Dirichlet series
| L(s) = 1 | − 6·2-s + 6·3-s + 7·4-s + 4·5-s − 36·6-s − 45·7-s + 20·8-s − 63·9-s − 24·10-s − 121·11-s + 42·12-s + 270·14-s + 24·15-s − 57·16-s + 265·17-s + 378·18-s − 127·19-s + 28·20-s − 270·21-s + 726·22-s + 42·23-s + 120·24-s − 311·25-s − 481·27-s − 315·28-s + 435·29-s − 144·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 7/8·4-s + 0.357·5-s − 2.44·6-s − 2.42·7-s + 0.883·8-s − 7/3·9-s − 0.758·10-s − 3.31·11-s + 1.01·12-s + 5.15·14-s + 0.413·15-s − 0.890·16-s + 3.78·17-s + 4.94·18-s − 1.53·19-s + 0.313·20-s − 2.80·21-s + 7.03·22-s + 0.380·23-s + 1.02·24-s − 2.48·25-s − 3.42·27-s − 2.12·28-s + 2.78·29-s − 0.876·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{11} \cdot 13^{22}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{11} \cdot 13^{22}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(11^{11} \cdot 13^{22}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.76439\times 10^{22}\) |
| Root analytic conductor: | \(10.4730\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 11^{11} \cdot 13^{22} ,\ ( \ : [3/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(1.275760476\) |
| \(L(\frac12)\) | \(\approx\) | \(1.275760476\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + p T )^{11} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T + 29 T^{2} + 7 p^{4} T^{3} + 203 p T^{4} + 1253 T^{5} + 2159 p T^{6} + 13027 T^{7} + 20449 p T^{8} + 64221 p T^{9} + 103427 p^{2} T^{10} + 144401 p^{3} T^{11} + 103427 p^{5} T^{12} + 64221 p^{7} T^{13} + 20449 p^{10} T^{14} + 13027 p^{12} T^{15} + 2159 p^{16} T^{16} + 1253 p^{18} T^{17} + 203 p^{22} T^{18} + 7 p^{28} T^{19} + 29 p^{27} T^{20} + 3 p^{31} T^{21} + p^{33} T^{22} \) |
| 3 | \( 1 - 2 p T + 11 p^{2} T^{2} - 491 T^{3} + 1621 p T^{4} - 15094 T^{5} + 41768 p T^{6} - 41834 T^{7} + 1397875 T^{8} + 4486337 p T^{9} - 15533906 T^{10} + 532685564 T^{11} - 15533906 p^{3} T^{12} + 4486337 p^{7} T^{13} + 1397875 p^{9} T^{14} - 41834 p^{12} T^{15} + 41768 p^{16} T^{16} - 15094 p^{18} T^{17} + 1621 p^{22} T^{18} - 491 p^{24} T^{19} + 11 p^{29} T^{20} - 2 p^{31} T^{21} + p^{33} T^{22} \) | |
| 5 | \( 1 - 4 T + 327 T^{2} - 2302 T^{3} + 56242 T^{4} - 543066 T^{5} + 7588062 T^{6} - 90402988 T^{7} + 1199686581 T^{8} - 12006130546 T^{9} + 199047361427 T^{10} - 1476171417196 T^{11} + 199047361427 p^{3} T^{12} - 12006130546 p^{6} T^{13} + 1199686581 p^{9} T^{14} - 90402988 p^{12} T^{15} + 7588062 p^{15} T^{16} - 543066 p^{18} T^{17} + 56242 p^{21} T^{18} - 2302 p^{24} T^{19} + 327 p^{27} T^{20} - 4 p^{30} T^{21} + p^{33} T^{22} \) | |
