Dirichlet series
| L(s) = 1 | + 9·3-s + 4·4-s + 28·5-s + 12·7-s − 24·8-s + 27·9-s − 54·11-s + 36·12-s − 18·13-s + 252·15-s + 87·16-s − 80·17-s − 280·19-s + 112·20-s + 108·21-s − 392·23-s − 216·24-s + 618·25-s + 27·27-s + 48·28-s + 13·29-s + 413·31-s + 90·32-s − 486·33-s + 336·35-s + 108·36-s + 654·37-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1/2·4-s + 2.50·5-s + 0.647·7-s − 1.06·8-s + 9-s − 1.48·11-s + 0.866·12-s − 0.384·13-s + 4.33·15-s + 1.35·16-s − 1.14·17-s − 3.38·19-s + 1.25·20-s + 1.12·21-s − 3.55·23-s − 1.83·24-s + 4.94·25-s + 0.192·27-s + 0.323·28-s + 0.0832·29-s + 2.39·31-s + 0.497·32-s − 2.56·33-s + 1.62·35-s + 1/2·36-s + 2.90·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2968.65\) |
| Root analytic conductor: | \(1.39537\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 11^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(6.066376453\) |
| \(L(\frac12)\) | \(\approx\) | \(6.066376453\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{3} \) |
| 11 | \( 1 + 54 T + 7 T^{2} - 126342 T^{3} - 293882 p T^{4} + 588168 p^{2} T^{5} + 5611388 p^{3} T^{6} + 588168 p^{5} T^{7} - 293882 p^{7} T^{8} - 126342 p^{9} T^{9} + 7 p^{12} T^{10} + 54 p^{15} T^{11} + p^{18} T^{12} \) | |
| good | 2 | \( 1 - p^{2} T^{2} + 3 p^{3} T^{3} - 71 T^{4} - 141 p T^{5} + 991 T^{6} - 2439 T^{7} - 7897 T^{8} + 16893 p T^{9} - 10091 p^{2} T^{10} - 11955 p^{3} T^{11} + 74327 p^{4} T^{12} - 11955 p^{6} T^{13} - 10091 p^{8} T^{14} + 16893 p^{10} T^{15} - 7897 p^{12} T^{16} - 2439 p^{15} T^{17} + 991 p^{18} T^{18} - 141 p^{22} T^{19} - 71 p^{24} T^{20} + 3 p^{30} T^{21} - p^{32} T^{22} + p^{36} T^{24} \) |
| 5 | \( 1 - 28 T + 166 T^{2} + 2008 T^{3} - 6243 T^{4} - 39644 p T^{5} - 1179036 p T^{6} + 127925504 T^{7} - 250932132 T^{8} - 5175713236 T^{9} - 947298662 p T^{10} - 636606527596 T^{11} + 19293489775396 T^{12} - 636606527596 p^{3} T^{13} - 947298662 p^{7} T^{14} - 5175713236 p^{9} T^{15} - 250932132 p^{12} T^{16} + 127925504 p^{15} T^{17} - 1179036 p^{19} T^{18} - 39644 p^{22} T^{19} - 6243 p^{24} T^{20} + 2008 p^{27} T^{21} + 166 p^{30} T^{22} - 28 p^{33} T^{23} + p^{36} T^{24} \) | |
| 7 | \( 1 - 12 T - 292 T^{2} + 3268 T^{3} + 253084 T^{4} - 245580 p T^{5} - 15232479 p T^{6} + 798814860 T^{7} + 47875350316 T^{8} - 186188964726 T^{9} - 14936850681692 T^{10} - 2595426222344 T^{11} + 7067195488099391 T^{12} - 2595426222344 p^{3} T^{13} - 14936850681692 p^{6} T^{14} - 186188964726 p^{9} T^{15} + 47875350316 p^{12} T^{16} + 798814860 p^{15} T^{17} - 15232479 p^{19} T^{18} - 245580 p^{22} T^{19} + 253084 p^{24} T^{20} + 3268 p^{27} T^{21} - 292 p^{30} T^{22} - 12 p^{33} T^{23} + p^{36} T^{24} \) | |
