Dirichlet series
| L(s) = 1 | − 20·7-s + 6·8-s − 54·9-s + 6·11-s − 6·13-s + 16·16-s − 20·19-s + 4·27-s − 40·29-s − 32·31-s + 18·32-s − 158·37-s + 70·41-s + 130·47-s + 200·49-s − 216·53-s − 120·56-s + 132·59-s − 152·61-s + 1.08e3·63-s + 18·64-s − 178·67-s − 258·71-s − 324·72-s + 262·73-s − 120·77-s − 544·79-s + ⋯ |
| L(s) = 1 | − 2.85·7-s + 3/4·8-s − 6·9-s + 6/11·11-s − 0.461·13-s + 16-s − 1.05·19-s + 4/27·27-s − 1.37·29-s − 1.03·31-s + 9/16·32-s − 4.27·37-s + 1.70·41-s + 2.76·47-s + 4.08·49-s − 4.07·53-s − 2.14·56-s + 2.23·59-s − 2.49·61-s + 17.1·63-s + 9/32·64-s − 2.65·67-s − 3.63·71-s − 9/2·72-s + 3.58·73-s − 1.55·77-s − 6.88·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.79768\times 10^{18}\) |
| Root analytic conductor: | \(2.97583\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{40} \cdot 13^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.001433516026\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001433516026\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 6 T - 298 T^{2} + 1650 T^{3} + 91439 T^{4} - 90808 T^{5} - 12375044 T^{6} + 2467792 p T^{7} + 23669600 p^{2} T^{8} + 4810304 p^{3} T^{9} - 21585956 p^{4} T^{10} + 4810304 p^{5} T^{11} + 23669600 p^{6} T^{12} + 2467792 p^{7} T^{13} - 12375044 p^{8} T^{14} - 90808 p^{10} T^{15} + 91439 p^{12} T^{16} + 1650 p^{14} T^{17} - 298 p^{16} T^{18} + 6 p^{18} T^{19} + p^{20} T^{20} \) | |
| good | 2 | \( 1 - 3 p T^{3} - p^{4} T^{4} - 9 p T^{5} + 9 p T^{6} + 117 p T^{7} + 11 p T^{8} + 231 p T^{9} - 477 p T^{10} - 465 p T^{11} - 53 p^{6} T^{12} + 2841 p T^{13} + 4491 p T^{14} - 4221 p T^{15} + 14585 T^{16} - 62727 p T^{17} + 31617 p T^{18} - 7413 p^{3} T^{19} + 46855 p^{5} T^{20} - 7413 p^{5} T^{21} + 31617 p^{5} T^{22} - 62727 p^{7} T^{23} + 14585 p^{8} T^{24} - 4221 p^{11} T^{25} + 4491 p^{13} T^{26} + 2841 p^{15} T^{27} - 53 p^{22} T^{28} - 465 p^{19} T^{29} - 477 p^{21} T^{30} + 231 p^{23} T^{31} + 11 p^{25} T^{32} + 117 p^{27} T^{33} + 9 p^{29} T^{34} - 9 p^{31} T^{35} - p^{36} T^{36} - 3 p^{35} T^{37} + p^{40} T^{40} \) |
| 3 | \( ( 1 + p^{3} T^{2} - 2 T^{3} + 140 p T^{4} + 58 T^{5} + 1741 p T^{6} + 812 p T^{7} + 17743 p T^{8} + 11732 p T^{9} + 159220 p T^{10} + 11732 p^{3} T^{11} + 17743 p^{5} T^{12} + 812 p^{7} T^{13} + 1741 p^{9} T^{14} + 58 p^{10} T^{15} + 140 p^{13} T^{16} - 2 p^{14} T^{17} + p^{19} T^{18} + p^{20} T^{20} )^{2} \) | |
