Dirichlet series
| L(s) = 1 | − 2-s + 33·3-s − 28·4-s − 17·5-s − 33·6-s + 13·8-s + 594·9-s + 17·10-s + 7·11-s − 924·12-s + 143·13-s − 561·15-s + 351·16-s + 20·17-s − 594·18-s − 242·19-s + 476·20-s − 7·22-s − 122·23-s + 429·24-s − 439·25-s − 143·26-s + 7.72e3·27-s − 177·29-s + 561·30-s − 493·31-s + 81·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 6.35·3-s − 7/2·4-s − 1.52·5-s − 2.24·6-s + 0.574·8-s + 22·9-s + 0.537·10-s + 0.191·11-s − 22.2·12-s + 3.05·13-s − 9.65·15-s + 5.48·16-s + 0.285·17-s − 7.77·18-s − 2.92·19-s + 5.32·20-s − 0.0678·22-s − 1.10·23-s + 3.64·24-s − 3.51·25-s − 1.07·26-s + 55.0·27-s − 1.13·29-s + 3.41·30-s − 2.85·31-s + 0.447·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 7^{22} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 7^{22} \cdot 13^{11}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(3^{11} \cdot 7^{22} \cdot 13^{11}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.74450\times 10^{22}\) |
| Root analytic conductor: | \(10.6185\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(11\) |
| Selberg data: | \((22,\ 3^{11} \cdot 7^{22} \cdot 13^{11} ,\ ( \ : [3/2]^{11} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T )^{11} \) |
| 7 | \( 1 \) | |
| 13 | \( ( 1 - p T )^{11} \) | |
| good | 2 | \( 1 + T + 29 T^{2} + 11 p^{2} T^{3} + 123 p^{2} T^{4} + 915 T^{5} + 6057 T^{6} + 6571 p T^{7} + 62117 T^{8} + 70281 p T^{9} + 69491 p^{3} T^{10} + 154509 p^{3} T^{11} + 69491 p^{6} T^{12} + 70281 p^{7} T^{13} + 62117 p^{9} T^{14} + 6571 p^{13} T^{15} + 6057 p^{15} T^{16} + 915 p^{18} T^{17} + 123 p^{23} T^{18} + 11 p^{26} T^{19} + 29 p^{27} T^{20} + p^{30} T^{21} + p^{33} T^{22} \) |
| 5 | \( 1 + 17 T + 728 T^{2} + 9369 T^{3} + 235876 T^{4} + 2259323 T^{5} + 43921276 T^{6} + 281480851 T^{7} + 1044024523 p T^{8} + 14974772456 T^{9} + 96325670696 p T^{10} + 233545403392 T^{11} + 96325670696 p^{4} T^{12} + 14974772456 p^{6} T^{13} + 1044024523 p^{10} T^{14} + 281480851 p^{12} T^{15} + 43921276 p^{15} T^{16} + 2259323 p^{18} T^{17} + 235876 p^{21} T^{18} + 9369 p^{24} T^{19} + 728 p^{27} T^{20} + 17 p^{30} T^{21} + p^{33} T^{22} \) | |
| 11 | \( 1 - 7 T + 7688 T^{2} - 95047 T^{3} + 28583343 T^{4} - 490202434 T^{5} + 70626988618 T^{6} - 1442713585592 T^{7} + 134516678956959 T^{8} - 2887396845474683 T^{9} + 211999021798474700 T^{10} - 4348216987707977845 T^{11} + 211999021798474700 p^{3} T^{12} - 2887396845474683 p^{6} T^{13} + 134516678956959 p^{9} T^{14} - 1442713585592 p^{12} T^{15} + 70626988618 p^{15} T^{16} - 490202434 p^{18} T^{17} + 28583343 p^{21} T^{18} - 95047 p^{24} T^{19} + 7688 p^{27} T^{20} - 7 p^{30} T^{21} + p^{33} T^{22} \) | |
