Dirichlet series
| L(s) = 1 | − 4·2-s + 6·4-s − 4·5-s + 2·7-s − 4·8-s + 22·9-s + 16·10-s − 2·11-s − 8·14-s + 2·16-s − 6·17-s − 88·18-s + 2·19-s − 24·20-s + 8·22-s − 2·23-s + 11·25-s + 12·28-s − 14·29-s − 4·32-s + 24·34-s − 8·35-s + 132·36-s + 8·37-s − 8·38-s + 16·40-s − 44·43-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 1.78·5-s + 0.755·7-s − 1.41·8-s + 22/3·9-s + 5.05·10-s − 0.603·11-s − 2.13·14-s + 1/2·16-s − 1.45·17-s − 20.7·18-s + 0.458·19-s − 5.36·20-s + 1.70·22-s − 0.417·23-s + 11/5·25-s + 2.26·28-s − 2.59·29-s − 0.707·32-s + 4.11·34-s − 1.35·35-s + 22·36-s + 1.31·37-s − 1.29·38-s + 2.52·40-s − 6.70·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{72} \cdot 5^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.000313767\) |
| Root analytic conductor: | \(0.799251\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{72} \cdot 5^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07268145532\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07268145532\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{2} T + 5 p T^{2} + 5 p^{2} T^{3} + 17 p T^{4} + 13 p^{2} T^{5} + 19 p^{2} T^{6} + 3 p^{5} T^{7} + 29 p^{2} T^{8} + 19 p^{3} T^{9} + 29 p^{3} T^{10} + 3 p^{7} T^{11} + 19 p^{5} T^{12} + 13 p^{6} T^{13} + 17 p^{6} T^{14} + 5 p^{8} T^{15} + 5 p^{8} T^{16} + p^{10} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 4 T + p T^{2} + 4 T^{4} + 48 T^{5} + 4 p T^{6} - 384 T^{7} - 986 T^{8} - 1576 T^{9} - 986 p T^{10} - 384 p^{2} T^{11} + 4 p^{4} T^{12} + 48 p^{4} T^{13} + 4 p^{5} T^{14} + p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 3 | \( 1 - 22 T^{2} + 83 p T^{4} - 1952 T^{6} + 12020 T^{8} - 20680 p T^{10} + 277684 T^{12} - 1097312 T^{14} + 3868246 T^{16} - 12237572 T^{18} + 3868246 p^{2} T^{20} - 1097312 p^{4} T^{22} + 277684 p^{6} T^{24} - 20680 p^{9} T^{26} + 12020 p^{10} T^{28} - 1952 p^{12} T^{30} + 83 p^{15} T^{32} - 22 p^{16} T^{34} + p^{18} T^{36} \) |
| 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 | \( ( 1 + 61 T^{2} + 16 T^{3} + 1908 T^{4} + 928 T^{5} + 40580 T^{6} + 27056 T^{7} + 660406 T^{8} + 451904 T^{9} + 660406 p T^{10} + 27056 p^{2} T^{11} + 40580 p^{3} T^{12} + 928 p^{4} T^{13} + 1908 p^{5} T^{14} + 16 p^{6} T^{15} + 61 p^{7} T^{16} + p^{9} T^{18} )^{2} \) | |
| 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 | \( ( 1 - 4 T + 197 T^{2} - 1024 T^{3} + 20644 T^{4} - 111280 T^{5} + 1450996 T^{6} - 7367808 T^{7} + 72944518 T^{8} - 328519384 T^{9} + 72944518 p T^{10} - 7367808 p^{2} T^{11} + 1450996 p^{3} T^{12} - 111280 p^{4} T^{13} + 20644 p^{5} T^{14} - 1024 p^{6} T^{15} + 197 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 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 | \( ( 1 + 22 T + 459 T^{2} + 5948 T^{3} + 73264 T^{4} + 695720 T^{5} + 6433304 T^{6} + 49524844 T^{7} + 378530410 T^{8} + 2482332748 T^{9} + 378530410 p T^{10} + 49524844 p^{2} T^{11} + 6433304 p^{3} T^{12} + 695720 p^{4} T^{13} + 73264 p^{5} T^{14} + 5948 p^{6} T^{15} + 459 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 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 | \( 1 - 646 T^{2} + 204289 T^{4} - 42062000 T^{6} + 6331181988 T^{8} - 742044595240 T^{10} + 70453884825524 T^{12} - 5564449393062864 T^{14} + 372091290097786238 T^{16} - 21285258439059863076 T^{18} + 372091290097786238 p^{2} T^{20} - 5564449393062864 p^{4} T^{22} + 70453884825524 p^{6} T^{24} - 742044595240 p^{8} T^{26} + 6331181988 p^{10} T^{28} - 42062000 p^{12} T^{30} + 204289 p^{14} T^{32} - 646 p^{16} T^{34} + p^{18} T^{36} \) | |