| 7 | \( 1 + 20 T + 200 T^{2} + 2434 T^{3} + 18692 T^{4} + 1956 p T^{5} - 502382 T^{6} - 11503104 T^{7} - 24983666 p T^{8} - 1093390822 T^{9} - 5404979400 T^{10} - 25326730092 T^{11} + 44171296486 p T^{12} + 4454165786736 T^{13} + 29958699590892 T^{14} + 272264822879082 T^{15} + 1284732364125953 T^{16} - 2318435203859696 T^{17} - 40453768337294710 T^{18} - 572612177039962490 T^{19} - 6293377644501829972 T^{20} - 572612177039962490 p^{2} T^{21} - 40453768337294710 p^{4} T^{22} - 2318435203859696 p^{6} T^{23} + 1284732364125953 p^{8} T^{24} + 272264822879082 p^{10} T^{25} + 29958699590892 p^{12} T^{26} + 4454165786736 p^{14} T^{27} + 44171296486 p^{17} T^{28} - 25326730092 p^{18} T^{29} - 5404979400 p^{20} T^{30} - 1093390822 p^{22} T^{31} - 24983666 p^{25} T^{32} - 11503104 p^{26} T^{33} - 502382 p^{28} T^{34} + 1956 p^{31} T^{35} + 18692 p^{32} T^{36} + 2434 p^{34} T^{37} + 200 p^{36} T^{38} + 20 p^{38} T^{39} + p^{40} T^{40} \) | |
| 11 | \( 1 - 6 T + 18 T^{2} - 2804 T^{3} + 26314 T^{4} - 322546 T^{5} + 5392832 T^{6} - 71958178 T^{7} + 521049219 T^{8} - 505961790 p T^{9} + 135896303120 T^{10} - 42551717790 p T^{11} + 2120415018696 T^{12} - 83701045853472 T^{13} - 225257076568242 T^{14} + 9337511465424586 T^{15} - 101055485882403699 T^{16} + 924918958225873004 T^{17} - 30062525920424760368 T^{18} + \)\(35\!\cdots\!16\)\( T^{19} - \)\(15\!\cdots\!42\)\( T^{20} + \)\(35\!\cdots\!16\)\( p^{2} T^{21} - 30062525920424760368 p^{4} T^{22} + 924918958225873004 p^{6} T^{23} - 101055485882403699 p^{8} T^{24} + 9337511465424586 p^{10} T^{25} - 225257076568242 p^{12} T^{26} - 83701045853472 p^{14} T^{27} + 2120415018696 p^{16} T^{28} - 42551717790 p^{19} T^{29} + 135896303120 p^{20} T^{30} - 505961790 p^{23} T^{31} + 521049219 p^{24} T^{32} - 71958178 p^{26} T^{33} + 5392832 p^{28} T^{34} - 322546 p^{30} T^{35} + 26314 p^{32} T^{36} - 2804 p^{34} T^{37} + 18 p^{36} T^{38} - 6 p^{38} T^{39} + p^{40} T^{40} \) | |
| 17 | \( 1 - 2988 T^{2} + 4570530 T^{4} - 4756667872 T^{6} + 3771385321163 T^{8} - 2417042889609248 T^{10} + 1296949168613911328 T^{12} - \)\(59\!\cdots\!80\)\( T^{14} + \)\(23\!\cdots\!73\)\( T^{16} - \)\(83\!\cdots\!32\)\( T^{18} + \)\(25\!\cdots\!90\)\( T^{20} - \)\(83\!\cdots\!32\)\( p^{4} T^{22} + \)\(23\!\cdots\!73\)\( p^{8} T^{24} - \)\(59\!\cdots\!80\)\( p^{12} T^{26} + 1296949168613911328 p^{16} T^{28} - 2417042889609248 p^{20} T^{30} + 3771385321163 p^{24} T^{32} - 4756667872 p^{28} T^{34} + 4570530 p^{32} T^{36} - 2988 p^{36} T^{38} + p^{40} T^{40} \) | |