| 17 | \( 1 - 20 T + 28150 T^{2} - 438754 T^{3} + 411905823 T^{4} - 5430227612 T^{5} + 4130653831676 T^{6} - 46877088345630 T^{7} + 31401339888776125 T^{8} - 310739974263927062 T^{9} + 11182782788891413066 p T^{10} - \)\(16\!\cdots\!18\)\( T^{11} + 11182782788891413066 p^{4} T^{12} - 310739974263927062 p^{6} T^{13} + 31401339888776125 p^{9} T^{14} - 46877088345630 p^{12} T^{15} + 4130653831676 p^{15} T^{16} - 5430227612 p^{18} T^{17} + 411905823 p^{21} T^{18} - 438754 p^{24} T^{19} + 28150 p^{27} T^{20} - 20 p^{30} T^{21} + p^{33} T^{22} \) | |
| 19 | \( 1 + 242 T + 56526 T^{2} + 8568616 T^{3} + 1230774288 T^{4} + 146147018630 T^{5} + 16871516002495 T^{6} + 1751664686379480 T^{7} + 177446197049274523 T^{8} + 16589111440027544426 T^{9} + \)\(15\!\cdots\!64\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{11} + \)\(15\!\cdots\!64\)\( p^{3} T^{12} + 16589111440027544426 p^{6} T^{13} + 177446197049274523 p^{9} T^{14} + 1751664686379480 p^{12} T^{15} + 16871516002495 p^{15} T^{16} + 146147018630 p^{18} T^{17} + 1230774288 p^{21} T^{18} + 8568616 p^{24} T^{19} + 56526 p^{27} T^{20} + 242 p^{30} T^{21} + p^{33} T^{22} \) | |
| 23 | \( 1 + 122 T + 74145 T^{2} + 10035508 T^{3} + 2909963940 T^{4} + 17414201802 p T^{5} + 77738819838801 T^{6} + 10333165223702372 T^{7} + 1549766721884056086 T^{8} + \)\(19\!\cdots\!08\)\( T^{9} + \)\(23\!\cdots\!69\)\( T^{10} + \)\(26\!\cdots\!60\)\( T^{11} + \)\(23\!\cdots\!69\)\( p^{3} T^{12} + \)\(19\!\cdots\!08\)\( p^{6} T^{13} + 1549766721884056086 p^{9} T^{14} + 10333165223702372 p^{12} T^{15} + 77738819838801 p^{15} T^{16} + 17414201802 p^{19} T^{17} + 2909963940 p^{21} T^{18} + 10035508 p^{24} T^{19} + 74145 p^{27} T^{20} + 122 p^{30} T^{21} + p^{33} T^{22} \) | |
| 29 | \( 1 + 177 T + 148495 T^{2} + 24160922 T^{3} + 10615761677 T^{4} + 1714424476439 T^{5} + 506324396395862 T^{6} + 83843054408153119 T^{7} + 18367235764494095482 T^{8} + \)\(30\!\cdots\!03\)\( T^{9} + \)\(53\!\cdots\!46\)\( T^{10} + \)\(84\!\cdots\!35\)\( T^{11} + \)\(53\!\cdots\!46\)\( p^{3} T^{12} + \)\(30\!\cdots\!03\)\( p^{6} T^{13} + 18367235764494095482 p^{9} T^{14} + 83843054408153119 p^{12} T^{15} + 506324396395862 p^{15} T^{16} + 1714424476439 p^{18} T^{17} + 10615761677 p^{21} T^{18} + 24160922 p^{24} T^{19} + 148495 p^{27} T^{20} + 177 p^{30} T^{21} + p^{33} T^{22} \) | |
| 31 | \( 1 + 493 T + 318185 T^{2} + 107607230 T^{3} + 41511867366 T^{4} + 11031138734710 T^{5} + 3227716589342259 T^{6} + 717821730224875069 T^{7} + \)\(17\!\cdots\!52\)\( T^{8} + \)\(33\!\cdots\!41\)\( T^{9} + \)\(68\!\cdots\!79\)\( T^{10} + \)\(11\!\cdots\!98\)\( T^{11} + \)\(68\!\cdots\!79\)\( p^{3} T^{12} + \)\(33\!\cdots\!41\)\( p^{6} T^{13} + \)\(17\!\cdots\!52\)\( p^{9} T^{14} + 717821730224875069 p^{12} T^{15} + 3227716589342259 p^{15} T^{16} + 11031138734710 p^{18} T^{17} + 41511867366 p^{21} T^{18} + 107607230 p^{24} T^{19} + 318185 p^{27} T^{20} + 493 p^{30} T^{21} + p^{33} T^{22} \) | |
