Dirichlet series
| L(s) = 1 | + 2·3-s − 7-s + 9·9-s − 10·13-s + 2·19-s − 2·21-s + 27·23-s + 10·27-s + 9·29-s + 8·31-s − 22·37-s − 20·39-s + 54·41-s + 44·43-s − 108·47-s + 174·49-s + 4·57-s + 9·59-s − 55·61-s − 9·63-s − 28·67-s + 54·69-s + 86·73-s + 11·79-s − 4·81-s − 306·83-s + 18·87-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 1/7·7-s + 9-s − 0.769·13-s + 2/19·19-s − 0.0952·21-s + 1.17·23-s + 0.370·27-s + 9/29·29-s + 8/31·31-s − 0.594·37-s − 0.512·39-s + 1.31·41-s + 1.02·43-s − 2.29·47-s + 3.55·49-s + 4/57·57-s + 9/59·59-s − 0.901·61-s − 1/7·63-s − 0.417·67-s + 0.782·69-s + 1.17·73-s + 0.139·79-s − 0.0493·81-s − 3.68·83-s + 6/29·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.71096\times 10^{22}\) |
| Root analytic conductor: | \(4.95209\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.334044439\) |
| \(L(\frac12)\) | \(\approx\) | \(1.334044439\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T - 5 T^{2} + 2 p^{2} T^{3} + 11 p T^{4} + p^{2} T^{5} - 95 p^{2} T^{6} - 43 p^{3} T^{7} + 56 p^{5} T^{8} - 43 p^{5} T^{9} - 95 p^{6} T^{10} + p^{8} T^{11} + 11 p^{9} T^{12} + 2 p^{12} T^{13} - 5 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T - 173 T^{2} - 76 T^{3} + 1796 p T^{4} + 995 T^{5} - 503676 T^{6} - 482400 T^{7} + 15954660 T^{8} + 85373277 T^{9} - 102023283 p T^{10} - 5257963569 T^{11} + 6326818878 p T^{12} + 136495028703 T^{13} - 2930129682423 T^{14} - 162268608897 p T^{15} + 3388256670933 p^{2} T^{16} - 162268608897 p^{3} T^{17} - 2930129682423 p^{4} T^{18} + 136495028703 p^{6} T^{19} + 6326818878 p^{9} T^{20} - 5257963569 p^{10} T^{21} - 102023283 p^{13} T^{22} + 85373277 p^{14} T^{23} + 15954660 p^{16} T^{24} - 482400 p^{18} T^{25} - 503676 p^{20} T^{26} + 995 p^{22} T^{27} + 1796 p^{25} T^{28} - 76 p^{26} T^{29} - 173 p^{28} T^{30} + p^{30} T^{31} + p^{32} T^{32} \) |
| 11 | \( 1 + 458 T^{2} + 92712 T^{4} + 146529 T^{5} + 12038416 T^{6} + 3218535 p T^{7} + 1375665020 T^{8} - 936533178 T^{9} + 172148293599 T^{10} - 1510922280609 T^{11} + 16721317926007 T^{12} - 315164769976692 T^{13} + 686859727339688 T^{14} - 42040141678654524 T^{15} - 5156924675770584 T^{16} - 42040141678654524 p^{2} T^{17} + 686859727339688 p^{4} T^{18} - 315164769976692 p^{6} T^{19} + 16721317926007 p^{8} T^{20} - 1510922280609 p^{10} T^{21} + 172148293599 p^{12} T^{22} - 936533178 p^{14} T^{23} + 1375665020 p^{16} T^{24} + 3218535 p^{19} T^{25} + 12038416 p^{20} T^{26} + 146529 p^{22} T^{27} + 92712 p^{24} T^{28} + 458 p^{28} T^{30} + p^{32} T^{32} \) | |
