Dirichlet series
| L(s) = 1 | + 8·3-s + 55·5-s + 38·7-s − 58·9-s + 112·11-s + 20·13-s + 440·15-s + 122·17-s + 344·19-s + 304·21-s + 254·23-s + 1.65e3·25-s − 632·27-s − 319·29-s + 136·31-s + 896·33-s + 2.09e3·35-s + 140·37-s + 160·39-s + 382·41-s + 564·43-s − 3.19e3·45-s + 448·47-s − 898·49-s + 976·51-s − 260·53-s + 6.16e3·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 4.91·5-s + 2.05·7-s − 2.14·9-s + 3.06·11-s + 0.426·13-s + 7.57·15-s + 1.74·17-s + 4.15·19-s + 3.15·21-s + 2.30·23-s + 66/5·25-s − 4.50·27-s − 2.04·29-s + 0.787·31-s + 4.72·33-s + 10.0·35-s + 0.622·37-s + 0.656·39-s + 1.45·41-s + 2.00·43-s − 10.5·45-s + 1.39·47-s − 2.61·49-s + 2.67·51-s − 0.673·53-s + 15.1·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 5^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 5^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{33} \cdot 5^{11} \cdot 29^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.54370\times 10^{20}\) |
| Root analytic conductor: | \(8.27298\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{33} \cdot 5^{11} \cdot 29^{11} ,\ ( \ : [3/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(5557.049598\) |
| \(L(\frac12)\) | \(\approx\) | \(5557.049598\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T )^{11} \) | |
| 29 | \( ( 1 + p T )^{11} \) | |
| good | 3 | \( 1 - 8 T + 122 T^{2} - 808 T^{3} + 2549 p T^{4} - 44980 T^{5} + 327128 T^{6} - 1752976 T^{7} + 10949491 T^{8} - 54473156 T^{9} + 35059565 p^{2} T^{10} - 1513350608 T^{11} + 35059565 p^{5} T^{12} - 54473156 p^{6} T^{13} + 10949491 p^{9} T^{14} - 1752976 p^{12} T^{15} + 327128 p^{15} T^{16} - 44980 p^{18} T^{17} + 2549 p^{22} T^{18} - 808 p^{24} T^{19} + 122 p^{27} T^{20} - 8 p^{30} T^{21} + p^{33} T^{22} \) |
| 7 | \( 1 - 38 T + 2342 T^{2} - 73530 T^{3} + 377857 p T^{4} - 1407008 p^{2} T^{5} + 1867972416 T^{6} - 41685273620 T^{7} + 945209910835 T^{8} - 18729268967034 T^{9} + 379860043549293 T^{10} - 6930272395104644 T^{11} + 379860043549293 p^{3} T^{12} - 18729268967034 p^{6} T^{13} + 945209910835 p^{9} T^{14} - 41685273620 p^{12} T^{15} + 1867972416 p^{15} T^{16} - 1407008 p^{20} T^{17} + 377857 p^{22} T^{18} - 73530 p^{24} T^{19} + 2342 p^{27} T^{20} - 38 p^{30} T^{21} + p^{33} T^{22} \) | |
| 11 | \( 1 - 112 T + 11993 T^{2} - 909688 T^{3} + 62370263 T^{4} - 3697216944 T^{5} + 201021983863 T^{6} - 9914476700896 T^{7} + 456691008228362 T^{8} - 19467809136724832 T^{9} + 782165924048398602 T^{10} - 29367254293673937488 T^{11} + 782165924048398602 p^{3} T^{12} - 19467809136724832 p^{6} T^{13} + 456691008228362 p^{9} T^{14} - 9914476700896 p^{12} T^{15} + 201021983863 p^{15} T^{16} - 3697216944 p^{18} T^{17} + 62370263 p^{21} T^{18} - 909688 p^{24} T^{19} + 11993 p^{27} T^{20} - 112 p^{30} T^{21} + p^{33} T^{22} \) | |