| 37 | \( 1 + 660 T + 393836 T^{2} + 148001684 T^{3} + 52194808430 T^{4} + 14725939687292 T^{5} + 4080134161123772 T^{6} + 1011653891194991310 T^{7} + \)\(25\!\cdots\!05\)\( T^{8} + \)\(60\!\cdots\!16\)\( T^{9} + \)\(14\!\cdots\!52\)\( T^{10} + \)\(33\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!52\)\( p^{3} T^{12} + \)\(60\!\cdots\!16\)\( p^{6} T^{13} + \)\(25\!\cdots\!05\)\( p^{9} T^{14} + 1011653891194991310 p^{12} T^{15} + 4080134161123772 p^{15} T^{16} + 14725939687292 p^{18} T^{17} + 52194808430 p^{21} T^{18} + 148001684 p^{24} T^{19} + 393836 p^{27} T^{20} + 660 p^{30} T^{21} + p^{33} T^{22} \) | |
| 41 | \( 1 + 854 T + 770457 T^{2} + 455346204 T^{3} + 253302177914 T^{4} + 115628084997700 T^{5} + 49295640895519667 T^{6} + 18427469136753217814 T^{7} + \)\(64\!\cdots\!58\)\( T^{8} + \)\(20\!\cdots\!38\)\( T^{9} + \)\(60\!\cdots\!19\)\( T^{10} + \)\(16\!\cdots\!12\)\( T^{11} + \)\(60\!\cdots\!19\)\( p^{3} T^{12} + \)\(20\!\cdots\!38\)\( p^{6} T^{13} + \)\(64\!\cdots\!58\)\( p^{9} T^{14} + 18427469136753217814 p^{12} T^{15} + 49295640895519667 p^{15} T^{16} + 115628084997700 p^{18} T^{17} + 253302177914 p^{21} T^{18} + 455346204 p^{24} T^{19} + 770457 p^{27} T^{20} + 854 p^{30} T^{21} + p^{33} T^{22} \) | |
| 43 | \( 1 + 1018 T + 955025 T^{2} + 594136362 T^{3} + 339780158416 T^{4} + 3665543492424 p T^{5} + 68625640652045147 T^{6} + 25958768106737334592 T^{7} + \)\(93\!\cdots\!00\)\( T^{8} + \)\(30\!\cdots\!42\)\( T^{9} + \)\(95\!\cdots\!73\)\( T^{10} + \)\(27\!\cdots\!76\)\( T^{11} + \)\(95\!\cdots\!73\)\( p^{3} T^{12} + \)\(30\!\cdots\!42\)\( p^{6} T^{13} + \)\(93\!\cdots\!00\)\( p^{9} T^{14} + 25958768106737334592 p^{12} T^{15} + 68625640652045147 p^{15} T^{16} + 3665543492424 p^{19} T^{17} + 339780158416 p^{21} T^{18} + 594136362 p^{24} T^{19} + 955025 p^{27} T^{20} + 1018 p^{30} T^{21} + p^{33} T^{22} \) | |
| 47 | \( 1 + 414 T + 12515 p T^{2} + 192240390 T^{3} + 3337509606 p T^{4} + 40560715872534 T^{5} + 25937947522556906 T^{6} + 5221040253771978696 T^{7} + \)\(31\!\cdots\!93\)\( T^{8} + \)\(49\!\cdots\!64\)\( T^{9} + \)\(32\!\cdots\!81\)\( T^{10} + \)\(45\!\cdots\!08\)\( T^{11} + \)\(32\!\cdots\!81\)\( p^{3} T^{12} + \)\(49\!\cdots\!64\)\( p^{6} T^{13} + \)\(31\!\cdots\!93\)\( p^{9} T^{14} + 5221040253771978696 p^{12} T^{15} + 25937947522556906 p^{15} T^{16} + 40560715872534 p^{18} T^{17} + 3337509606 p^{22} T^{18} + 192240390 p^{24} T^{19} + 12515 p^{28} T^{20} + 414 p^{30} T^{21} + p^{33} T^{22} \) | |