| 13 | \( 1 + 10 T - 566 T^{2} - 6790 T^{3} + 175913 T^{4} + 2649575 T^{5} - 32343867 T^{6} - 693138465 T^{7} + 2600674452 T^{8} + 134164256145 T^{9} + 524697112311 T^{10} - 19638956724015 T^{11} - 265022070717210 T^{12} + 2092089813816945 T^{13} + 65251680279938697 T^{14} - 117058814340687495 T^{15} - 12252909706713728001 T^{16} - 117058814340687495 p^{2} T^{17} + 65251680279938697 p^{4} T^{18} + 2092089813816945 p^{6} T^{19} - 265022070717210 p^{8} T^{20} - 19638956724015 p^{10} T^{21} + 524697112311 p^{12} T^{22} + 134164256145 p^{14} T^{23} + 2600674452 p^{16} T^{24} - 693138465 p^{18} T^{25} - 32343867 p^{20} T^{26} + 2649575 p^{22} T^{27} + 175913 p^{24} T^{28} - 6790 p^{26} T^{29} - 566 p^{28} T^{30} + 10 p^{30} T^{31} + p^{32} T^{32} \) | |
| 17 | \( 1 - 2038 T^{2} + 2096727 T^{4} - 1457979128 T^{6} + 775079950643 T^{8} - 337823552802273 T^{10} + 126529171900793275 T^{12} - 42115050983461695265 T^{14} + \)\(12\!\cdots\!76\)\( T^{16} - 42115050983461695265 p^{4} T^{18} + 126529171900793275 p^{8} T^{20} - 337823552802273 p^{12} T^{22} + 775079950643 p^{16} T^{24} - 1457979128 p^{20} T^{26} + 2096727 p^{24} T^{28} - 2038 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 - T + 1410 T^{2} + 823 T^{3} + 968846 T^{4} + 3060759 T^{5} + 460539217 T^{6} + 2470256684 T^{7} + 178951279446 T^{8} + 2470256684 p^{2} T^{9} + 460539217 p^{4} T^{10} + 3060759 p^{6} T^{11} + 968846 p^{8} T^{12} + 823 p^{10} T^{13} + 1410 p^{12} T^{14} - p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 27 T + 2390 T^{2} - 57969 T^{3} + 2815986 T^{4} - 64940670 T^{5} + 2118860017 T^{6} - 46791596592 T^{7} + 1066814470400 T^{8} - 22271881955571 T^{9} + 297605591638755 T^{10} - 4467680643855537 T^{11} - 64967907031482224 T^{12} + 3216997603170014265 T^{13} - \)\(14\!\cdots\!97\)\( T^{14} + \)\(39\!\cdots\!55\)\( T^{15} - \)\(97\!\cdots\!67\)\( T^{16} + \)\(39\!\cdots\!55\)\( p^{2} T^{17} - \)\(14\!\cdots\!97\)\( p^{4} T^{18} + 3216997603170014265 p^{6} T^{19} - 64967907031482224 p^{8} T^{20} - 4467680643855537 p^{10} T^{21} + 297605591638755 p^{12} T^{22} - 22271881955571 p^{14} T^{23} + 1066814470400 p^{16} T^{24} - 46791596592 p^{18} T^{25} + 2118860017 p^{20} T^{26} - 64940670 p^{22} T^{27} + 2815986 p^{24} T^{28} - 57969 p^{26} T^{29} + 2390 p^{28} T^{30} - 27 p^{30} T^{31} + p^{32} T^{32} \) | |
| 29 | \( 1 - 9 T + 3869 T^{2} - 34578 T^{3} + 7607838 T^{4} - 87088491 T^{5} + 9843450676 T^{6} - 140144406552 T^{7} + 9182468839598 T^{8} - 4987936766187 p T^{9} + 6445267048588155 T^{10} - 79376641074392607 T^{11} + 3332230316199648880 T^{12} + 11384266603718401191 T^{13} + \)\(11\!\cdots\!63\)\( T^{14} + \)\(64\!\cdots\!79\)\( T^{15} + \)\(44\!\cdots\!01\)\( T^{16} + \)\(64\!\cdots\!79\)\( p^{2} T^{17} + \)\(11\!\cdots\!63\)\( p^{4} T^{18} + 11384266603718401191 p^{6} T^{19} + 3332230316199648880 p^{8} T^{20} - 79376641074392607 p^{10} T^{21} + 6445267048588155 p^{12} T^{22} - 4987936766187 p^{15} T^{23} + 9182468839598 p^{16} T^{24} - 140144406552 p^{18} T^{25} + 9843450676 p^{20} T^{26} - 87088491 p^{22} T^{27} + 7607838 p^{24} T^{28} - 34578 p^{26} T^{29} + 3869 p^{28} T^{30} - 9 p^{30} T^{31} + p^{32} T^{32} \) | |