| 19 | \( 1 + 20 T + 200 T^{2} + 432 T^{3} - 133036 T^{4} - 5227628 T^{5} - 77852048 T^{6} - 1384501004 T^{7} + 7115318566 T^{8} + 701177925136 T^{9} + 15309312551256 T^{10} + 362640766156564 T^{11} + 5576738993915162 T^{12} - 27614010295636836 T^{13} - 1627411205996070488 T^{14} - 54845518191378878016 T^{15} - \)\(10\!\cdots\!87\)\( T^{16} - \)\(59\!\cdots\!16\)\( T^{17} + \)\(66\!\cdots\!80\)\( T^{18} + \)\(53\!\cdots\!88\)\( T^{19} + \)\(18\!\cdots\!48\)\( T^{20} + \)\(53\!\cdots\!88\)\( p^{2} T^{21} + \)\(66\!\cdots\!80\)\( p^{4} T^{22} - \)\(59\!\cdots\!16\)\( p^{6} T^{23} - \)\(10\!\cdots\!87\)\( p^{8} T^{24} - 54845518191378878016 p^{10} T^{25} - 1627411205996070488 p^{12} T^{26} - 27614010295636836 p^{14} T^{27} + 5576738993915162 p^{16} T^{28} + 362640766156564 p^{18} T^{29} + 15309312551256 p^{20} T^{30} + 701177925136 p^{22} T^{31} + 7115318566 p^{24} T^{32} - 1384501004 p^{26} T^{33} - 77852048 p^{28} T^{34} - 5227628 p^{30} T^{35} - 133036 p^{32} T^{36} + 432 p^{34} T^{37} + 200 p^{36} T^{38} + 20 p^{38} T^{39} + p^{40} T^{40} \) | |
| 23 | \( 1 - 5622 T^{2} + 14940775 T^{4} - 25181062394 T^{6} + 30673037463157 T^{8} - 29432877530856940 T^{10} + 23842133477870840296 T^{12} - \)\(17\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(69\!\cdots\!04\)\( T^{18} + \)\(38\!\cdots\!50\)\( T^{20} - \)\(69\!\cdots\!04\)\( p^{4} T^{22} + \)\(11\!\cdots\!86\)\( p^{8} T^{24} - \)\(17\!\cdots\!40\)\( p^{12} T^{26} + 23842133477870840296 p^{16} T^{28} - 29432877530856940 p^{20} T^{30} + 30673037463157 p^{24} T^{32} - 25181062394 p^{28} T^{34} + 14940775 p^{32} T^{36} - 5622 p^{36} T^{38} + p^{40} T^{40} \) | |
| 29 | \( ( 1 + 20 T + 3731 T^{2} + 47596 T^{3} + 6992200 T^{4} + 78678768 T^{5} + 9713449057 T^{6} + 110509597876 T^{7} + 10578633434939 T^{8} + 114043513173340 T^{9} + 9520278543622184 T^{10} + 114043513173340 p^{2} T^{11} + 10578633434939 p^{4} T^{12} + 110509597876 p^{6} T^{13} + 9713449057 p^{8} T^{14} + 78678768 p^{10} T^{15} + 6992200 p^{12} T^{16} + 47596 p^{14} T^{17} + 3731 p^{16} T^{18} + 20 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 31 | \( 1 + 32 T + 512 T^{2} - 11746 T^{3} - 750128 T^{4} + 593952 T^{5} + 472056258 T^{6} + 12317539340 T^{7} + 1102187942422 T^{8} + 19318797226350 T^{9} + 263478473825672 T^{10} - 16591321846094440 T^{11} - 932072021751554234 T^{12} - 16537566162583476980 T^{13} - 92416384813786441172 T^{14} + \)\(95\!