Dirichlet series
| L(s) = 1 | + 2·4-s − 2·5-s + 2·7-s + 4·8-s + 2·11-s + 2·16-s + 6·17-s − 2·19-s − 4·20-s + 2·23-s − 25-s + 4·28-s − 14·29-s + 4·32-s − 4·35-s − 8·40-s + 4·44-s − 38·47-s + 2·49-s − 12·53-s − 4·55-s + 8·56-s − 10·59-s + 14·61-s + 12·64-s + 12·68-s − 24·71-s + ⋯ |
| L(s) = 1 | + 4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s + 0.603·11-s + 1/2·16-s + 1.45·17-s − 0.458·19-s − 0.894·20-s + 0.417·23-s − 1/5·25-s + 0.755·28-s − 2.59·29-s + 0.707·32-s − 0.676·35-s − 1.26·40-s + 0.603·44-s − 5.54·47-s + 2/7·49-s − 1.64·53-s − 0.539·55-s + 1.06·56-s − 1.30·59-s + 1.79·61-s + 3/2·64-s + 1.45·68-s − 2.84·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{72} \cdot 3^{36} \cdot 5^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.70947\times 10^{13}\) |
| Root analytic conductor: | \(2.39775\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{72} \cdot 3^{36} \cdot 5^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.550408435\) |
| \(L(\frac12)\) | \(\approx\) | \(8.550408435\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T^{2} - p^{2} T^{3} + p T^{4} + 3 p^{2} T^{5} + p^{2} T^{6} - p^{3} T^{7} - 3 p^{2} T^{8} + p^{3} T^{9} - 3 p^{3} T^{10} - p^{5} T^{11} + p^{5} T^{12} + 3 p^{6} T^{13} + p^{6} T^{14} - p^{8} T^{15} - p^{8} T^{16} + p^{9} T^{18} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 + 2 T + p T^{2} + 16 T^{3} + 36 T^{4} + 104 T^{5} + 308 T^{6} + 688 T^{7} + 302 p T^{8} + 2572 T^{9} + 302 p^{2} T^{10} + 688 p^{2} T^{11} + 308 p^{3} T^{12} + 104 p^{4} T^{13} + 36 p^{5} T^{14} + 16 p^{6} T^{15} + p^{8} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 7 | \( 1 - 2 T + 2 T^{2} + 18 T^{3} - 31 T^{4} - 232 T^{5} + 688 T^{6} - 1720 T^{7} - 4140 T^{8} + 1608 p T^{9} + 3048 T^{10} - 91832 T^{11} + 432692 T^{12} - 46104 T^{13} - 615536 T^{14} + 3870968 T^{15} - 2489450 T^{16} - 35542124 T^{17} + 73581516 T^{18} - 35542124 p T^{19} - 2489450 p^{2} T^{20} + 3870968 p^{3} T^{21} - 615536 p^{4} T^{22} - 46104 p^{5} T^{23} + 432692 p^{6} T^{24} - 91832 p^{7} T^{25} + 3048 p^{8} T^{26} + 1608 p^{10} T^{27} - 4140 p^{10} T^{28} - 1720 p^{11} T^{29} + 688 p^{12} T^{30} - 232 p^{13} T^{31} - 31 p^{14} T^{32} + 18 p^{15} T^{33} + 2 p^{16} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) |
| 11 | \( 1 - 2 T + 2 T^{2} + 58 T^{3} - 175 T^{4} - 288 T^{5} + 2608 T^{6} - 8640 T^{7} - 7884 T^{8} + 120024 T^{9} - 227528 T^{10} + 65960 T^{11} + 5424292 T^{12} - 10553216 T^{13} - 2414512 T^{14} + 173878688 T^{15} - 3577314 p^{2} T^{16} - 74110004 p T^{17} + 5890482188 T^{18} - 74110004 p^{2} T^{19} - 3577314 p^{4} T^{20} + 173878688 p^{3} T^{21} - 2414512 p^{4} T^{22} - 10553216 p^{5} T^{23} + 5424292 p^{6} T^{24} + 65960 p^{7} T^{25} - 227528 p^{8} T^{26} + 120024 p^{9} T^{27} - 7884 p^{10} T^{28} - 8640 p^{11} T^{29} + 2608 p^{12} T^{30} - 288 p^{13} T^{31} - 175 p^{14} T^{32} + 58 p^{15} T^{33} + 2 p^{16} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 - 122 T^{2} + 7537 T^{4} - 313680 T^{6} + 760180 p T^{8} - 250866392 T^{10} + 5327881268 T^{12} - 96936565808 T^{14} + 1533425252030 T^{16} - 21263532733660 T^{18} + 1533425252030 p^{2} T^{20} - 96936565808 p^{4} T^{22} + 5327881268 p^{6} T^{24} - 250866392 p^{8} T^{26} + 760180 p^{11} T^{28} - 313680 p^{12} T^{30} + 7537 p^{14} T^{32} - 122 p^{16} T^{34} + p^{18} T^{36} \) | |