| 7 | \( 1 + 45 T + 2020 T^{2} + 49928 T^{3} + 1326538 T^{4} + 3626817 p T^{5} + 660801198 T^{6} + 13349009881 T^{7} + 334810464858 T^{8} + 5901111566312 T^{9} + 124381811756875 T^{10} + 1993981340938726 T^{11} + 124381811756875 p^{3} T^{12} + 5901111566312 p^{6} T^{13} + 334810464858 p^{9} T^{14} + 13349009881 p^{12} T^{15} + 660801198 p^{15} T^{16} + 3626817 p^{19} T^{17} + 1326538 p^{21} T^{18} + 49928 p^{24} T^{19} + 2020 p^{27} T^{20} + 45 p^{30} T^{21} + p^{33} T^{22} \) | |
| 17 | \( 1 - 265 T + 52657 T^{2} - 6311882 T^{3} + 587447357 T^{4} - 32306095349 T^{5} + 928683111725 T^{6} + 46861212946120 T^{7} - 1335598727621982 T^{8} - 601143514367212130 T^{9} + \)\(11\!\cdots\!62\)\( T^{10} - \)\(10\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!62\)\( p^{3} T^{12} - 601143514367212130 p^{6} T^{13} - 1335598727621982 p^{9} T^{14} + 46861212946120 p^{12} T^{15} + 928683111725 p^{15} T^{16} - 32306095349 p^{18} T^{17} + 587447357 p^{21} T^{18} - 6311882 p^{24} T^{19} + 52657 p^{27} T^{20} - 265 p^{30} T^{21} + p^{33} T^{22} \) | |
| 19 | \( 1 + 127 T + 38939 T^{2} + 4031089 T^{3} + 701075191 T^{4} + 62520835898 T^{5} + 8343928769238 T^{6} + 700141484014350 T^{7} + 80448226828274611 T^{8} + 6569779952920627287 T^{9} + \)\(66\!\cdots\!70\)\( T^{10} + \)\(50\!\cdots\!14\)\( T^{11} + \)\(66\!\cdots\!70\)\( p^{3} T^{12} + 6569779952920627287 p^{6} T^{13} + 80448226828274611 p^{9} T^{14} + 700141484014350 p^{12} T^{15} + 8343928769238 p^{15} T^{16} + 62520835898 p^{18} T^{17} + 701075191 p^{21} T^{18} + 4031089 p^{24} T^{19} + 38939 p^{27} T^{20} + 127 p^{30} T^{21} + p^{33} T^{22} \) | |
| 23 | \( 1 - 42 T + 70251 T^{2} - 1550471 T^{3} + 2455338397 T^{4} - 14346603638 T^{5} + 58596889466082 T^{6} + 228699335868934 T^{7} + 1086455462752823389 T^{8} + 8015336815068971983 T^{9} + \)\(16\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!06\)\( p^{3} T^{12} + 8015336815068971983 p^{6} T^{13} + 1086455462752823389 p^{9} T^{14} + 228699335868934 p^{12} T^{15} + 58596889466082 p^{15} T^{16} - 14346603638 p^{18} T^{17} + 2455338397 p^{21} T^{18} - 1550471 p^{24} T^{19} + 70251 p^{27} T^{20} - 42 p^{30} T^{21} + p^{33} T^{22} \) | |
| 29 | \( 1 - 15 p T + 210185 T^{2} - 60543274 T^{3} + 17502518681 T^{4} - 132067584011 p T^{5} + 829615502992809 T^{6} - 147716085195031800 T^{7} + 26528579793605101602 T^{8} - \)\(41\!\cdots\!58\)\( T^{9} + \)\(67\!\cdots\!94\)\( T^{10} - \)\(10\!\cdots\!36\)\( T^{11} + \)\(67\!\cdots\!94\)\( p^{3} T^{12} - \)\(41\!\cdots\!58\)\( p^{6} T^{13} + 26528579793605101602 p^{9} T^{14} - 147716085195031800 p^{12} T^{15} + 829615502992809 p^{15} T^{16} - 132067584011 p^{19} T^{17} + 17502518681 p^{21} T^{18} - 60543274 p^{24} T^{19} + 210185 p^{27} T^{20} - 15 p^{31} T^{21} + p^{33} T^{22} \) | |