| 53 | \( 1 + 961 T + 965796 T^{2} + 608144639 T^{3} + 385742942961 T^{4} + 187987429606926 T^{5} + 91777997164257354 T^{6} + 36229344555773413828 T^{7} + \)\(14\!\cdots\!13\)\( T^{8} + \)\(50\!\cdots\!13\)\( T^{9} + \)\(19\!\cdots\!94\)\( T^{10} + \)\(67\!\cdots\!97\)\( T^{11} + \)\(19\!\cdots\!94\)\( p^{3} T^{12} + \)\(50\!\cdots\!13\)\( p^{6} T^{13} + \)\(14\!\cdots\!13\)\( p^{9} T^{14} + 36229344555773413828 p^{12} T^{15} + 91777997164257354 p^{15} T^{16} + 187987429606926 p^{18} T^{17} + 385742942961 p^{21} T^{18} + 608144639 p^{24} T^{19} + 965796 p^{27} T^{20} + 961 p^{30} T^{21} + p^{33} T^{22} \) | |
| 59 | \( 1 + 2053 T + 2563228 T^{2} + 39988831 p T^{3} + 1867718167293 T^{4} + 1321028354626708 T^{5} + 864735948460365164 T^{6} + \)\(52\!\cdots\!22\)\( T^{7} + \)\(29\!\cdots\!19\)\( T^{8} + \)\(15\!\cdots\!57\)\( T^{9} + \)\(76\!\cdots\!84\)\( T^{10} + \)\(35\!\cdots\!29\)\( T^{11} + \)\(76\!\cdots\!84\)\( p^{3} T^{12} + \)\(15\!\cdots\!57\)\( p^{6} T^{13} + \)\(29\!\cdots\!19\)\( p^{9} T^{14} + \)\(52\!\cdots\!22\)\( p^{12} T^{15} + 864735948460365164 p^{15} T^{16} + 1321028354626708 p^{18} T^{17} + 1867718167293 p^{21} T^{18} + 39988831 p^{25} T^{19} + 2563228 p^{27} T^{20} + 2053 p^{30} T^{21} + p^{33} T^{22} \) | |
| 61 | \( 1 + 916 T + 2316520 T^{2} + 1727599742 T^{3} + 2407486920721 T^{4} + 1504929469289936 T^{5} + 1507865826862440768 T^{6} + \)\(80\!\cdots\!24\)\( T^{7} + \)\(64\!\cdots\!13\)\( T^{8} + \)\(29\!\cdots\!18\)\( T^{9} + \)\(19\!\cdots\!44\)\( T^{10} + \)\(78\!\cdots\!32\)\( T^{11} + \)\(19\!\cdots\!44\)\( p^{3} T^{12} + \)\(29\!\cdots\!18\)\( p^{6} T^{13} + \)\(64\!\cdots\!13\)\( p^{9} T^{14} + \)\(80\!\cdots\!24\)\( p^{12} T^{15} + 1507865826862440768 p^{15} T^{16} + 1504929469289936 p^{18} T^{17} + 2407486920721 p^{21} T^{18} + 1727599742 p^{24} T^{19} + 2316520 p^{27} T^{20} + 916 p^{30} T^{21} + p^{33} T^{22} \) | |
| 67 | \( 1 + 2788 T + 5060708 T^{2} + 6749007174 T^{3} + 7465195866163 T^{4} + 7042476601709130 T^{5} + 5889406265610516510 T^{6} + \)\(44\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!29\)\( T^{8} + \)\(19\!\cdots\!54\)\( T^{9} + \)\(11\!\cdots\!12\)\( T^{10} + \)\(65\!\cdots\!52\)\( T^{11} + \)\(11\!\cdots\!12\)\( p^{3} T^{12} + \)\(19\!\cdots\!54\)\( p^{6} T^{13} + \)\(30\!\cdots\!29\)\( p^{9} T^{14} + \)\(44\!\cdots\!72\)\( p^{12} T^{15} + 5889406265610516510 p^{15} T^{16} + 7042476601709130 p^{18} T^{17} + 7465195866163 p^{21} T^{18} + 6749007174 p^{24} T^{19} + 5060708 p^{27} T^{20} + 2788 p^{30} T^{21} + p^{33} T^{22} \) | |