| 17 | \( 1 - 6 T + 18 T^{2} - 134 T^{3} + 1017 T^{4} - 5360 T^{5} + 22832 T^{6} - 149296 T^{7} + 676660 T^{8} - 8712 p^{2} T^{9} + 14077048 T^{10} - 72254792 T^{11} + 312308676 T^{12} - 71484880 p T^{13} + 6226069072 T^{14} - 26188169744 T^{15} + 97051677358 T^{16} - 475514965764 T^{17} + 2098086654444 T^{18} - 475514965764 p T^{19} + 97051677358 p^{2} T^{20} - 26188169744 p^{3} T^{21} + 6226069072 p^{4} T^{22} - 71484880 p^{6} T^{23} + 312308676 p^{6} T^{24} - 72254792 p^{7} T^{25} + 14077048 p^{8} T^{26} - 8712 p^{11} T^{27} + 676660 p^{10} T^{28} - 149296 p^{11} T^{29} + 22832 p^{12} T^{30} - 5360 p^{13} T^{31} + 1017 p^{14} T^{32} - 134 p^{15} T^{33} + 18 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 19 | \( 1 + 2 T + 2 T^{2} + 102 T^{3} + 641 T^{4} + 1264 T^{5} + 6448 T^{6} + 94800 T^{7} + 201588 T^{8} + 18344 p T^{9} + 348904 p T^{10} + 35889000 T^{11} + 72631588 T^{12} + 371841296 T^{13} + 3644025680 T^{14} + 12117184176 T^{15} + 34196073198 T^{16} + 241157330252 T^{17} + 1442387298060 T^{18} + 241157330252 p T^{19} + 34196073198 p^{2} T^{20} + 12117184176 p^{3} T^{21} + 3644025680 p^{4} T^{22} + 371841296 p^{5} T^{23} + 72631588 p^{6} T^{24} + 35889000 p^{7} T^{25} + 348904 p^{9} T^{26} + 18344 p^{10} T^{27} + 201588 p^{10} T^{28} + 94800 p^{11} T^{29} + 6448 p^{12} T^{30} + 1264 p^{13} T^{31} + 641 p^{14} T^{32} + 102 p^{15} T^{33} + 2 p^{16} T^{34} + 2 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 - 2 T + 2 T^{2} - 70 T^{3} - 591 T^{4} + 2960 T^{5} - 2288 T^{6} + 63760 T^{7} + 89092 T^{8} - 1178512 T^{9} + 744232 T^{10} - 17437280 T^{11} - 6233124 p T^{12} + 676178128 T^{13} - 657066064 T^{14} + 23455281104 T^{15} + 63771483110 T^{16} - 805933417604 T^{17} + 643942666316 T^{18} - 805933417604 p T^{19} + 63771483110 p^{2} T^{20} + 23455281104 p^{3} T^{21} - 657066064 p^{4} T^{22} + 676178128 p^{5} T^{23} - 6233124 p^{7} T^{24} - 17437280 p^{7} T^{25} + 744232 p^{8} T^{26} - 1178512 p^{9} T^{27} + 89092 p^{10} T^{28} + 63760 p^{11} T^{29} - 2288 p^{12} T^{30} + 2960 p^{13} T^{31} - 591 p^{14} T^{32} - 70 p^{15} T^{33} + 2 p^{16} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 + 14 T + 98 T^{2} + 726 T^{3} + 6401 T^{4} + 42960 T^{5} + 237680 T^{6} + 1513232 T^{7} + 10244180 T^{8} + 56892360 T^{9} + 292560952 T^{10} + 1753452328 T^{11} + 10724984644 T^{12} + 1897307120 p T^{13} + 273776558096 T^{14} + 1576335370736 T^{15} + 8981682474190 T^{16} + 44280736253716 T^{17} + 221031341187660 T^{18} + 44280736253716 p T^{19} + 8981682474190 p^{2} T^{20} + 1576335370736 p^{3} T^{21} + 273776558096 p^{4} T^{22} + 1897307120 p^{6} T^{23} + 10724984644 p^{6} T^{24} + 1753452328 p^{7} T^{25} + 292560952 p^{8} T^{26} + 56892360 p^{9} T^{27} + 10244180 p^{10} T^{28} + 1513232 p^{11} T^{29} + 237680 p^{12} T^{30} + 42960 p^{13} T^{31} + 6401 p^{14} T^{32} + 726 p^{15} T^{33} + 98 p^{16} T^{34} + 14 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 - 362 T^{2} + 62649 T^{4} - 6948784 T^{6} + 560274836 T^{8} - 35376015000 T^{10} + 1837904876644 T^{12} - 81110278812112 T^{14} + 3093814373899278 T^{16} - 102716502829267196 T^{18} + 3093814373899278 p^{2} T^{20} - 81110278812112 p^{4} T^{22} + 1837904876644 p^{6} T^{24} - 35376015000 p^{8} T^{26} + 560274836 p^{10} T^{28} - 6948784 p^{12} T^{30} + 62649 p^{14} T^{32} - 362 p^{16} T^{34} + p^{18} T^{36} \) | |