| 31 | \( 1 - 8 T - 4202 T^{2} - 34726 T^{3} + 9633311 T^{4} + 177584303 T^{5} - 12195759429 T^{6} - 363755999007 T^{7} + 7960628209122 T^{8} + 375060758498313 T^{9} - 95953813832757 T^{10} - 152981213606528241 T^{11} - 2916180501891627912 T^{12} - 75111210005300864421 T^{13} - \)\(35\!\cdots\!69\)\( T^{14} + \)\(71\!\cdots\!87\)\( T^{15} + \)\(28\!\cdots\!31\)\( T^{16} + \)\(71\!\cdots\!87\)\( p^{2} T^{17} - \)\(35\!\cdots\!69\)\( p^{4} T^{18} - 75111210005300864421 p^{6} T^{19} - 2916180501891627912 p^{8} T^{20} - 152981213606528241 p^{10} T^{21} - 95953813832757 p^{12} T^{22} + 375060758498313 p^{14} T^{23} + 7960628209122 p^{16} T^{24} - 363755999007 p^{18} T^{25} - 12195759429 p^{20} T^{26} + 177584303 p^{22} T^{27} + 9633311 p^{24} T^{28} - 34726 p^{26} T^{29} - 4202 p^{28} T^{30} - 8 p^{30} T^{31} + p^{32} T^{32} \) | |
| 37 | \( ( 1 + 11 T + 4755 T^{2} + 137716 T^{3} + 12722987 T^{4} + 456370782 T^{5} + 27462452767 T^{6} + 843053873741 T^{7} + 44915869623024 T^{8} + 843053873741 p^{2} T^{9} + 27462452767 p^{4} T^{10} + 456370782 p^{6} T^{11} + 12722987 p^{8} T^{12} + 137716 p^{10} T^{13} + 4755 p^{12} T^{14} + 11 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 41 | \( 1 - 54 T + 9569 T^{2} - 464238 T^{3} + 1103553 p T^{4} - 2220207921 T^{5} + 155117403976 T^{6} - 7877120405832 T^{7} + 444676057615853 T^{8} - 22464179763746418 T^{9} + 1119164115332852640 T^{10} - 53884205877007006992 T^{11} + \)\(25\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!54\)\( T^{13} + \)\(50\!\cdots\!13\)\( T^{14} - \)\(21\!\cdots\!51\)\( T^{15} + \)\(90\!\cdots\!06\)\( T^{16} - \)\(21\!\cdots\!51\)\( p^{2} T^{17} + \)\(50\!\cdots\!13\)\( p^{4} T^{18} - \)\(11\!\cdots\!54\)\( p^{6} T^{19} + \)\(25\!\cdots\!20\)\( p^{8} T^{20} - 53884205877007006992 p^{10} T^{21} + 1119164115332852640 p^{12} T^{22} - 22464179763746418 p^{14} T^{23} + 444676057615853 p^{16} T^{24} - 7877120405832 p^{18} T^{25} + 155117403976 p^{20} T^{26} - 2220207921 p^{22} T^{27} + 1103553 p^{25} T^{28} - 464238 p^{26} T^{29} + 9569 p^{28} T^{30} - 54 p^{30} T^{31} + p^{32} T^{32} \) | |