| 31 | \( 1 - 174 T + 107434 T^{2} - 10890268 T^{3} + 5843086910 T^{4} - 140806325622 T^{5} + 179388437401923 T^{6} + 18757993613192336 T^{7} + 3156651490559727178 T^{8} + \)\(14\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!00\)\( T^{10} + \)\(54\!\cdots\!36\)\( T^{11} + \)\(20\!\cdots\!00\)\( p^{3} T^{12} + \)\(14\!\cdots\!80\)\( p^{6} T^{13} + 3156651490559727178 p^{9} T^{14} + 18757993613192336 p^{12} T^{15} + 179388437401923 p^{15} T^{16} - 140806325622 p^{18} T^{17} + 5843086910 p^{21} T^{18} - 10890268 p^{24} T^{19} + 107434 p^{27} T^{20} - 174 p^{30} T^{21} + p^{33} T^{22} \) | |
| 37 | \( 1 + 187 T + 390142 T^{2} + 67181275 T^{3} + 71945280894 T^{4} + 11152448423395 T^{5} + 8348884218652609 T^{6} + 1153588754759557484 T^{7} + \)\(69\!\cdots\!62\)\( T^{8} + \)\(85\!\cdots\!66\)\( T^{9} + \)\(43\!\cdots\!00\)\( T^{10} + \)\(48\!\cdots\!78\)\( T^{11} + \)\(43\!\cdots\!00\)\( p^{3} T^{12} + \)\(85\!\cdots\!66\)\( p^{6} T^{13} + \)\(69\!\cdots\!62\)\( p^{9} T^{14} + 1153588754759557484 p^{12} T^{15} + 8348884218652609 p^{15} T^{16} + 11152448423395 p^{18} T^{17} + 71945280894 p^{21} T^{18} + 67181275 p^{24} T^{19} + 390142 p^{27} T^{20} + 187 p^{30} T^{21} + p^{33} T^{22} \) | |
| 41 | \( 1 + 128 T + 372374 T^{2} + 41676045 T^{3} + 70863790779 T^{4} + 5425845964329 T^{5} + 8956161133768072 T^{6} + 347381703490711003 T^{7} + \)\(84\!\cdots\!41\)\( T^{8} + \)\(67\!\cdots\!35\)\( T^{9} + \)\(16\!\cdots\!37\)\( p T^{10} - \)\(30\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!37\)\( p^{4} T^{12} + \)\(67\!\cdots\!35\)\( p^{6} T^{13} + \)\(84\!\cdots\!41\)\( p^{9} T^{14} + 347381703490711003 p^{12} T^{15} + 8956161133768072 p^{15} T^{16} + 5425845964329 p^{18} T^{17} + 70863790779 p^{21} T^{18} + 41676045 p^{24} T^{19} + 372374 p^{27} T^{20} + 128 p^{30} T^{21} + p^{33} T^{22} \) | |
| 43 | \( 1 - 696 T + 622370 T^{2} - 255708586 T^{3} + 129023028046 T^{4} - 35289127690560 T^{5} + 14211742731901535 T^{6} - 3158873616083396296 T^{7} + \)\(14\!\cdots\!30\)\( T^{8} - \)\(35\!\cdots\!32\)\( T^{9} + \)\(15\!\cdots\!92\)\( T^{10} - \)\(35\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!92\)\( p^{3} T^{12} - \)\(35\!\cdots\!32\)\( p^{6} T^{13} + \)\(14\!\cdots\!30\)\( p^{9} T^{14} - 3158873616083396296 p^{12} T^{15} + 14211742731901535 p^{15} T^{16} - 35289127690560 p^{18} T^{17} + 129023028046 p^{21} T^{18} - 255708586 p^{24} T^{19} + 622370 p^{27} T^{20} - 696 p^{30} T^{21} + p^{33} T^{22} \) | |
| 47 | \( 1 - 355 T + 623689 T^{2} - 192302986 T^{3} + 202638420235 T^{4} - 56000292998455 T^{5} + 44935172284044595 T^{6} - 11226977907788205720 T^{7} + \)\(74\!\cdots\!94\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{9} + \)\(97\!\cdots\!98\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(97\!\cdots\!98\)\( p^{3} T^{12} - \)\(16\!\cdots\!02\)\( p^{6} T^{13} + \)\(74\!\cdots\!94\)\( p^{9} T^{14} - 11226977907788205720 p^{12} T^{15} + 44935172284044595 p^{15} T^{16} - 56000292998455 p^{18} T^{17} + 202638420235 p^{21} T^{18} - 192302986 p^{24} T^{19} + 623689 p^{27} T^{20} - 355 p^{30} T^{21} + p^{33} T^{22} \) | |