| 37 | \( 1 - 378 T^{2} + 71905 T^{4} - 9096912 T^{6} + 856912292 T^{8} - 63946093016 T^{10} + 3934223155252 T^{12} - 205214014299312 T^{14} + 9255275087166462 T^{16} - 365303320505490396 T^{18} + 9255275087166462 p^{2} T^{20} - 205214014299312 p^{4} T^{22} + 3934223155252 p^{6} T^{24} - 63946093016 p^{8} T^{26} + 856912292 p^{10} T^{28} - 9096912 p^{12} T^{30} + 71905 p^{14} T^{32} - 378 p^{16} T^{34} + p^{18} T^{36} \) | |
| 41 | \( 1 - 450 T^{2} + 100441 T^{4} - 14781456 T^{6} + 1610393956 T^{8} - 138340541496 T^{10} + 9747518002260 T^{12} - 578326413361776 T^{14} + 29398753828363198 T^{16} - 1293459229377566220 T^{18} + 29398753828363198 p^{2} T^{20} - 578326413361776 p^{4} T^{22} + 9747518002260 p^{6} T^{24} - 138340541496 p^{8} T^{26} + 1610393956 p^{10} T^{28} - 14781456 p^{12} T^{30} + 100441 p^{14} T^{32} - 450 p^{16} T^{34} + p^{18} T^{36} \) | |
| 43 | \( 1 - 434 T^{2} + 95497 T^{4} - 14132576 T^{6} + 1575069332 T^{8} - 140307196616 T^{10} + 10347976565460 T^{12} - 645785319162016 T^{14} + 34557565852762614 T^{16} - 1597142837246964908 T^{18} + 34557565852762614 p^{2} T^{20} - 645785319162016 p^{4} T^{22} + 10347976565460 p^{6} T^{24} - 140307196616 p^{8} T^{26} + 1575069332 p^{10} T^{28} - 14132576 p^{12} T^{30} + 95497 p^{14} T^{32} - 434 p^{16} T^{34} + p^{18} T^{36} \) | |
| 47 | \( 1 + 38 T + 722 T^{2} + 214 p T^{3} + 122161 T^{4} + 1328456 T^{5} + 12862768 T^{6} + 113701112 T^{7} + 943490484 T^{8} + 7442197960 T^{9} + 56278247848 T^{10} + 412597571480 T^{11} + 2981952610452 T^{12} + 21538353011032 T^{13} + 155091730527504 T^{14} + 1106483733730792 T^{15} + 7843790068463798 T^{16} + 55222405019152308 T^{17} + 382643757483978924 T^{18} + 55222405019152308 p T^{19} + 7843790068463798 p^{2} T^{20} + 1106483733730792 p^{3} T^{21} + 155091730527504 p^{4} T^{22} + 21538353011032 p^{5} T^{23} + 2981952610452 p^{6} T^{24} + 412597571480 p^{7} T^{25} + 56278247848 p^{8} T^{26} + 7442197960 p^{9} T^{27} + 943490484 p^{10} T^{28} + 113701112 p^{11} T^{29} + 12862768 p^{12} T^{30} + 1328456 p^{13} T^{31} + 122161 p^{14} T^{32} + 214 p^{16} T^{33} + 722 p^{16} T^{34} + 38 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( ( 1 + 6 T + 341 T^{2} + 2064 T^{3} + 56388 T^{4} + 330040 T^{5} + 5912980 T^{6} + 32036656 T^{7} + 432884038 T^{8} + 2061378468 T^{9} + 432884038 p T^{10} + 32036656 p^{2} T^{11} + 5912980 p^{3} T^{12} + 330040 p^{4} T^{13} + 56388 p^{5} T^{14} + 2064 p^{6} T^{15} + 341 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 59 | \( 1 + 10 T + 50 T^{2} + 494 T^{3} - 1743 T^{4} - 33136 T^{5} - 122192 T^{6} - 748688 T^{7} + 22720756 T^{8} + 207434808 T^{9} + 904475000 T^{10} + 8738584488 T^{11} - 73057283740 T^{12} - 1079521529168 T^{13} - 5065988762672 T^{14} - 53579997360688 T^{15} - 92361378765970 T^{16} + 1839910849923292 T^{17} + 8984420762513708 T^{18} + 1839910849923292 p T^{19} - 92361378765970 p^{2} T^{20} - 53579997360688 