| 13 | \( 1 - 20 T + 9970 T^{2} - 318384 T^{3} + 56967159 T^{4} - 2230120850 T^{5} + 232992498764 T^{6} - 771209629612 p T^{7} + 756252584296655 T^{8} - 32414183866631562 T^{9} + 2008814237517846871 T^{10} - 80669519965198208040 T^{11} + 2008814237517846871 p^{3} T^{12} - 32414183866631562 p^{6} T^{13} + 756252584296655 p^{9} T^{14} - 771209629612 p^{13} T^{15} + 232992498764 p^{15} T^{16} - 2230120850 p^{18} T^{17} + 56967159 p^{21} T^{18} - 318384 p^{24} T^{19} + 9970 p^{27} T^{20} - 20 p^{30} T^{21} + p^{33} T^{22} \) | |
| 17 | \( 1 - 122 T + 34094 T^{2} - 3448358 T^{3} + 571345539 T^{4} - 49703876566 T^{5} + 6303922911544 T^{6} - 483753385929640 T^{7} + 51148178156188875 T^{8} - 3496504778438933232 T^{9} + \)\(32\!\cdots\!43\)\( T^{10} - \)\(19\!\cdots\!32\)\( T^{11} + \)\(32\!\cdots\!43\)\( p^{3} T^{12} - 3496504778438933232 p^{6} T^{13} + 51148178156188875 p^{9} T^{14} - 483753385929640 p^{12} T^{15} + 6303922911544 p^{15} T^{16} - 49703876566 p^{18} T^{17} + 571345539 p^{21} T^{18} - 3448358 p^{24} T^{19} + 34094 p^{27} T^{20} - 122 p^{30} T^{21} + p^{33} T^{22} \) | |
| 19 | \( 1 - 344 T + 103929 T^{2} - 22335848 T^{3} + 4173590399 T^{4} - 661796366104 T^{5} + 93464883448111 T^{6} - 11764489875658208 T^{7} + 1340816232492151642 T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(13\!\cdots\!14\)\( T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!14\)\( p^{3} T^{12} - \)\(13\!\cdots\!28\)\( p^{6} T^{13} + 1340816232492151642 p^{9} T^{14} - 11764489875658208 p^{12} T^{15} + 93464883448111 p^{15} T^{16} - 661796366104 p^{18} T^{17} + 4173590399 p^{21} T^{18} - 22335848 p^{24} T^{19} + 103929 p^{27} T^{20} - 344 p^{30} T^{21} + p^{33} T^{22} \) | |
| 23 | \( 1 - 254 T + 96810 T^{2} - 17734266 T^{3} + 4190456907 T^{4} - 627739984480 T^{5} + 115416184330896 T^{6} - 14866603804445716 T^{7} + 2305203223255405027 T^{8} - \)\(26\!\cdots\!46\)\( T^{9} + \)\(35\!\cdots\!49\)\( T^{10} - \)\(35\!\cdots\!96\)\( T^{11} + \)\(35\!\cdots\!49\)\( p^{3} T^{12} - \)\(26\!\cdots\!46\)\( p^{6} T^{13} + 2305203223255405027 p^{9} T^{14} - 14866603804445716 p^{12} T^{15} + 115416184330896 p^{15} T^{16} - 627739984480 p^{18} T^{17} + 4190456907 p^{21} T^{18} - 17734266 p^{24} T^{19} + 96810 p^{27} T^{20} - 254 p^{30} T^{21} + p^{33} T^{22} \) | |
| 31 | \( 1 - 136 T + 3730 p T^{2} - 4284192 T^{3} + 6478079595 T^{4} + 149049407124 T^{5} + 298051146464696 T^{6} + 14976921776793952 T^{7} + 12422865565528711971 T^{8} + \)\(64\!\cdots\!92\)\( T^{9} + \)\(43\!\cdots\!21\)\( T^{10} + \)\(22\!\cdots\!48\)\( T^{11} + \)\(43\!\cdots\!21\)\( p^{3} T^{12} + \)\(64\!\cdots\!92\)\( p^{6} T^{13} + 12422865565528711971 p^{9} T^{14} + 14976921776793952 p^{12} T^{15} + 298051146464696 p^{15} T^{16} + 149049407124 p^{18} T^{17} + 6478079595 p^{21} T^{18} - 4284192 p^{24} T^{19} + 3730 p^{28} T^{20} - 136 p^{30} T^{21} + p^{33} T^{22} \) | |