| 13 | \( 1 + 18 T + 2580 T^{2} + 28134 T^{3} + 9792792 T^{4} - 132508020 T^{5} + 18118966899 T^{6} - 845007146202 T^{7} + 41552489371800 T^{8} - 3032707536653446 T^{9} + 27211290556502316 T^{10} - 10062545621894725254 T^{11} + 99326497359265182237 T^{12} - 10062545621894725254 p^{3} T^{13} + 27211290556502316 p^{6} T^{14} - 3032707536653446 p^{9} T^{15} + 41552489371800 p^{12} T^{16} - 845007146202 p^{15} T^{17} + 18118966899 p^{18} T^{18} - 132508020 p^{21} T^{19} + 9792792 p^{24} T^{20} + 28134 p^{27} T^{21} + 2580 p^{30} T^{22} + 18 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 + 80 T + 5981 T^{2} + 1271624 T^{3} + 6779602 p T^{4} + 10488840088 T^{5} + 1050907094986 T^{6} + 88631392584416 T^{7} + 7529874686088563 T^{8} + 593783963038148776 T^{9} + 47965315498679770506 T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!87\)\( T^{12} + \)\(35\!\cdots\!00\)\( p^{3} T^{13} + 47965315498679770506 p^{6} T^{14} + 593783963038148776 p^{9} T^{15} + 7529874686088563 p^{12} T^{16} + 88631392584416 p^{15} T^{17} + 1050907094986 p^{18} T^{18} + 10488840088 p^{21} T^{19} + 6779602 p^{25} T^{20} + 1271624 p^{27} T^{21} + 5981 p^{30} T^{22} + 80 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 280 T + 35745 T^{2} + 193110 p T^{3} + 447314010 T^{4} + 44109561030 T^{5} + 2823999409126 T^{6} + 165028979914510 T^{7} + 11066477722204575 T^{8} - 31812268720570 T^{9} - \)\(10\!\cdots\!30\)\( T^{10} - \)\(10\!\cdots\!90\)\( T^{11} - \)\(79\!\cdots\!69\)\( T^{12} - \)\(10\!\cdots\!90\)\( p^{3} T^{13} - \)\(10\!\cdots\!30\)\( p^{6} T^{14} - 31812268720570 p^{9} T^{15} + 11066477722204575 p^{12} T^{16} + 165028979914510 p^{15} T^{17} + 2823999409126 p^{18} T^{18} + 44109561030 p^{21} T^{19} + 447314010 p^{24} T^{20} + 193110 p^{28} T^{21} + 35745 p^{30} T^{22} + 280 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( ( 1 + 196 T + 50798 T^{2} + 7868182 T^{3} + 1226347462 T^{4} + 145519235908 T^{5} + 18400674137554 T^{6} + 145519235908 p^{3} T^{7} + 1226347462 p^{6} T^{8} + 7868182 p^{9} T^{9} + 50798 p^{12} T^{10} + 196 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( 1 - 13 T - 22361 T^{2} - 7268261 T^{3} + 1074887025 T^{4} - 8519956096 T^{5} + 9520176357252 T^{6} - 4111853450195056 T^{7} + 1165191373084224204 T^{8} - \)\(11\!\cdots\!16\)\( T^{9} - \)\(92\!\cdots\!90\)\( T^{10} - \)\(28\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!18\)\( T^{12} - \)\(28\!\cdots\!98\)\( p^{3} T^{13} - \)\(92\!\cdots\!90\)\( p^{6} T^{14} - \)\(11\!\cdots\!16\)\( p^{9} T^{15} + 1165191373084224204 p^{12} T^{16} - 4111853450195056 p^{15} T^{17} + 9520176357252 p^{18} T^{18} - 8519956096 p^{21} T^{19} + 1074887025 p^{24} T^{20} - 7268261 p^{27} T^{21} - 22361 p^{30} T^{22} - 13 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 413 T + 23883 T^{2} + 9132189 T^{3} - 2225395551 T^{4} + 772309998732 T^{5} - 105670038412940 T^{6} - 21115198783619612 T^{7} + 5412938378503579212 T^{8} - \)\(44\!