| 43 | \( 1 - 44 T - 6644 T^{2} + 371588 T^{3} + 18190880 T^{4} - 1380116794 T^{5} - 26507347596 T^{6} + 3249327060882 T^{7} + 20653446638007 T^{8} - 5374104569173848 T^{9} - 24468427595855244 T^{10} + 5781109733583287676 T^{11} + \)\(15\!\cdots\!80\)\( T^{12} - \)\(38\!\cdots\!74\)\( T^{13} - \)\(53\!\cdots\!72\)\( T^{14} + \)\(16\!\cdots\!94\)\( T^{15} + \)\(11\!\cdots\!52\)\( T^{16} + \)\(16\!\cdots\!94\)\( p^{2} T^{17} - \)\(53\!\cdots\!72\)\( p^{4} T^{18} - \)\(38\!\cdots\!74\)\( p^{6} T^{19} + \)\(15\!\cdots\!80\)\( p^{8} T^{20} + 5781109733583287676 p^{10} T^{21} - 24468427595855244 p^{12} T^{22} - 5374104569173848 p^{14} T^{23} + 20653446638007 p^{16} T^{24} + 3249327060882 p^{18} T^{25} - 26507347596 p^{20} T^{26} - 1380116794 p^{22} T^{27} + 18190880 p^{24} T^{28} + 371588 p^{26} T^{29} - 6644 p^{28} T^{30} - 44 p^{30} T^{31} + p^{32} T^{32} \) | |
| 47 | \( 1 + 108 T + 18449 T^{2} + 1572588 T^{3} + 167464920 T^{4} + 12318161076 T^{5} + 1009201833871 T^{6} + 66333493090422 T^{7} + 4548180687798086 T^{8} + 273722899919474358 T^{9} + 16399267985958921618 T^{10} + \)\(91\!\cdots\!60\)\( T^{11} + \)\(49\!\cdots\!52\)\( T^{12} + \)\(55\!\cdots\!12\)\( p T^{13} + \)\(13\!\cdots\!80\)\( T^{14} + \)\(13\!\cdots\!58\)\( p T^{15} + \)\(30\!\cdots\!67\)\( T^{16} + \)\(13\!\cdots\!58\)\( p^{3} T^{17} + \)\(13\!\cdots\!80\)\( p^{4} T^{18} + \)\(55\!\cdots\!12\)\( p^{7} T^{19} + \)\(49\!\cdots\!52\)\( p^{8} T^{20} + \)\(91\!\cdots\!60\)\( p^{10} T^{21} + 16399267985958921618 p^{12} T^{22} + 273722899919474358 p^{14} T^{23} + 4548180687798086 p^{16} T^{24} + 66333493090422 p^{18} T^{25} + 1009201833871 p^{20} T^{26} + 12318161076 p^{22} T^{27} + 167464920 p^{24} T^{28} + 1572588 p^{26} T^{29} + 18449 p^{28} T^{30} + 108 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 - 12928 T^{2} + 83844978 T^{4} - 402609798377 T^{6} + 1688853204078047 T^{8} - 6352297600325006553 T^{10} + \)\(21\!\cdots\!42\)\( T^{12} - \)\(69\!\cdots\!90\)\( T^{14} + \)\(20\!\cdots\!04\)\( T^{16} - \)\(69\!\cdots\!90\)\( p^{4} T^{18} + \)\(21\!\cdots\!42\)\( p^{8} T^{20} - 6352297600325006553 p^{12} T^{22} + 1688853204078047 p^{16} T^{24} - 402609798377 p^{20} T^{26} + 83844978 p^{24} T^{28} - 12928 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( 1 - 9 T + 9221 T^{2} - 82746 T^{3} + 24816945 T^{4} - 917898714 T^{5} + 21445173691 T^{6} - 7557920893404 T^{7} + 253376559727169 T^{8} - 28211053236897393 T^{9} + 2122191315532993398 T^{10} - 50491034100868868001 T^{11} + \)\(78\!\cdots\!14\)\( T^{12} - \)\(26\!\cdots\!21\)\( T^{13} + \)\(13\!\cdots\!46\)\( T^{14} - \)\(20\!\cdots\!56\)\( T^{15} + \)\(16\!\cdots\!18\)\( T^{16} - \)\(20\!\cdots\!56\)\( p^{2} T^{17} + \)\(13\!\cdots\!46\)\( p^{4} T^{18} - \)\(26\!\cdots\!21\)\( p^{6} T^{19} + \)\(78\!\cdots\!14\)\( p^{8} T^{20} - 50491034100868868001 p^{10} T^{21} + 2122191315532993398 p^{12} T^{22} - 28211053236897393 p^{14} T^{23} + 253376559727169 p^{16} T^{24} - 7557920893404 p^{18} T^{25} + 21445173691 p^{20} T^{26} - 917898714 p^{22} T^{27} + 24816945 p^{24} T^{28} - 82746 p^{26} T^{29} + 9221 p^{28} T^{30} - 9 p^{30} T^{31} + p^{32} T^{32} \) | |