| 37 | \( 1 - 140 T + 263947 T^{2} - 37796384 T^{3} + 35952361947 T^{4} - 5276979057308 T^{5} + 3333488665185929 T^{6} - 510741079606890752 T^{7} + \)\(23\!\cdots\!30\)\( T^{8} - \)\(37\!\cdots\!04\)\( T^{9} + \)\(14\!\cdots\!14\)\( T^{10} - \)\(21\!\cdots\!92\)\( T^{11} + \)\(14\!\cdots\!14\)\( p^{3} T^{12} - \)\(37\!\cdots\!04\)\( p^{6} T^{13} + \)\(23\!\cdots\!30\)\( p^{9} T^{14} - 510741079606890752 p^{12} T^{15} + 3333488665185929 p^{15} T^{16} - 5276979057308 p^{18} T^{17} + 35952361947 p^{21} T^{18} - 37796384 p^{24} T^{19} + 263947 p^{27} T^{20} - 140 p^{30} T^{21} + p^{33} T^{22} \) | |
| 41 | \( 1 - 382 T + 473531 T^{2} - 121413420 T^{3} + 93463575679 T^{4} - 15859230031910 T^{5} + 10950086872218333 T^{6} - 1070274823025064144 T^{7} + \)\(91\!\cdots\!70\)\( T^{8} - \)\(33\!\cdots\!08\)\( T^{9} + \)\(64\!\cdots\!34\)\( T^{10} - \)\(59\!\cdots\!84\)\( T^{11} + \)\(64\!\cdots\!34\)\( p^{3} T^{12} - \)\(33\!\cdots\!08\)\( p^{6} T^{13} + \)\(91\!\cdots\!70\)\( p^{9} T^{14} - 1070274823025064144 p^{12} T^{15} + 10950086872218333 p^{15} T^{16} - 15859230031910 p^{18} T^{17} + 93463575679 p^{21} T^{18} - 121413420 p^{24} T^{19} + 473531 p^{27} T^{20} - 382 p^{30} T^{21} + p^{33} T^{22} \) | |
| 43 | \( 1 - 564 T + 509486 T^{2} - 237252732 T^{3} + 125448918283 T^{4} - 51773297918572 T^{5} + 20866000920562408 T^{6} - 7647040383650150296 T^{7} + \)\(26\!\cdots\!27\)\( T^{8} - \)\(84\!\cdots\!92\)\( T^{9} + \)\(25\!\cdots\!01\)\( T^{10} - \)\(75\!\cdots\!64\)\( T^{11} + \)\(25\!\cdots\!01\)\( p^{3} T^{12} - \)\(84\!\cdots\!92\)\( p^{6} T^{13} + \)\(26\!\cdots\!27\)\( p^{9} T^{14} - 7647040383650150296 p^{12} T^{15} + 20866000920562408 p^{15} T^{16} - 51773297918572 p^{18} T^{17} + 125448918283 p^{21} T^{18} - 237252732 p^{24} T^{19} + 509486 p^{27} T^{20} - 564 p^{30} T^{21} + p^{33} T^{22} \) | |
| 47 | \( 1 - 448 T + 794721 T^{2} - 349276584 T^{3} + 313641286011 T^{4} - 130947844009728 T^{5} + 80197310698240395 T^{6} - 31021391204641499040 T^{7} + \)\(14\!\cdots\!30\)\( T^{8} - \)\(51\!\cdots\!64\)\( T^{9} + \)\(19\!\cdots\!58\)\( T^{10} - \)\(62\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!58\)\( p^{3} T^{12} - \)\(51\!\cdots\!64\)\( p^{6} T^{13} + \)\(14\!\cdots\!30\)\( p^{9} T^{14} - 31021391204641499040 p^{12} T^{15} + 80197310698240395 p^{15} T^{16} - 130947844009728 p^{18} T^{17} + 313641286011 p^{21} T^{18} - 349276584 p^{24} T^{19} + 794721 p^{27} T^{20} - 448 p^{30} T^{21} + p^{33} T^{22} \) | |