\cdots\!72\)\( T^{9} + \)\(77\!\cdots\!30\)\( T^{10} + \)\(13\!\cdots\!22\)\( T^{11} - \)\(73\!\cdots\!62\)\( T^{12} + \)\(13\!\cdots\!22\)\( p^{3} T^{13} + \)\(77\!\cdots\!30\)\( p^{6} T^{14} - \)\(44\!\cdots\!72\)\( p^{9} T^{15} + 5412938378503579212 p^{12} T^{16} - 21115198783619612 p^{15} T^{17} - 105670038412940 p^{18} T^{18} + 772309998732 p^{21} T^{19} - 2225395551 p^{24} T^{20} + 9132189 p^{27} T^{21} + 23883 p^{30} T^{22} - 413 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( 1 - 654 T + 74760 T^{2} + 21496536 T^{3} + 3172278300 T^{4} - 4613962451016 T^{5} + 914242818447663 T^{6} + 42769681130442984 T^{7} - 18021863201800227252 T^{8} - \)\(13\!\cdots\!24\)\( T^{9} + \)\(48\!\cdots\!04\)\( T^{10} - \)\(23\!\cdots\!54\)\( T^{11} - \)\(91\!\cdots\!35\)\( T^{12} - \)\(23\!\cdots\!54\)\( p^{3} T^{13} + \)\(48\!\cdots\!04\)\( p^{6} T^{14} - \)\(13\!\cdots\!24\)\( p^{9} T^{15} - 18021863201800227252 p^{12} T^{16} + 42769681130442984 p^{15} T^{17} + 914242818447663 p^{18} T^{18} - 4613962451016 p^{21} T^{19} + 3172278300 p^{24} T^{20} + 21496536 p^{27} T^{21} + 74760 p^{30} T^{22} - 654 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 + 1490 T + 880858 T^{2} + 231230716 T^{3} - 1247845038 T^{4} - 21538077143224 T^{5} - 7213863330589185 T^{6} - 376292589237375262 T^{7} + \)\(59\!\cdots\!24\)\( T^{8} + \)\(23\!\cdots\!26\)\( T^{9} + \)\(25\!\cdots\!30\)\( T^{10} - \)\(93\!\cdots\!70\)\( T^{11} - \)\(44\!\cdots\!67\)\( T^{12} - \)\(93\!\cdots\!70\)\( p^{3} T^{13} + \)\(25\!\cdots\!30\)\( p^{6} T^{14} + \)\(23\!\cdots\!26\)\( p^{9} T^{15} + \)\(59\!\cdots\!24\)\( p^{12} T^{16} - 376292589237375262 p^{15} T^{17} - 7213863330589185 p^{18} T^{18} - 21538077143224 p^{21} T^{19} - 1247845038 p^{24} T^{20} + 231230716 p^{27} T^{21} + 880858 p^{30} T^{22} + 1490 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( ( 1 - 208 T + 292746 T^{2} - 28880738 T^{3} + 37781127122 T^{4} - 1726024887592 T^{5} + 3403809232886930 T^{6} - 1726024887592 p^{3} T^{7} + 37781127122 p^{6} T^{8} - 28880738 p^{9} T^{9} + 292746 p^{12} T^{10} - 208 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( 1 + 150 T - 229834 T^{2} - 24484236 T^{3} + 4788515779 T^{4} - 1166873288142 T^{5} + 4483635535981786 T^{6} + 613830793032007656 T^{7} - \)\(52\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!34\)\( T^{9} - \)\(21\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!50\)\( T^{11} + \)\(78\!\cdots\!12\)\( T^{12} + \)\(28\!\cdots\!50\)\( p^{3} T^{13} - \)\(21\!\cdots\!84\)\( p^{6} T^{14} - \)\(66\!\cdots\!34\)\( p^{9} T^{15} - \)\(52\!