| 61 | \( 1 + 55 T - 20948 T^{2} - 969337 T^{3} + 243845990 T^{4} + 8951361278 T^{5} - 2082762713853 T^{6} - 57644048591208 T^{7} + 14373812538574596 T^{8} + 285008023497630165 T^{9} - 83533486064837896515 T^{10} - \)\(10\!\cdots\!47\)\( T^{11} + \)\(41\!\cdots\!96\)\( T^{12} + \)\(30\!\cdots\!79\)\( T^{13} - \)\(18\!\cdots\!21\)\( T^{14} - \)\(42\!\cdots\!15\)\( T^{15} + \)\(73\!\cdots\!69\)\( T^{16} - \)\(42\!\cdots\!15\)\( p^{2} T^{17} - \)\(18\!\cdots\!21\)\( p^{4} T^{18} + \)\(30\!\cdots\!79\)\( p^{6} T^{19} + \)\(41\!\cdots\!96\)\( p^{8} T^{20} - \)\(10\!\cdots\!47\)\( p^{10} T^{21} - 83533486064837896515 p^{12} T^{22} + 285008023497630165 p^{14} T^{23} + 14373812538574596 p^{16} T^{24} - 57644048591208 p^{18} T^{25} - 2082762713853 p^{20} T^{26} + 8951361278 p^{22} T^{27} + 243845990 p^{24} T^{28} - 969337 p^{26} T^{29} - 20948 p^{28} T^{30} + 55 p^{30} T^{31} + p^{32} T^{32} \) | |
| 67 | \( 1 + 28 T - 27038 T^{2} + 5048 T^{3} + 415566458 T^{4} - 7125205363 T^{5} - 4231890977526 T^{6} + 121715150854446 T^{7} + 31894280371417644 T^{8} - 1115207854388550894 T^{9} - \)\(19\!\cdots\!50\)\( T^{10} + \)\(66\!\cdots\!10\)\( T^{11} + \)\(97\!\cdots\!57\)\( T^{12} - \)\(26\!\cdots\!70\)\( T^{13} - \)\(45\!\cdots\!22\)\( T^{14} + \)\(46\!\cdots\!39\)\( T^{15} + \)\(20\!\cdots\!76\)\( T^{16} + \)\(46\!\cdots\!39\)\( p^{2} T^{17} - \)\(45\!\cdots\!22\)\( p^{4} T^{18} - \)\(26\!\cdots\!70\)\( p^{6} T^{19} + \)\(97\!\cdots\!57\)\( p^{8} T^{20} + \)\(66\!\cdots\!10\)\( p^{10} T^{21} - \)\(19\!\cdots\!50\)\( p^{12} T^{22} - 1115207854388550894 p^{14} T^{23} + 31894280371417644 p^{16} T^{24} + 121715150854446 p^{18} T^{25} - 4231890977526 p^{20} T^{26} - 7125205363 p^{22} T^{27} + 415566458 p^{24} T^{28} + 5048 p^{26} T^{29} - 27038 p^{28} T^{30} + 28 p^{30} T^{31} + p^{32} T^{32} \) | |
| 71 | \( 1 - 34468 T^{2} + 609411090 T^{4} - 7504814130221 T^{6} + 72822008723353751 T^{8} - \)\(59\!\cdots\!53\)\( T^{10} + \)\(41\!\cdots\!38\)\( T^{12} - \)\(25\!\cdots\!34\)\( T^{14} + \)\(13\!\cdots\!28\)\( T^{16} - \)\(25\!\cdots\!34\)\( p^{4} T^{18} + \)\(41\!\cdots\!38\)\( p^{8} T^{20} - \)\(59\!