\cdots\!42\)\( T^{15} + \)\(39\!\cdots\!93\)\( T^{16} - \)\(33\!\cdots\!76\)\( p T^{17} - \)\(48\!\cdots\!78\)\( T^{18} - \)\(21\!\cdots\!78\)\( T^{19} - \)\(65\!\cdots\!84\)\( T^{20} - \)\(21\!\cdots\!78\)\( p^{2} T^{21} - \)\(48\!\cdots\!78\)\( p^{4} T^{22} - \)\(33\!\cdots\!76\)\( p^{7} T^{23} + \)\(39\!\cdots\!93\)\( p^{8} T^{24} + \)\(95\!\cdots\!42\)\( p^{10} T^{25} - 92416384813786441172 p^{12} T^{26} - 16537566162583476980 p^{14} T^{27} - 932072021751554234 p^{16} T^{28} - 16591321846094440 p^{18} T^{29} + 263478473825672 p^{20} T^{30} + 19318797226350 p^{22} T^{31} + 1102187942422 p^{24} T^{32} + 12317539340 p^{26} T^{33} + 472056258 p^{28} T^{34} + 593952 p^{30} T^{35} - 750128 p^{32} T^{36} - 11746 p^{34} T^{37} + 512 p^{36} T^{38} + 32 p^{38} T^{39} + p^{40} T^{40} \) | |
| 37 | \( 1 + 158 T + 12482 T^{2} + 708934 T^{3} + 34644430 T^{4} + 1457039542 T^{5} + 49074180554 T^{6} + 1244442483966 T^{7} + 15255172051393 T^{8} - 980751388777024 T^{9} - 101811088578934312 T^{10} - 5881654060441746384 T^{11} - 7369002183978099584 p T^{12} - \)\(10\!\cdots\!28\)\( T^{13} - \)\(33\!\cdots\!16\)\( T^{14} - \)\(70\!\cdots\!64\)\( T^{15} - \)\(12\!\cdots\!94\)\( T^{16} + \)\(95\!\cdots\!08\)\( T^{17} + \)\(69\!\cdots\!92\)\( T^{18} + \)\(33\!\cdots\!84\)\( T^{19} + \)\(13\!\cdots\!76\)\( T^{20} + \)\(33\!\cdots\!84\)\( p^{2} T^{21} + \)\(69\!\cdots\!92\)\( p^{4} T^{22} + \)\(95\!\cdots\!08\)\( p^{6} T^{23} - \)\(12\!\cdots\!94\)\( p^{8} T^{24} - \)\(70\!\cdots\!64\)\( p^{10} T^{25} - \)\(33\!\cdots\!16\)\( p^{12} T^{26} - \)\(10\!\cdots\!28\)\( p^{14} T^{27} - 7369002183978099584 p^{17} T^{28} - 5881654060441746384 p^{18} T^{29} - 101811088578934312 p^{20} T^{30} - 980751388777024 p^{22} T^{31} + 15255172051393 p^{24} T^{32} + 1244442483966 p^{26} T^{33} + 49074180554 p^{28} T^{34} + 1457039542 p^{30} T^{35} + 34644430 p^{32} T^{36} + 708934 p^{34} T^{37} + 12482 p^{36} T^{38} + 158 p^{38} T^{39} + p^{40} T^{40} \) | |
| 41 | \( 1 - 70 T + 2450 T^{2} - 212230 T^{3} + 6409708 T^{4} + 250658318 T^{5} - 10729080410 T^{6} + 1078443157190 T^{7} - 101780165969874 T^{8} + 2535903094613558 T^{9} - 56844732519276618 T^{10} + 3005493092498107478 T^{11} + \)\(22\!\cdots\!30\)\( T^{12} - \)\(16\!\cdots\!66\)\( T^{13} + \)\(45\!\cdots\!78\)\( T^{14} - \)\(32\!\cdots\!86\)\( T^{15} + \)\(23\!