\cdots\!02\)\( p^{12} T^{16} + 613830793032007656 p^{15} T^{17} + 4483635535981786 p^{18} T^{18} - 1166873288142 p^{21} T^{19} + 4788515779 p^{24} T^{20} - 24484236 p^{27} T^{21} - 229834 p^{30} T^{22} + 150 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( 1 - 1359 T + 701543 T^{2} - 211527147 T^{3} + 104475729361 T^{4} - 64065057259272 T^{5} + 21267785855167432 T^{6} - 4359429544881114288 T^{7} + \)\(22\!\cdots\!92\)\( T^{8} - \)\(13\!\cdots\!96\)\( T^{9} + \)\(51\!\cdots\!14\)\( T^{10} - \)\(35\!\cdots\!30\)\( p T^{11} + \)\(79\!\cdots\!70\)\( T^{12} - \)\(35\!\cdots\!30\)\( p^{4} T^{13} + \)\(51\!\cdots\!14\)\( p^{6} T^{14} - \)\(13\!\cdots\!96\)\( p^{9} T^{15} + \)\(22\!\cdots\!92\)\( p^{12} T^{16} - 4359429544881114288 p^{15} T^{17} + 21267785855167432 p^{18} T^{18} - 64065057259272 p^{21} T^{19} + 104475729361 p^{24} T^{20} - 211527147 p^{27} T^{21} + 701543 p^{30} T^{22} - 1359 p^{33} T^{23} + p^{36} T^{24} \) | |
| 59 | \( 1 - 1262 T + 275411 T^{2} + 319635658 T^{3} - 142409552824 T^{4} - 89740709300188 T^{5} + 73592642874959518 T^{6} + 1626297918888671692 T^{7} - \)\(12\!\cdots\!87\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} + \)\(33\!\cdots\!74\)\( T^{10} - \)\(27\!\cdots\!40\)\( T^{11} - \)\(45\!\cdots\!91\)\( T^{12} - \)\(27\!\cdots\!40\)\( p^{3} T^{13} + \)\(33\!\cdots\!74\)\( p^{6} T^{14} - \)\(11\!\cdots\!48\)\( p^{9} T^{15} - \)\(12\!\cdots\!87\)\( p^{12} T^{16} + 1626297918888671692 p^{15} T^{17} + 73592642874959518 p^{18} T^{18} - 89740709300188 p^{21} T^{19} - 142409552824 p^{24} T^{20} + 319635658 p^{27} T^{21} + 275411 p^{30} T^{22} - 1262 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1044 T + 504986 T^{2} + 424821958 T^{3} + 336901340533 T^{4} + 216760940872116 T^{5} + 139959802763217936 T^{6} + 77515379143789767996 T^{7} + \)\(43\!\cdots\!44\)\( T^{8} + \)\(24\!\cdots\!24\)\( T^{9} + \)\(13\!\cdots\!26\)\( T^{10} + \)\(64\!\cdots\!58\)\( T^{11} + \)\(30\!\cdots\!16\)\( T^{12} + \)\(64\!\cdots\!58\)\( p^{3} T^{13} + \)\(13\!\cdots\!26\)\( p^{6} T^{14} + \)\(24\!\cdots\!24\)\( p^{9} T^{15} + \)\(43\!\cdots\!44\)\( p^{12} T^{16} + 77515379143789767996 p^{15} T^{17} + 139959802763217936 p^{18} T^{18} + 216760940872116 p^{21} T^{19} + 336901340533 p^{24} T^{20} + 424821958 p^{27} T^{21} + 504986 p^{30} T^{22} + 1044 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( ( 1 + 264 T - 23619 T^{2} - 79408030 T^{3} + 61162456014 T^{4} + 20942991876474 T^{5} + 29398858296591696 T^{6} + 20942991876474 p^{3} T^{7} + 61162456014 p^{6} T^{8} - 79408030 p^{9} T^{9} - 23619 p^{12} T^{10} + 264 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( 1 - 558 T - 214976 T^{2} + 306079236 T^{3} + 184582139233 T^{4} - 28441797589062 T^{5} - 159756033892200858 T^{6} + 77696498899965080124 T^{7} + \)\(51\!