| 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 | \( ( 1 - 6 T + 339 T^{2} - 2332 T^{3} + 57824 T^{4} - 422744 T^{5} + 6773640 T^{6} - 47574572 T^{7} + 596885338 T^{8} - 3741287532 T^{9} + 596885338 p T^{10} - 47574572 p^{2} T^{11} + 6773640 p^{3} T^{12} - 422744 p^{4} T^{13} + 57824 p^{5} T^{14} - 2332 p^{6} T^{15} + 339 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 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 | \( 1 - 822 T^{2} + 339161 T^{4} - 93784032 T^{6} + 19542251380 T^{8} - 3265482130776 T^{10} + 454015681415604 T^{12} - 53737884583975712 T^{14} + 5490944378360489558 T^{16} - \)\(48\!\cdots\!20\)\( T^{18} + 5490944378360489558 p^{2} T^{20} - 53737884583975712 p^{4} T^{22} + 454015681415604 p^{6} T^{24} - 3265482130776 p^{8} T^{26} + 19542251380 p^{10} T^{28} - 93784032 p^{12} T^{30} + 339161 p^{14} T^{32} - 822 p^{16} T^{34} + p^{18} T^{36} \) | |
| 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
−4.19306731332274611733261548386, −4.17535766716212241244810693878, −4.03832024927086418200604841773, −4.02379646555656430446560006042, −3.93047030287500441640760317790, −3.90143342478921247978563678283, −3.71059470283862852366981585869, −3.65101419125624917640561548549, −3.59452952640116843501973373663, −3.56168342163255562777626767352, −3.20186047640124971935993350644, −3.17617200714762836863130478872, −3.16493363326819554594006276673, −2.98832188762392502769356523350, −2.82140471547993004938655216879, −2.59908451284629552398595266078, −2.17416218862061596664810628327, −2.14103448986340814153092380725, −1.95813183540523251651850835743, −1.94286648681342546356333279966, −1.79651105004497366420481368959, −1.62712533632086963832074203688, −1.54532472257734829377951785688, −1.22425056110779022660606694733, −1.07650061040305668188564730603, 1.07650061040305668188564730603, 1.22425056110779022660606694733, 1.54532472257734829377951785688, 1.62712533632086963832074203688, 1.79651105004497366420481368959, 1.94286648681342546356333279966, 1.95813183540523251651850835743, 2.14103448986340814153092380725, 2.17416218862061596664810628327, 2.59908451284629552398595266078, 2.82140471547993004938655216879, 2.98832188762392502769356523350, 3.16493363326819554594006276673, 3.17617200714762836863130478872, 3.20186047640124971935993350644, 3.56168342163255562777626767352, 3.59452952640116843501973373663, 3.65101419125624917640561548549, 3.71059470283862852366981585869, 3.90143342478921247978563678283, 3.93047030287500441640760317790, 4.02379646555656430446560006042, 4.03832024927086418200604841773, 4.17535766716212241244810693878, 4.19306731332274611733261548386
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.