\cdots\!13\)\( p T^{16} + \)\(18\!\cdots\!72\)\( T^{17} - \)\(75\!\cdots\!52\)\( T^{18} + \)\(68\!\cdots\!60\)\( T^{19} - \)\(55\!\cdots\!40\)\( T^{20} + \)\(68\!\cdots\!60\)\( p^{2} T^{21} - \)\(75\!\cdots\!52\)\( p^{4} T^{22} + \)\(18\!\cdots\!72\)\( p^{6} T^{23} + \)\(23\!\cdots\!13\)\( p^{9} T^{24} - \)\(32\!\cdots\!86\)\( p^{10} T^{25} + \)\(45\!\cdots\!78\)\( p^{12} T^{26} - \)\(16\!\cdots\!66\)\( p^{14} T^{27} + \)\(22\!\cdots\!30\)\( p^{16} T^{28} + 3005493092498107478 p^{18} T^{29} - 56844732519276618 p^{20} T^{30} + 2535903094613558 p^{22} T^{31} - 101780165969874 p^{24} T^{32} + 1078443157190 p^{26} T^{33} - 10729080410 p^{28} T^{34} + 250658318 p^{30} T^{35} + 6409708 p^{32} T^{36} - 212230 p^{34} T^{37} + 2450 p^{36} T^{38} - 70 p^{38} T^{39} + p^{40} T^{40} \) | |
| 43 | \( 1 - 16998 T^{2} + 3222785 p T^{4} - 733193021986 T^{6} + 2894960588932789 T^{8} - 9340352587407572204 T^{10} + \)\(26\!\cdots\!20\)\( T^{12} - \)\(66\!\cdots\!76\)\( T^{14} + \)\(15\!\cdots\!82\)\( T^{16} - \)\(32\!\cdots\!36\)\( T^{18} + \)\(61\!\cdots\!06\)\( T^{20} - \)\(32\!\cdots\!36\)\( p^{4} T^{22} + \)\(15\!\cdots\!82\)\( p^{8} T^{24} - \)\(66\!\cdots\!76\)\( p^{12} T^{26} + \)\(26\!\cdots\!20\)\( p^{16} T^{28} - 9340352587407572204 p^{20} T^{30} + 2894960588932789 p^{24} T^{32} - 733193021986 p^{28} T^{34} + 3222785 p^{33} T^{36} - 16998 p^{36} T^{38} + p^{40} T^{40} \) | |
| 47 | \( 1 - 130 T + 8450 T^{2} - 9948 p T^{3} + 34139352 T^{4} - 2564715986 T^{5} + 154239860348 T^{6} - 8879701176294 T^{7} + 563577481495334 T^{8} - 33755927052785648 T^{9} + 1801033520980641006 T^{10} - 97878759299150014390 T^{11} + \)\(56\!\cdots\!14\)\( T^{12} - \)\(29\!\cdots\!14\)\( T^{13} + \)\(14\!\cdots\!62\)\( T^{14} - \)\(77\!\cdots\!88\)\( T^{15} + \)\(40\!\cdots\!25\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{17} + \)\(94\!\cdots\!94\)\( T^{18} - \)\(46\!\cdots\!70\)\( T^{19} + \)\(22\!\cdots\!68\)\( T^{20} - \)\(46\!\cdots\!70\)\( p^{2} T^{21} + \)\(94\!\cdots\!94\)\( p^{4} T^{22} - \)\(20\!\cdots\!80\)\( p^{6} T^{23} + \)\(40\!\cdots\!25\)\( p^{8} T^{24} - \)\(77\!\cdots\!88\)\( p^{10} T^{25} + \)\(14\!\cdots\!62\)\( p^{12} T^{26} - \)\(29\!\cdots\!14\)\( p^{14} T^{27} + \)\(56\!\cdots\!14\)\( p^{16} T^{28} - 97878759299150014390 p^{18} T^{29} + 1801033520980641006 p^{20} T^{30} - 33755927052785648 p^{22} T^{31} + 563577481495334 p^{24} T^{32} - 8879701176294 p^{26} T^{33} + 154239860348 p^{28} T^{34} - 2564715986 p^{30} T^{35} + 34139352 p^{32} T^{36} - 9948 p^{35} T^{37} + 8450 p^{36} T^{38} - 130 p^{38} T^{39} + p^{40} T^{40} \) | |