\cdots\!50\)\( T^{8} - \)\(94\!\cdots\!34\)\( T^{9} - \)\(18\!\cdots\!78\)\( T^{10} - \)\(22\!\cdots\!30\)\( T^{11} + \)\(12\!\cdots\!84\)\( T^{12} - \)\(22\!\cdots\!30\)\( p^{3} T^{13} - \)\(18\!\cdots\!78\)\( p^{6} T^{14} - \)\(94\!\cdots\!34\)\( p^{9} T^{15} + \)\(51\!\cdots\!50\)\( p^{12} T^{16} + 77696498899965080124 p^{15} T^{17} - 159756033892200858 p^{18} T^{18} - 28441797589062 p^{21} T^{19} + 184582139233 p^{24} T^{20} + 306079236 p^{27} T^{21} - 214976 p^{30} T^{22} - 558 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( 1 + 699 T + 552765 T^{2} - 109264537 T^{3} + 81592008873 T^{4} - 205792245455572 T^{5} - 26056419673978324 T^{6} - 97921969488226725576 T^{7} + \)\(54\!\cdots\!68\)\( T^{8} - \)\(20\!\cdots\!12\)\( T^{9} + \)\(18\!\cdots\!70\)\( T^{10} - \)\(13\!\cdots\!18\)\( T^{11} + \)\(93\!\cdots\!74\)\( T^{12} - \)\(13\!\cdots\!18\)\( p^{3} T^{13} + \)\(18\!\cdots\!70\)\( p^{6} T^{14} - \)\(20\!\cdots\!12\)\( p^{9} T^{15} + \)\(54\!\cdots\!68\)\( p^{12} T^{16} - 97921969488226725576 p^{15} T^{17} - 26056419673978324 p^{18} T^{18} - 205792245455572 p^{21} T^{19} + 81592008873 p^{24} T^{20} - 109264537 p^{27} T^{21} + 552765 p^{30} T^{22} + 699 p^{33} T^{23} + p^{36} T^{24} \) | |
| 79 | \( 1 + 1252 T - 706116 T^{2} - 920771466 T^{3} + 605427548280 T^{4} + 160463564258004 T^{5} - 635347097378782703 T^{6} - 61745950226810786696 T^{7} + \)\(19\!\cdots\!84\)\( T^{8} - \)\(60\!\cdots\!66\)\( T^{9} - \)\(29\!\cdots\!20\)\( T^{10} + \)\(43\!\cdots\!92\)\( T^{11} + \)\(29\!\cdots\!03\)\( T^{12} + \)\(43\!\cdots\!92\)\( p^{3} T^{13} - \)\(29\!\cdots\!20\)\( p^{6} T^{14} - \)\(60\!\cdots\!66\)\( p^{9} T^{15} + \)\(19\!\cdots\!84\)\( p^{12} T^{16} - 61745950226810786696 p^{15} T^{17} - 635347097378782703 p^{18} T^{18} + 160463564258004 p^{21} T^{19} + 605427548280 p^{24} T^{20} - 920771466 p^{27} T^{21} - 706116 p^{30} T^{22} + 1252 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 + 4464 T + 7776076 T^{2} + 5918441448 T^{3} + 358006292308 T^{4} - 2898431576154492 T^{5} - 2332832226440682945 T^{6} - 81475410547409243304 T^{7} + \)\(19\!\cdots\!76\)\( T^{8} + \)\(21\!\cdots\!98\)\( T^{9} + \)\(67\!\cdots\!60\)\( T^{10} - \)\(75\!\cdots\!60\)\( T^{11} - \)\(10\!\cdots\!57\)\( T^{12} - \)\(75\!\cdots\!60\)\( p^{3} T^{13} + \)\(67\!\cdots\!60\)\( p^{6} T^{14} + \)\(21\!\cdots\!98\)\( p^{9} T^{15} + \)\(19\!