p^{3} T^{21} - 5065988762672 p^{4} T^{22} - 1079521529168 p^{5} T^{23} - 73057283740 p^{6} T^{24} + 8738584488 p^{7} T^{25} + 904475000 p^{8} T^{26} + 207434808 p^{9} T^{27} + 22720756 p^{10} T^{28} - 748688 p^{11} T^{29} - 122192 p^{12} T^{30} - 33136 p^{13} T^{31} - 1743 p^{14} T^{32} + 494 p^{15} T^{33} + 50 p^{16} T^{34} + 10 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 - 14 T + 98 T^{2} - 134 T^{3} + 5041 T^{4} - 83168 T^{5} + 679312 T^{6} - 36608 T^{7} + 160260 T^{8} - 147217176 T^{9} + 2084286776 T^{10} + 4953407144 T^{11} - 49660935404 T^{12} - 134974970304 T^{13} + 7721336684016 T^{14} + 9084322150944 T^{15} - 105587914401314 T^{16} - 961048211302948 T^{17} + 38943687185882188 T^{18} - 961048211302948 p T^{19} - 105587914401314 p^{2} T^{20} + 9084322150944 p^{3} T^{21} + 7721336684016 p^{4} T^{22} - 134974970304 p^{5} T^{23} - 49660935404 p^{6} T^{24} + 4953407144 p^{7} T^{25} + 2084286776 p^{8} T^{26} - 147217176 p^{9} T^{27} + 160260 p^{10} T^{28} - 36608 p^{11} T^{29} + 679312 p^{12} T^{30} - 83168 p^{13} T^{31} + 5041 p^{14} T^{32} - 134 p^{15} T^{33} + 98 p^{16} T^{34} - 14 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 642 T^{2} + 202585 T^{4} - 42240800 T^{6} + 6587340692 T^{8} - 822533069448 T^{10} + 85863586532372 T^{12} - 7723657296997344 T^{14} + 611883598063154038 T^{16} - 43282699253224131468 T^{18} + 611883598063154038 p^{2} T^{20} - 7723657296997344 p^{4} T^{22} + 85863586532372 p^{6} T^{24} - 822533069448 p^{8} T^{26} + 6587340692 p^{10} T^{28} - 42240800 p^{12} T^{30} + 202585 p^{14} T^{32} - 642 p^{16} T^{34} + p^{18} T^{36} \) | |
| 71 | \( ( 1 + 12 T + 487 T^{2} + 5408 T^{3} + 109276 T^{4} + 1117104 T^{5} + 15152068 T^{6} + 140510560 T^{7} + 1464429734 T^{8} + 11951169672 T^{9} + 1464429734 p T^{10} + 140510560 p^{2} T^{11} + 15152068 p^{3} T^{12} + 1117104 p^{4} T^{13} + 109276 p^{5} T^{14} + 5408 p^{6} T^{15} + 487 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 73 | \( 1 + 14 T + 98 T^{2} + 478 T^{3} + 18985 T^{4} + 253168 T^{5} + 1798064 T^{6} + 6514992 T^{7} + 139557044 T^{8} + 2029936680 T^{9} + 15767462456 T^{10} + 33693552040 T^{11} + 292899083780 T^{12} + 8285915235920 T^{13} + 88264019259856 T^{14} + 73937299816080 T^{15} - 2231393636040914 T^{16} + 15114080754166388 T^{17} + 440374853480206540 T^{18} + 15114080754166388 p T^{19} - 2231393636040914 p^{2} T^{20} + 73937299816080 p^{3} T^{21} + 88264019259856 p^{4} T^{22} + 8285915235920 p^{5} T^{23} + 292899083780 p^{6} T^{24} + 33693552040 p^{7} T^{25} + 15767462456 p^{8} T^{26} + 2029936680 p^{9} T^{27} + 139557044 p^{10} T^{28} + 6514992 p^{11} T^{29} + 1798064 p^{12} T^{30} + 253168 p^{13} T^{31} + 18985 p^{14} T^{32} + 478 p^{15} T^{33} + 98 p^{16} T^{34} + 14 p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( ( 1 + 8 T + 391 T^{2} + 2080 T^{3} + 75204 T^{4} + 283424 T^{5} + 124660 p T^{6} + 28571616 T^{7} + 990116222 T^{8} + 2449460656 T^{9} + 990116222 p T^{10} + 28571616 p^{2} T^{11} + 124660 p^{4} T^{12} + 283424 p^{4} T^{13} + 75204 p^{5} T^{14} + 2080 p^{6} T^{15} + 391 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 83 | \( ( 1 + 20 T + 611 T^{2} + 9668 T^{3} + 176280 T^{4} + 2256568 T^{5} + 31051408 