\cdots\!53\)\( p^{12} T^{22} + 72822008723353751 p^{16} T^{24} - 7504814130221 p^{20} T^{26} + 609411090 p^{24} T^{28} - 34468 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 43 T + 11865 T^{2} - 372143 T^{3} + 120121058 T^{4} - 2983986129 T^{5} + 720847308175 T^{6} - 16190095293604 T^{7} + 62682255233676 p T^{8} - 16190095293604 p^{2} T^{9} + 720847308175 p^{4} T^{10} - 2983986129 p^{6} T^{11} + 120121058 p^{8} T^{12} - 372143 p^{10} T^{13} + 11865 p^{12} T^{14} - 43 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 - 11 T - 17798 T^{2} + 931427 T^{3} + 159953438 T^{4} - 13872105958 T^{5} - 155052131427 T^{6} + 109581403934988 T^{7} - 8866700904498624 T^{8} - 115235770661149701 T^{9} + 96114835101632346789 T^{10} - \)\(47\!\cdots\!59\)\( T^{11} - \)\(28\!\cdots\!96\)\( T^{12} + \)\(49\!\cdots\!07\)\( T^{13} - \)\(18\!\cdots\!01\)\( T^{14} - \)\(13\!\cdots\!49\)\( T^{15} + \)\(24\!\cdots\!09\)\( T^{16} - \)\(13\!\cdots\!49\)\( p^{2} T^{17} - \)\(18\!\cdots\!01\)\( p^{4} T^{18} + \)\(49\!\cdots\!07\)\( p^{6} T^{19} - \)\(28\!\cdots\!96\)\( p^{8} T^{20} - \)\(47\!\cdots\!59\)\( p^{10} T^{21} + 96114835101632346789 p^{12} T^{22} - 115235770661149701 p^{14} T^{23} - 8866700904498624 p^{16} T^{24} + 109581403934988 p^{18} T^{25} - 155052131427 p^{20} T^{26} - 13872105958 p^{22} T^{27} + 159953438 p^{24} T^{28} + 931427 p^{26} T^{29} - 17798 p^{28} T^{30} - 11 p^{30} T^{31} + p^{32} T^{32} \) | |
| 83 | \( 1 + 306 T + 63782 T^{2} + 9966420 T^{3} + 1301571390 T^{4} + 150941837061 T^{5} + 15328986985285 T^{6} + 1396301388096627 T^{7} + 110909016254544236 T^{8} + 7625597345221493124 T^{9} + \)\(43\!\cdots\!38\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} - \)\(14\!\cdots\!97\)\( T^{12} - \)\(12\!\cdots\!39\)\( T^{13} - \)\(16\!\cdots\!29\)\( T^{14} - \)\(16\!\cdots\!43\)\( T^{15} - \)\(14\!\cdots\!96\)\( T^{16} - \)\(16\!\cdots\!43\)\( p^{2} T^{17} - \)\(16\!\cdots\!29\)\( p^{4} T^{18} - \)\(12\!\cdots\!39\)\( p^{6} T^{19} - \)\(14\!\cdots\!97\)\( p^{8} T^{20} + \)\(16\!\cdots\!80\)\( p^{10} T^{21} + \)\(43\!\cdots\!38\)\( p^{12} T^{22} + 7625597345221493124 p^{14} T^{23} + 110909016254544236 p^{16} T^{24} + 1396301388096627 p^{18} T^{25} + 15328986985285 p^{20} T^{26} + 150941837061 p^{22} T^{27} + 1301571390 p^{24} T^{28} + 9966420 p^{26} T^{29} + 63782 p^{28} T^{30} + 306 p^{30} T^{31} + p^{32} T^{32} \) | |
| 89 | \( 1 - 47563 T^{2} + 1188755265 T^{4} - 21260580063086 T^{6} + 305421676006002176 T^{8} - \)\(37\!\cdots\!53\)\( T^{10} + \)\(39\!\cdots\!48\)\( T^{12} - \)\(36\!\cdots\!69\)\( T^{14} + \)\(30\!\cdots\!23\)\( T^{16} - \)\(36\!\cdots\!69\)\( p^{4} T^{18} + \)\(39\!\cdots\!48\)\( p^{8} T^{20} - \)\(37\!\cdots\!53\)\( p^{12} T^{22} + 305421676006002176 p^{16} T^{24} - 21260580063086 p^{20} T^{26} + 1188755265 p^{24} T^{28} - 47563 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( 1 - 41 T - 33140 T^{2} + 1711643 T^{3} + 486478427 T^{4} - 33780286021 T^{5} - 4022134173783 T^{6} + 453845299750965 T^{7} + 17126012214628317 T^{8} - 5261492745179603022 T^{9} + 53794952466489086730 T^{10} + \)\(52\!