| 53 | \( 1 + 260 T + 679234 T^{2} + 208126888 T^{3} + 279297852447 T^{4} + 79752160270210 T^{5} + 81640402922923316 T^{6} + 21637797080383778364 T^{7} + \)\(18\!\cdots\!79\)\( T^{8} + \)\(44\!\cdots\!14\)\( T^{9} + \)\(33\!\cdots\!63\)\( T^{10} + \)\(73\!\cdots\!88\)\( T^{11} + \)\(33\!\cdots\!63\)\( p^{3} T^{12} + \)\(44\!\cdots\!14\)\( p^{6} T^{13} + \)\(18\!\cdots\!79\)\( p^{9} T^{14} + 21637797080383778364 p^{12} T^{15} + 81640402922923316 p^{15} T^{16} + 79752160270210 p^{18} T^{17} + 279297852447 p^{21} T^{18} + 208126888 p^{24} T^{19} + 679234 p^{27} T^{20} + 260 p^{30} T^{21} + p^{33} T^{22} \) | |
| 59 | \( 1 - 1060 T + 1653954 T^{2} - 1407018508 T^{3} + 1316064738243 T^{4} - 931915408467844 T^{5} + 667486983796065720 T^{6} - \)\(40\!\cdots\!24\)\( T^{7} + \)\(24\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{9} + \)\(64\!\cdots\!21\)\( T^{10} - \)\(29\!\cdots\!12\)\( T^{11} + \)\(64\!\cdots\!21\)\( p^{3} T^{12} - \)\(12\!\cdots\!40\)\( p^{6} T^{13} + \)\(24\!\cdots\!75\)\( p^{9} T^{14} - \)\(40\!\cdots\!24\)\( p^{12} T^{15} + 667486983796065720 p^{15} T^{16} - 931915408467844 p^{18} T^{17} + 1316064738243 p^{21} T^{18} - 1407018508 p^{24} T^{19} + 1653954 p^{27} T^{20} - 1060 p^{30} T^{21} + p^{33} T^{22} \) | |
| 61 | \( 1 - 336 T + 1558734 T^{2} - 415447440 T^{3} + 1214680849099 T^{4} - 268005374058086 T^{5} + 626836269385303460 T^{6} - \)\(11\!\cdots\!08\)\( T^{7} + \)\(23\!\cdots\!91\)\( T^{8} - \)\(38\!\cdots\!10\)\( T^{9} + \)\(68\!\cdots\!99\)\( T^{10} - \)\(98\!\cdots\!28\)\( T^{11} + \)\(68\!\cdots\!99\)\( p^{3} T^{12} - \)\(38\!\cdots\!10\)\( p^{6} T^{13} + \)\(23\!\cdots\!91\)\( p^{9} T^{14} - \)\(11\!\cdots\!08\)\( p^{12} T^{15} + 626836269385303460 p^{15} T^{16} - 268005374058086 p^{18} T^{17} + 1214680849099 p^{21} T^{18} - 415447440 p^{24} T^{19} + 1558734 p^{27} T^{20} - 336 p^{30} T^{21} + p^{33} T^{22} \) | |
| 67 | \( 1 - 850 T + 1602657 T^{2} - 884358548 T^{3} + 1034647141799 T^{4} - 448574711674234 T^{5} + 474938461988944415 T^{6} - \)\(19\!\cdots\!24\)\( T^{7} + \)\(19\!\cdots\!82\)\( T^{8} - \)\(80\!\cdots\!20\)\( T^{9} + \)\(70\!\cdots\!22\)\( T^{10} - \)\(26\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!22\)\( p^{3} T^{12} - \)\(80\!\cdots\!20\)\( p^{6} T^{13} + \)\(19\!\cdots\!82\)\( p^{9} T^{14} - \)\(19\!\cdots\!24\)\( p^{12} T^{15} + 474938461988944415 p^{15} T^{16} - 448574711674234 p^{18} T^{17} + 1034647141799 p^{21} T^{18} - 884358548 p^{24} T^{19} + 1602657 p^{27} T^{20} - 850 p^{30} T^{21} + p^{33} T^{22} \) | |