T^{6} + 330148068 T^{7} + 3680856986 T^{8} + 32879739720 T^{9} + 3680856986 p T^{10} + 330148068 p^{2} T^{11} + 31051408 p^{3} T^{12} + 2256568 p^{4} T^{13} + 176280 p^{5} T^{14} + 9668 p^{6} T^{15} + 611 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 89 | \( ( 1 + 6 T + 353 T^{2} + 1520 T^{3} + 61156 T^{4} + 2152 p T^{5} + 7861620 T^{6} + 23163024 T^{7} + 859994622 T^{8} + 2479282468 T^{9} + 859994622 p T^{10} + 23163024 p^{2} T^{11} + 7861620 p^{3} T^{12} + 2152 p^{5} T^{13} + 61156 p^{5} T^{14} + 1520 p^{6} T^{15} + 353 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 97 | \( 1 - 18 T + 162 T^{2} - 1234 T^{3} + 9625 T^{4} - 177840 T^{5} + 2403248 T^{6} - 340912 p T^{7} + 465762612 T^{8} - 3963438712 T^{9} + 29377109048 T^{10} - 206755021688 T^{11} + 2522153752132 T^{12} - 44692244379088 T^{13} + 500529043592144 T^{14} - 5501619757610192 T^{15} + 44798006883295918 T^{16} - 270735800376809964 T^{17} + 2482816968072160076 T^{18} - 270735800376809964 p T^{19} + 44798006883295918 p^{2} T^{20} - 5501619757610192 p^{3} T^{21} + 500529043592144 p^{4} T^{22} - 44692244379088 p^{5} T^{23} + 2522153752132 p^{6} T^{24} - 206755021688 p^{7} T^{25} + 29377109048 p^{8} T^{26} - 3963438712 p^{9} T^{27} + 465762612 p^{10} T^{28} - 340912 p^{12} T^{29} + 2403248 p^{12} T^{30} - 177840 p^{13} T^{31} + 9625 p^{14} T^{32} - 1234 p^{15} T^{33} + 162 p^{16} T^{34} - 18 p^{17} T^{35} + p^{18} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.71924991682344631732368469011, −2.48384258031328905305246645501, −2.42529080253536476815135331010, −2.30001520835118670424065807324, −2.12582546639631723730163891947, −2.06377308222490913915247213303, −2.04637133957110228225353267028, −1.85881385036400464265293690147, −1.82209836319717618781399543043, −1.78096817855320364286018023669, −1.65934967960976533025809081225, −1.65442082248678146133020744321, −1.64582762771007513126847414122, −1.51619365351225217441680747555, −1.48930584823784622082293184812, −1.41911897526617904150632197472, −1.29023126328936466492519661417, −1.14073892554907882156992967011, −1.12591427189849845333061736068, −1.10152792793602178871218149934, −0.61087239565339544816742762933, −0.51443373386913311930646162525, −0.38938034209023880542467143473, −0.29298373649545751405285094752, −0.23470889926297489077378308303, 0.23470889926297489077378308303, 0.29298373649545751405285094752, 0.38938034209023880542467143473, 0.51443373386913311930646162525, 0.61087239565339544816742762933, 1.10152792793602178871218149934, 1.12591427189849845333061736068, 1.14073892554907882156992967011, 1.29023126328936466492519661417, 1.41911897526617904150632197472, 1.48930584823784622082293184812, 1.51619365351225217441680747555, 1.64582762771007513126847414122, 1.65442082248678146133020744321, 1.65934967960976533025809081225, 1.78096817855320364286018023669, 1.82209836319717618781399543043, 1.85881385036400464265293690147, 2.04637133957110228225353267028, 2.06377308222490913915247213303, 2.12582546639631723730163891947, 2.30001520835118670424065807324, 2.42529080253536476815135331010, 2.48384258031328905305246645501, 2.71924991682344631732368469011
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.