| 53 | \( ( 1 + 108 T + 21781 T^{2} + 2008404 T^{3} + 225550464 T^{4} + 17889558956 T^{5} + 1462054279311 T^{6} + 100585094443444 T^{7} + 6587954701722699 T^{8} + 392885142279367328 T^{9} + 21574561265842441728 T^{10} + 392885142279367328 p^{2} T^{11} + 6587954701722699 p^{4} T^{12} + 100585094443444 p^{6} T^{13} + 1462054279311 p^{8} T^{14} + 17889558956 p^{10} T^{15} + 225550464 p^{12} T^{16} + 2008404 p^{14} T^{17} + 21781 p^{16} T^{18} + 108 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 59 | \( 1 - 132 T + 8712 T^{2} - 730294 T^{3} + 44263056 T^{4} - 1150446108 T^{5} + 32903805602 T^{6} - 443885568080 T^{7} - 141126832581554 T^{8} + 4213388910281290 T^{9} + 202836730080517808 T^{10} - 32590959527044871724 T^{11} + \)\(27\!\cdots\!70\)\( T^{12} - \)\(11\!\cdots\!44\)\( T^{13} + \)\(58\!\cdots\!64\)\( T^{14} - \)\(29\!\cdots\!26\)\( T^{15} - \)\(93\!\cdots\!75\)\( T^{16} + \)\(17\!\cdots\!00\)\( T^{17} - \)\(94\!\cdots\!26\)\( T^{18} + \)\(96\!\cdots\!98\)\( T^{19} - \)\(85\!\cdots\!36\)\( T^{20} + \)\(96\!\cdots\!98\)\( p^{2} T^{21} - \)\(94\!\cdots\!26\)\( p^{4} T^{22} + \)\(17\!\cdots\!00\)\( p^{6} T^{23} - \)\(93\!\cdots\!75\)\( p^{8} T^{24} - \)\(29\!\cdots\!26\)\( p^{10} T^{25} + \)\(58\!\cdots\!64\)\( p^{12} T^{26} - \)\(11\!\cdots\!44\)\( p^{14} T^{27} + \)\(27\!\cdots\!70\)\( p^{16} T^{28} - 32590959527044871724 p^{18} T^{29} + 202836730080517808 p^{20} T^{30} + 4213388910281290 p^{22} T^{31} - 141126832581554 p^{24} T^{32} - 443885568080 p^{26} T^{33} + 32903805602 p^{28} T^{34} - 1150446108 p^{30} T^{35} + 44263056 p^{32} T^{36} - 730294 p^{34} T^{37} + 8712 p^{36} T^{38} - 132 p^{38} T^{39} + p^{40} T^{40} \) | |
| 61 | \( ( 1 + 76 T + 15413 T^{2} + 953248 T^{3} + 98954994 T^{4} + 5545624036 T^{5} + 405451996297 T^{6} + 24054864783264 T^{7} + 1524760285213295 T^{8} + 97070998685324948 T^{9} + 5833117829685972186 T^{10} + 97070998685324948 p^{2} T^{11} + 1524760285213295 p^{4} T^{12} + 24054864783264 p^{6} T^{13} + 405451996297 p^{8} T^{14} + 5545624036 p^{10} T^{15} + 98954994 p^{12} T^{16} + 953248 p^{14} T^{17} + 15413 p^{16} T^{18} + 76 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 67 | \( 1 + 178 T + 15842 T^{2} + 1396484 T^{3} + 171971114 T^{4} + 16858719566 T^{5} + 1251569475888 T^{6} + 97554957719838 T^{7} + 8671063019977667 T^{8} + 667167559285498278 T^{9} + 44497198452422857408 T^{10} + \)\(31\!