| 53 | \( 1 + 693 T + 1218793 T^{2} + 599531081 T^{3} + 625077234147 T^{4} + 236851665421456 T^{5} + 197044058401293296 T^{6} + 62116315175061196968 T^{7} + \)\(46\!\cdots\!77\)\( T^{8} + \)\(12\!\cdots\!47\)\( T^{9} + \)\(86\!\cdots\!86\)\( T^{10} + \)\(21\!\cdots\!54\)\( T^{11} + \)\(86\!\cdots\!86\)\( p^{3} T^{12} + \)\(12\!\cdots\!47\)\( p^{6} T^{13} + \)\(46\!\cdots\!77\)\( p^{9} T^{14} + 62116315175061196968 p^{12} T^{15} + 197044058401293296 p^{15} T^{16} + 236851665421456 p^{18} T^{17} + 625077234147 p^{21} T^{18} + 599531081 p^{24} T^{19} + 1218793 p^{27} T^{20} + 693 p^{30} T^{21} + p^{33} T^{22} \) | |
| 59 | \( 1 - 609 T + 1232196 T^{2} - 817970345 T^{3} + 825046243092 T^{4} - 521411014894881 T^{5} + 385513565350096159 T^{6} - \)\(21\!\cdots\!84\)\( T^{7} + \)\(22\!\cdots\!86\)\( p T^{8} - \)\(65\!\cdots\!78\)\( T^{9} + \)\(35\!\cdots\!92\)\( T^{10} - \)\(15\!\cdots\!70\)\( T^{11} + \)\(35\!\cdots\!92\)\( p^{3} T^{12} - \)\(65\!\cdots\!78\)\( p^{6} T^{13} + \)\(22\!\cdots\!86\)\( p^{10} T^{14} - \)\(21\!\cdots\!84\)\( p^{12} T^{15} + 385513565350096159 p^{15} T^{16} - 521411014894881 p^{18} T^{17} + 825046243092 p^{21} T^{18} - 817970345 p^{24} T^{19} + 1232196 p^{27} T^{20} - 609 p^{30} T^{21} + p^{33} T^{22} \) | |
| 61 | \( 1 - 1625 T + 2507479 T^{2} - 2707529654 T^{3} + 2647910856495 T^{4} - 2197868635705805 T^{5} + 1665561808345197217 T^{6} - \)\(11\!\cdots\!04\)\( T^{7} + \)\(71\!\cdots\!34\)\( T^{8} - \)\(41\!\cdots\!66\)\( T^{9} + \)\(21\!\cdots\!78\)\( T^{10} - \)\(10\!\cdots\!12\)\( T^{11} + \)\(21\!\cdots\!78\)\( p^{3} T^{12} - \)\(41\!\cdots\!66\)\( p^{6} T^{13} + \)\(71\!\cdots\!34\)\( p^{9} T^{14} - \)\(11\!\cdots\!04\)\( p^{12} T^{15} + 1665561808345197217 p^{15} T^{16} - 2197868635705805 p^{18} T^{17} + 2647910856495 p^{21} T^{18} - 2707529654 p^{24} T^{19} + 2507479 p^{27} T^{20} - 1625 p^{30} T^{21} + p^{33} T^{22} \) | |
| 67 | \( 1 + 633 T + 1831598 T^{2} + 1344323689 T^{3} + 1786831105912 T^{4} + 1284039321721769 T^{5} + 1220267947798555581 T^{6} + \)\(78\!\cdots\!64\)\( T^{7} + \)\(61\!\cdots\!80\)\( T^{8} + \)\(35\!\cdots\!34\)\( T^{9} + \)\(23\!\cdots\!62\)\( T^{10} + \)\(12\!\cdots\!30\)\( T^{11} + \)\(23\!\cdots\!62\)\( p^{3} T^{12} + \)\(35\!\cdots\!34\)\( p^{6} T^{13} + \)\(61\!\cdots\!80\)\( p^{9} T^{14} + \)\(78\!\cdots\!64\)\( p^{12} T^{15} + 1220267947798555581 p^{15} T^{16} + 1284039321721769 p^{18} T^{17} + 1786831105912 p^{21} T^{18} + 1344323689 p^{24} T^{19} + 1831598 p^{27} T^{20} + 633 p^{30} T^{21} + p^{33} T^{22} \) | |