| 71 | \( 1 - 886 T + 2871464 T^{2} - 2048808946 T^{3} + 3843591855193 T^{4} - 2323655869380050 T^{5} + 3268752370277772894 T^{6} - \)\(17\!\cdots\!90\)\( T^{7} + \)\(19\!\cdots\!99\)\( T^{8} - \)\(92\!\cdots\!04\)\( T^{9} + \)\(91\!\cdots\!90\)\( T^{10} - \)\(37\!\cdots\!10\)\( T^{11} + \)\(91\!\cdots\!90\)\( p^{3} T^{12} - \)\(92\!\cdots\!04\)\( p^{6} T^{13} + \)\(19\!\cdots\!99\)\( p^{9} T^{14} - \)\(17\!\cdots\!90\)\( p^{12} T^{15} + 3268752370277772894 p^{15} T^{16} - 2323655869380050 p^{18} T^{17} + 3843591855193 p^{21} T^{18} - 2048808946 p^{24} T^{19} + 2871464 p^{27} T^{20} - 886 p^{30} T^{21} + p^{33} T^{22} \) | |
| 73 | \( 1 + 1916 T + 4060593 T^{2} + 5499755918 T^{3} + 7290389428346 T^{4} + 7732675607171318 T^{5} + 7870815362399698757 T^{6} + \)\(68\!\cdots\!40\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} + \)\(42\!\cdots\!26\)\( T^{9} + \)\(30\!\cdots\!23\)\( T^{10} + \)\(19\!\cdots\!60\)\( T^{11} + \)\(30\!\cdots\!23\)\( p^{3} T^{12} + \)\(42\!\cdots\!26\)\( p^{6} T^{13} + \)\(57\!\cdots\!20\)\( p^{9} T^{14} + \)\(68\!\cdots\!40\)\( p^{12} T^{15} + 7870815362399698757 p^{15} T^{16} + 7732675607171318 p^{18} T^{17} + 7290389428346 p^{21} T^{18} + 5499755918 p^{24} T^{19} + 4060593 p^{27} T^{20} + 1916 p^{30} T^{21} + p^{33} T^{22} \) | |
| 79 | \( 1 - 651 T + 1859061 T^{2} - 842633698 T^{3} + 1970080046670 T^{4} - 758154366801570 T^{5} + 1585350297100096807 T^{6} - \)\(53\!\cdots\!99\)\( T^{7} + \)\(10\!\cdots\!08\)\( T^{8} - \)\(34\!\cdots\!83\)\( T^{9} + \)\(62\!\cdots\!51\)\( T^{10} - \)\(18\!\cdots\!30\)\( T^{11} + \)\(62\!\cdots\!51\)\( p^{3} T^{12} - \)\(34\!\cdots\!83\)\( p^{6} T^{13} + \)\(10\!\cdots\!08\)\( p^{9} T^{14} - \)\(53\!\cdots\!99\)\( p^{12} T^{15} + 1585350297100096807 p^{15} T^{16} - 758154366801570 p^{18} T^{17} + 1970080046670 p^{21} T^{18} - 842633698 p^{24} T^{19} + 1859061 p^{27} T^{20} - 651 p^{30} T^{21} + p^{33} T^{22} \) | |
| 83 | \( 1 + 185 T + 2386329 T^{2} + 1313737748 T^{3} + 3485757058826 T^{4} + 2363809758985612 T^{5} + 3884049234589443341 T^{6} + \)\(27\!\cdots\!13\)\( T^{7} + \)\(33\!\cdots\!74\)\( T^{8} + \)\(24\!\cdots\!55\)\( T^{9} + \)\(23\!\cdots\!99\)\( T^{10} + \)\(15\!\cdots\!14\)\( T^{11} + \)\(23\!\cdots\!99\)\( p^{3} T^{12} + \)\(24\!\cdots\!55\)\( p^{6} T^{13} + \)\(33\!\cdots\!74\)\( p^{9} T^{14} + \)\(27\!\cdots\!13\)\( p^{12} T^{15} + 3884049234589443341 p^{15} T^{16} + 2363809758985612 p^{18} T^{17} + 3485757058826 p^{21} T^{18} + 1313737748 p^{24} T^{19} + 2386329 p^{27} T^{20} + 185 p^{30} T^{21} + p^{33} T^{22} \) | |
| 89 | \( 1 + 1490 T + 5259110 T^{2} + 7114996458 T^{3} + 13969190600104 T^{4} + 16889863729178918 T^{5} + 24221853516755738734 T^{6} + \)\(25\!\cdots\!90\)\( T^{7} + \)\(30\!\cdots\!83\)\( T^{8} + \)\(28\!\cdots\!28\)\( T^{9} + \)\(27\!\cdots\!16\)\( T^{10} + \)\(23\!\cdots\!52\)\( T^{11} + \)\(27\!\cdots\!16\)\( p^{3} T^{12} + \)\(28\!\cdots\!28\)\( p^{6} T^{13} + \)\(30\!\cdots\!83\)\( p^{9} T^{14} + \)\(25\!\cdots\!90\)\( p^{12} T^{15} + 24221853516755738734 p^{15} T^{16} + 16889863729178918 p^{18} T^{17} + 13969190600104 p^{21} T^{18} + 7114996458 p^{24} T^{19} + 5259110 p^{27} T^{20} + 1490 p^{30} T^{21} + p^{33} T^{22} \) | |