\cdots\!76\)\( p^{12} T^{16} - 81475410547409243304 p^{15} T^{17} - 2332832226440682945 p^{18} T^{18} - 2898431576154492 p^{21} T^{19} + 358006292308 p^{24} T^{20} + 5918441448 p^{27} T^{21} + 7776076 p^{30} T^{22} + 4464 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( ( 1 - 158 T + 2359364 T^{2} - 294587366 T^{3} + 2380741274284 T^{4} - 254691947575022 T^{5} + 1709394429694985182 T^{6} - 254691947575022 p^{3} T^{7} + 2380741274284 p^{6} T^{8} - 294587366 p^{9} T^{9} + 2359364 p^{12} T^{10} - 158 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( 1 - 1608 T + 2107194 T^{2} - 526452544 T^{3} - 693523220127 T^{4} + 2283715532884784 T^{5} - 1050434593951340620 T^{6} - \)\(60\!\cdots\!08\)\( T^{7} + \)\(29\!\cdots\!96\)\( T^{8} - \)\(26\!\cdots\!56\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} - \)\(23\!\cdots\!12\)\( T^{12} + \)\(16\!\cdots\!80\)\( p^{3} T^{13} + \)\(10\!\cdots\!10\)\( p^{6} T^{14} - \)\(26\!\cdots\!56\)\( p^{9} T^{15} + \)\(29\!\cdots\!96\)\( p^{12} T^{16} - \)\(60\!\cdots\!08\)\( p^{15} T^{17} - 1050434593951340620 p^{18} T^{18} + 2283715532884784 p^{21} T^{19} - 693523220127 p^{24} T^{20} - 526452544 p^{27} T^{21} + 2107194 p^{30} T^{22} - 1608 p^{33} T^{23} + p^{36} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.08888964094089180518060638183, −6.05259101946214510209683846554, −5.77708058162761477179307767467, −5.42368404494586509073436703633, −5.39077542762661344446445747360, −5.37100274233153274389910985992, −5.36801250442759266170957059512, −4.75935449459932389256205528145, −4.52886337626886367500816656837, −4.48069246512469574362811245238, −4.39016487474488471701468751592, −4.34674395697612340356973556155, −4.02944129016999110590153255201, −3.83985007288244693883992186482, −3.45564765013432837834607068994, −3.01855952416486627062960827742, −2.91306726018415625311837657410, −2.90834818557474824855845871293, −2.66913916292836820110596499036, −2.30345123669569882701715729987, −2.21276575554957768119308440266, −2.13414026844156928401540389062, −1.74163632182615716172540996269, −1.36877990919721438725607690042, −0.43282054496099308948135698084, 0.43282054496099308948135698084, 1.36877990919721438725607690042, 1.74163632182615716172540996269, 2.13414026844156928401540389062, 2.21276575554957768119308440266, 2.30345123669569882701715729987, 2.66913916292836820110596499036, 2.90834818557474824855845871293, 2.91306726018415625311837657410, 3.01855952416486627062960827742, 3.45564765013432837834607068994, 3.83985007288244693883992186482, 4.02944129016999110590153255201, 4.34674395697612340356973556155, 4.39016487474488471701468751592, 4.48069246512469574362811245238, 4.52886337626886367500816656837, 4.75935449459932389256205528145, 5.36801250442759266170957059512, 5.37100274233153274389910985992, 5.39077542762661344446445747360, 5.42368404494586509073436703633, 5.77708058162761477179307767467, 6.05259101946214510209683846554, 6.08888964094089180518060638183
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.