| 71 | \( 1 - 1937 T + 3690754 T^{2} - 4380094113 T^{3} + 4911805433048 T^{4} - 4366298574254037 T^{5} + 3745002142585751661 T^{6} - \)\(28\!\cdots\!88\)\( T^{7} + \)\(21\!\cdots\!20\)\( T^{8} - \)\(20\!\cdots\!26\)\( p T^{9} + \)\(95\!\cdots\!74\)\( T^{10} - \)\(58\!\cdots\!90\)\( T^{11} + \)\(95\!\cdots\!74\)\( p^{3} T^{12} - \)\(20\!\cdots\!26\)\( p^{7} T^{13} + \)\(21\!\cdots\!20\)\( p^{9} T^{14} - \)\(28\!\cdots\!88\)\( p^{12} T^{15} + 3745002142585751661 p^{15} T^{16} - 4366298574254037 p^{18} T^{17} + 4911805433048 p^{21} T^{18} - 4380094113 p^{24} T^{19} + 3690754 p^{27} T^{20} - 1937 p^{30} T^{21} + p^{33} T^{22} \) | |
| 73 | \( 1 + 404 T + 2295534 T^{2} + 1071333023 T^{3} + 2641609078427 T^{4} + 1329833481597235 T^{5} + 2022081024374305216 T^{6} + \)\(10\!\cdots\!01\)\( T^{7} + \)\(11\!\cdots\!05\)\( T^{8} + \)\(59\!\cdots\!61\)\( T^{9} + \)\(53\!\cdots\!57\)\( T^{10} + \)\(25\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!57\)\( p^{3} T^{12} + \)\(59\!\cdots\!61\)\( p^{6} T^{13} + \)\(11\!\cdots\!05\)\( p^{9} T^{14} + \)\(10\!\cdots\!01\)\( p^{12} T^{15} + 2022081024374305216 p^{15} T^{16} + 1329833481597235 p^{18} T^{17} + 2641609078427 p^{21} T^{18} + 1071333023 p^{24} T^{19} + 2295534 p^{27} T^{20} + 404 p^{30} T^{21} + p^{33} T^{22} \) | |
| 79 | \( 1 - 1670 T + 4945161 T^{2} - 6239064900 T^{3} + 10617391644051 T^{4} - 10824951276438830 T^{5} + 13617126187691726939 T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!46\)\( T^{8} - \)\(88\!\cdots\!04\)\( T^{9} + \)\(77\!\cdots\!10\)\( T^{10} - \)\(50\!\cdots\!80\)\( T^{11} + \)\(77\!\cdots\!10\)\( p^{3} T^{12} - \)\(88\!\cdots\!04\)\( p^{6} T^{13} + \)\(11\!\cdots\!46\)\( p^{9} T^{14} - \)\(11\!\cdots\!72\)\( p^{12} T^{15} + 13617126187691726939 p^{15} T^{16} - 10824951276438830 p^{18} T^{17} + 10617391644051 p^{21} T^{18} - 6239064900 p^{24} T^{19} + 4945161 p^{27} T^{20} - 1670 p^{30} T^{21} + p^{33} T^{22} \) | |
| 83 | \( 1 + 785 T + 3628157 T^{2} + 2561291339 T^{3} + 6106733857403 T^{4} + 3838331407565426 T^{5} + 6277395658447349610 T^{6} + \)\(35\!\cdots\!74\)\( T^{7} + \)\(45\!\cdots\!91\)\( T^{8} + \)\(23\!\cdots\!45\)\( T^{9} + \)\(32\!\cdots\!92\)\( p T^{10} + \)\(13\!\cdots\!74\)\( T^{11} + \)\(32\!\cdots\!92\)\( p^{4} T^{12} + \)\(23\!\cdots\!45\)\( p^{6} T^{13} + \)\(45\!\cdots\!91\)\( p^{9} T^{14} + \)\(35\!\cdots\!74\)\( p^{12} T^{15} + 6277395658447349610 p^{15} T^{16} + 3838331407565426 p^{18} T^{17} + 6106733857403 p^{21} T^{18} + 2561291339 p^{24} T^{19} + 3628157 p^{27} T^{20} + 785 p^{30} T^{21} + p^{33} T^{22} \) | |
| 89 | \( 1 + 1464 T + 5443276 T^{2} + 6316683422 T^{3} + 13271648753948 T^{4} + 12743660757479464 T^{5} + 19860761721654995205 T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(21\!\cdots\!94\)\( T^{8} + \)\(15\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!12\)\( T^{10} + \)\(11\!\cdots\!80\)\( T^{11} + \)\(17\!\cdots\!12\)\( p^{3} T^{12} + \)\(15\!\cdots\!20\)\( p^{6} T^{13} + \)\(21\!\cdots\!94\)\( p^{9} T^{14} + \)\(16\!\cdots\!24\)\( p^{12} T^{15} + 19860761721654995205 p^{15} T^{16} + 12743660757479464 p^{18} T^{17} + 13271648753948 p^{21} T^{18} + 6316683422 p^{24} T^{19} + 5443276 p^{27} T^{20} + 1464 p^{30} T^{21} + p^{33} T^{22} \) | |