| 71 | \( 1 - 680 T + 3458589 T^{2} - 2008199760 T^{3} + 5530812253447 T^{4} - 2781091219887880 T^{5} + 5436844073306064387 T^{6} - \)\(23\!\cdots\!04\)\( T^{7} + \)\(36\!\cdots\!38\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!82\)\( T^{10} - \)\(58\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!82\)\( p^{3} T^{12} - \)\(13\!\cdots\!20\)\( p^{6} T^{13} + \)\(36\!\cdots\!38\)\( p^{9} T^{14} - \)\(23\!\cdots\!04\)\( p^{12} T^{15} + 5436844073306064387 p^{15} T^{16} - 2781091219887880 p^{18} T^{17} + 5530812253447 p^{21} T^{18} - 2008199760 p^{24} T^{19} + 3458589 p^{27} T^{20} - 680 p^{30} T^{21} + p^{33} T^{22} \) | |
| 73 | \( 1 + 850 T + 1980038 T^{2} + 2102840670 T^{3} + 2423843876451 T^{4} + 2348285703859114 T^{5} + 30042263506511320 p T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!31\)\( T^{8} + \)\(10\!\cdots\!32\)\( T^{9} + \)\(70\!\cdots\!23\)\( T^{10} + \)\(45\!\cdots\!20\)\( T^{11} + \)\(70\!\cdots\!23\)\( p^{3} T^{12} + \)\(10\!\cdots\!32\)\( p^{6} T^{13} + \)\(14\!\cdots\!31\)\( p^{9} T^{14} + \)\(17\!\cdots\!00\)\( p^{12} T^{15} + 30042263506511320 p^{16} T^{16} + 2348285703859114 p^{18} T^{17} + 2423843876451 p^{21} T^{18} + 2102840670 p^{24} T^{19} + 1980038 p^{27} T^{20} + 850 p^{30} T^{21} + p^{33} T^{22} \) | |
| 79 | \( 1 - 560 T + 3271682 T^{2} - 1941022520 T^{3} + 5270282531791 T^{4} - 3223265412790068 T^{5} + 5660639816272409928 T^{6} - \)\(33\!\cdots\!92\)\( T^{7} + \)\(45\!\cdots\!99\)\( T^{8} - \)\(25\!\cdots\!12\)\( T^{9} + \)\(28\!\cdots\!09\)\( T^{10} - \)\(14\!\cdots\!68\)\( T^{11} + \)\(28\!\cdots\!09\)\( p^{3} T^{12} - \)\(25\!\cdots\!12\)\( p^{6} T^{13} + \)\(45\!\cdots\!99\)\( p^{9} T^{14} - \)\(33\!\cdots\!92\)\( p^{12} T^{15} + 5660639816272409928 p^{15} T^{16} - 3223265412790068 p^{18} T^{17} + 5270282531791 p^{21} T^{18} - 1941022520 p^{24} T^{19} + 3271682 p^{27} T^{20} - 560 p^{30} T^{21} + p^{33} T^{22} \) | |
| 83 | \( 1 - 1102 T + 4082285 T^{2} - 3517349252 T^{3} + 7772418867227 T^{4} - 5662344332523878 T^{5} + 9641321264745020671 T^{6} - \)\(62\!\cdots\!92\)\( T^{7} + \)\(88\!\cdots\!34\)\( T^{8} - \)\(51\!\cdots\!76\)\( T^{9} + \)\(63\!\cdots\!26\)\( T^{10} - \)\(33\!\cdots\!44\)\( T^{11} + \)\(63\!\cdots\!26\)\( p^{3} T^{12} - \)\(51\!\cdots\!76\)\( p^{6} T^{13} + \)\(88\!\cdots\!34\)\( p^{9} T^{14} - \)\(62\!\cdots\!92\)\( p^{12} T^{15} + 9641321264745020671 p^{15} T^{16} - 5662344332523878 p^{18} T^{17} + 7772418867227 p^{21} T^{18} - 3517349252 p^{24} T^{19} + 4082285 p^{27} T^{20} - 1102 p^{30} T^{21} + p^{33} T^{22} \) | |
| 89 | \( 1 - 1398 T + 2598267 T^{2} - 3581928188 T^{3} + 5138079064559 T^{4} - 6203766415977470 T^{5} + 7203799173606315277 T^{6} - \)\(74\!\cdots\!72\)\( T^{7} + \)\(78\!\cdots\!90\)\( T^{8} - \)\(73\!\cdots\!96\)\( T^{9} + \)\(67\!\cdots\!42\)\( T^{10} - \)\(57\!\cdots\!72\)\( T^{11} + \)\(67\!\cdots\!42\)\( p^{3} T^{12} - \)\(73\!\cdots\!96\)\( p^{6} T^{13} + \)\(78\!\cdots\!90\)\( p^{9} T^{14} - \)\(74\!\cdots\!72\)\( p^{12} T^{15} + 7203799173606315277 p^{15} T^{16} - 6203766415977470 p^{18} T^{17} + 5138079064559 p^{21} T^{18} - 3581928188 p^{24} T^{19} + 2598267 p^{27} T^{20} - 1398 p^{30} T^{21} + p^{33} T^{22} \) | |