\cdots\!98\)\( T^{11} - \)\(23\!\cdots\!48\)\( T^{12} - \)\(39\!\cdots\!72\)\( T^{13} + \)\(35\!\cdots\!01\)\( T^{14} + \)\(14\!\cdots\!18\)\( T^{15} - \)\(38\!\cdots\!54\)\( T^{16} + \)\(14\!\cdots\!18\)\( p^{2} T^{17} + \)\(35\!\cdots\!01\)\( p^{4} T^{18} - \)\(39\!\cdots\!72\)\( p^{6} T^{19} - \)\(23\!\cdots\!48\)\( p^{8} T^{20} + \)\(52\!\cdots\!98\)\( p^{10} T^{21} + 53794952466489086730 p^{12} T^{22} - 5261492745179603022 p^{14} T^{23} + 17126012214628317 p^{16} T^{24} + 453845299750965 p^{18} T^{25} - 4022134173783 p^{20} T^{26} - 33780286021 p^{22} T^{27} + 486478427 p^{24} T^{28} + 1711643 p^{26} T^{29} - 33140 p^{28} T^{30} - 41 p^{30} T^{31} + p^{32} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.41712074339012642251313944626, −2.38480723155145808202763897639, −2.26032902943219693989807353710, −2.22565809083597926451817845502, −2.20468326872986413641267831960, −1.92360443936934764904561645289, −1.89503518154929343420038473108, −1.87779527002713071076893700053, −1.60661529356691451211249426557, −1.59547304915182623692398156951, −1.53992697888351586761264516686, −1.47384334173670784827957925898, −1.33722606751598919650679428761, −1.28659573571807007658762945034, −1.26142943374579077307071062784, −1.04118804392291780417269110128, −0.970973498607528176143474447452, −0.964187191071330229251618929039, −0.882686066088126004938369400264, −0.66376426777999778241456370491, −0.55046309178106700521461123805, −0.44434514475161591507282503891, −0.16162875493443962521169619435, −0.14571844318851328080005164984, −0.087362633138502227664420895158, 0.087362633138502227664420895158, 0.14571844318851328080005164984, 0.16162875493443962521169619435, 0.44434514475161591507282503891, 0.55046309178106700521461123805, 0.66376426777999778241456370491, 0.882686066088126004938369400264, 0.964187191071330229251618929039, 0.970973498607528176143474447452, 1.04118804392291780417269110128, 1.26142943374579077307071062784, 1.28659573571807007658762945034, 1.33722606751598919650679428761, 1.47384334173670784827957925898, 1.53992697888351586761264516686, 1.59547304915182623692398156951, 1.60661529356691451211249426557, 1.87779527002713071076893700053, 1.89503518154929343420038473108, 1.92360443936934764904561645289, 2.20468326872986413641267831960, 2.22565809083597926451817845502, 2.26032902943219693989807353710, 2.38480723155145808202763897639, 2.41712074339012642251313944626
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.