| 97 | \( 1 + 1184 T + 4957720 T^{2} + 6196310562 T^{3} + 13906616459220 T^{4} + 16067965778152232 T^{5} + 27379569641679938673 T^{6} + \)\(28\!\cdots\!88\)\( T^{7} + \)\(40\!\cdots\!86\)\( T^{8} + \)\(37\!\cdots\!76\)\( T^{9} + \)\(46\!\cdots\!24\)\( T^{10} + \)\(38\!\cdots\!48\)\( T^{11} + \)\(46\!\cdots\!24\)\( p^{3} T^{12} + \)\(37\!\cdots\!76\)\( p^{6} T^{13} + \)\(40\!\cdots\!86\)\( p^{9} T^{14} + \)\(28\!\cdots\!88\)\( p^{12} T^{15} + 27379569641679938673 p^{15} T^{16} + 16067965778152232 p^{18} T^{17} + 13906616459220 p^{21} T^{18} + 6196310562 p^{24} T^{19} + 4957720 p^{27} T^{20} + 1184 p^{30} T^{21} + p^{33} T^{22} \) | |
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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
−2.56103587971777888647895225344, −2.51479763467184788708093039341, −2.45073574928428306862085709275, −2.44841519528461437907868834501, −2.33494343591501595698675737256, −2.30236325061967710217384759854, −2.14253621915758030720896469976, −1.95673796018640755848089343447, −1.89785645645594271551510775176, −1.73573710651568320262262185881, −1.66661691083039426360732867849, −1.63741334925512298538449995310, −1.30963981824671546358533767301, −1.27616065729402246321671573654, −1.09081132858999051273434388320, −1.06573633355253555682397995696, −0.842229882166319846571970520690, −0.64602209567014149453714795269, −0.60757123828929473694772771136, −0.54206751388970193406496130733, −0.49704093264817287813424993915, −0.41057833914511877355750151860, −0.34086641397372680737368897935, −0.25434051860100545286118939762, −0.11761491436207687552123901019, 0.11761491436207687552123901019, 0.25434051860100545286118939762, 0.34086641397372680737368897935, 0.41057833914511877355750151860, 0.49704093264817287813424993915, 0.54206751388970193406496130733, 0.60757123828929473694772771136, 0.64602209567014149453714795269, 0.842229882166319846571970520690, 1.06573633355253555682397995696, 1.09081132858999051273434388320, 1.27616065729402246321671573654, 1.30963981824671546358533767301, 1.63741334925512298538449995310, 1.66661691083039426360732867849, 1.73573710651568320262262185881, 1.89785645645594271551510775176, 1.95673796018640755848089343447, 2.14253621915758030720896469976, 2.30236325061967710217384759854, 2.33494343591501595698675737256, 2.44841519528461437907868834501, 2.45073574928428306862085709275, 2.51479763467184788708093039341, 2.56103587971777888647895225344
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.