| 97 | \( 1 - 1126 T + 6718174 T^{2} - 6170884906 T^{3} + 20494749301611 T^{4} - 15732159102185426 T^{5} + 39518539652886430344 T^{6} - \)\(26\!\cdots\!88\)\( T^{7} + \)\(56\!\cdots\!71\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!59\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(63\!\cdots\!59\)\( p^{3} T^{12} - \)\(33\!\cdots\!92\)\( p^{6} T^{13} + \)\(56\!\cdots\!71\)\( p^{9} T^{14} - \)\(26\!\cdots\!88\)\( p^{12} T^{15} + 39518539652886430344 p^{15} T^{16} - 15732159102185426 p^{18} T^{17} + 20494749301611 p^{21} T^{18} - 6170884906 p^{24} T^{19} + 6718174 p^{27} T^{20} - 1126 p^{30} T^{21} + p^{33} T^{22} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.96717576473788503858469747890, −2.90557445287647189273690348082, −2.85237429002926219303720550008, −2.29905396646320903490601193967, −2.27833807220197659783448496231, −2.25395962409234759148618177753, −2.19831047064872742841044318357, −2.16820484457932834476072732310, −2.05305487559866316853293922275, −2.04730063617127622385288957539, −1.84373705978565891534082909518, −1.74976667164370701495059073951, −1.61928142912339424992078591900, −1.60109546575294656029351316506, −1.40030347431421034848539053090, −1.24364964619820645796853490474, −1.16949917196838207989750108961, −1.00177572167273827298634898148, −0.841282198443267306771481384348, −0.816534940151619076487862785072, −0.811638188735118786287773818508, −0.72485204683282458844248951597, −0.62988713870169292404662182787, −0.43565775747905442853853657991, −0.36833362995064170793521110560, 0.36833362995064170793521110560, 0.43565775747905442853853657991, 0.62988713870169292404662182787, 0.72485204683282458844248951597, 0.811638188735118786287773818508, 0.816534940151619076487862785072, 0.841282198443267306771481384348, 1.00177572167273827298634898148, 1.16949917196838207989750108961, 1.24364964619820645796853490474, 1.40030347431421034848539053090, 1.60109546575294656029351316506, 1.61928142912339424992078591900, 1.74976667164370701495059073951, 1.84373705978565891534082909518, 2.04730063617127622385288957539, 2.05305487559866316853293922275, 2.16820484457932834476072732310, 2.19831047064872742841044318357, 2.25395962409234759148618177753, 2.27833807220197659783448496231, 2.29905396646320903490601193967, 2.85237429002926219303720550008, 2.90557445287647189273690348082, 2.96717576473788503858469747890
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.