| 97 | \( 1 + 2705 T + 88576 p T^{2} + 17272666155 T^{3} + 33309653140730 T^{4} + 53642144174507247 T^{5} + 79668906394596945764 T^{6} + \)\(10\!\cdots\!83\)\( T^{7} + \)\(13\!\cdots\!91\)\( T^{8} + \)\(15\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!02\)\( T^{10} + \)\(15\!\cdots\!24\)\( T^{11} + \)\(16\!\cdots\!02\)\( p^{3} T^{12} + \)\(15\!\cdots\!96\)\( p^{6} T^{13} + \)\(13\!\cdots\!91\)\( p^{9} T^{14} + \)\(10\!\cdots\!83\)\( p^{12} T^{15} + 79668906394596945764 p^{15} T^{16} + 53642144174507247 p^{18} T^{17} + 33309653140730 p^{21} T^{18} + 17272666155 p^{24} T^{19} + 88576 p^{28} T^{20} + 2705 p^{30} T^{21} + p^{33} T^{22} \) | |
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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
−3.18796446284149830745603930556, −3.17739020331355684302314013290, −3.12137094456837279537621962819, −3.08257874425247636260465769657, −2.89717792817016115906854197192, −2.72804331053227660101225438495, −2.63472472066021368431823978482, −2.51831698610794369930598258764, −2.44078723319993233602426190102, −2.28617599335133304326060668072, −2.24193396828817649940683987359, −1.92924254711756243695445339070, −1.90696704288572648084337297173, −1.89854177054876100623051167352, −1.80235813760227871159486424818, −1.79056627755949044647827972037, −1.77451562773360665096257691023, −1.57482340749289864597819560438, −1.39885227839236888175192168964, −1.32448970002064000858211429139, −1.29908564975214440243715351157, −1.18451059047576010121644796744, −1.11477917592428180190122570715, −1.06412436332571189134371679885, −1.05951837945251632198716447576, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.05951837945251632198716447576, 1.06412436332571189134371679885, 1.11477917592428180190122570715, 1.18451059047576010121644796744, 1.29908564975214440243715351157, 1.32448970002064000858211429139, 1.39885227839236888175192168964, 1.57482340749289864597819560438, 1.77451562773360665096257691023, 1.79056627755949044647827972037, 1.80235813760227871159486424818, 1.89854177054876100623051167352, 1.90696704288572648084337297173, 1.92924254711756243695445339070, 2.24193396828817649940683987359, 2.28617599335133304326060668072, 2.44078723319993233602426190102, 2.51831698610794369930598258764, 2.63472472066021368431823978482, 2.72804331053227660101225438495, 2.89717792817016115906854197192, 3.08257874425247636260465769657, 3.12137094456837279537621962819, 3.17739020331355684302314013290, 3.18796446284149830745603930556
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.