643, 1.49108618240634153573658746524, 1.75921362187966111280090576783, 1.94400106817590540922421432321, 2.14923808116560336202324711305, 2.31241121641938364952898666074, 2.70446975359335587073669934009, 2.85598991471211841011020740978, 3.00820504982403643225610108791, 3.38441135977466307347827911875, 3.49739914711234879654348896596, 3.52108124210253391847801265695, 3.61797055365329499953029951058, 3.71812872251981761579952682874, 3.84552991494495652333892676751, 3.96475093978772444334821328697, 4.20827723092694388620753588808, 4.61903572998725857207479342440, 4.69006233743045664076635235280, 5.10647504780188132743307517862
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.