\cdots\!98\)\( T^{11} + \)\(23\!\cdots\!88\)\( T^{12} + \)\(16\!\cdots\!96\)\( T^{13} + \)\(10\!\cdots\!86\)\( T^{14} + \)\(71\!\cdots\!86\)\( T^{15} + \)\(49\!\cdots\!05\)\( T^{16} + \)\(34\!\cdots\!88\)\( T^{17} + \)\(22\!\cdots\!96\)\( T^{18} + \)\(15\!\cdots\!60\)\( T^{19} + \)\(10\!\cdots\!30\)\( T^{20} + \)\(15\!\cdots\!60\)\( p^{2} T^{21} + \)\(22\!\cdots\!96\)\( p^{4} T^{22} + \)\(34\!\cdots\!88\)\( p^{6} T^{23} + \)\(49\!\cdots\!05\)\( p^{8} T^{24} + \)\(71\!\cdots\!86\)\( p^{10} T^{25} + \)\(10\!\cdots\!86\)\( p^{12} T^{26} + \)\(16\!\cdots\!96\)\( p^{14} T^{27} + \)\(23\!\cdots\!88\)\( p^{16} T^{28} + \)\(31\!\cdots\!98\)\( p^{18} T^{29} + 44497198452422857408 p^{20} T^{30} + 667167559285498278 p^{22} T^{31} + 8671063019977667 p^{24} T^{32} + 97554957719838 p^{26} T^{33} + 1251569475888 p^{28} T^{34} + 16858719566 p^{30} T^{35} + 171971114 p^{32} T^{36} + 1396484 p^{34} T^{37} + 15842 p^{36} T^{38} + 178 p^{38} T^{39} + p^{40} T^{40} \) | |
| 71 | \( 1 + 258 T + 33282 T^{2} + 2928306 T^{3} + 262363086 T^{4} + 26652154538 T^{5} + 2431775657370 T^{6} + 162846682498682 T^{7} + 7828487188433597 T^{8} + 317131224068624416 T^{9} + 15297567476688769568 T^{10} - \)\(13\!\cdots\!48\)\( T^{11} - \)\(18\!\cdots\!84\)\( T^{12} - \)\(23\!\cdots\!40\)\( T^{13} - \)\(16\!\cdots\!88\)\( T^{14} - \)\(89\!\cdots\!24\)\( T^{15} - \)\(54\!\cdots\!14\)\( T^{16} - \)\(39\!\cdots\!24\)\( T^{17} - \)\(21\!\cdots\!64\)\( T^{18} - \)\(39\!\cdots\!40\)\( T^{19} + \)\(21\!\cdots\!84\)\( T^{20} - \)\(39\!\cdots\!40\)\( p^{2} T^{21} - \)\(21\!\cdots\!64\)\( p^{4} T^{22} - \)\(39\!\cdots\!24\)\( p^{6} T^{23} - \)\(54\!\cdots\!14\)\( p^{8} T^{24} - \)\(89\!\cdots\!24\)\( p^{10} T^{25} - \)\(16\!\cdots\!88\)\( p^{12} T^{26} - \)\(23\!\cdots\!40\)\( p^{14} T^{27} - \)\(18\!\cdots\!84\)\( p^{16} T^{28} - \)\(13\!\cdots\!48\)\( p^{18} T^{29} + 15297567476688769568 p^{20} T^{30} + 317131224068624416 p^{22} T^{31} + 7828487188433597 p^{24} T^{32} + 162846682498682 p^{26} T^{33} + 2431775657370 p^{28} T^{34} + 26652154538 p^{30} T^{35} + 262363086 p^{32} T^{36} + 2928306 p^{34} T^{37} + 33282 p^{36} T^{38} + 258 p^{38} T^{39} + p^{40} T^{40} \) | |
| 73 | \( 1 - 262 T + 34322 T^{2} - 3336918 T^{3} + 265025636 T^{4} - 19303994522 T^{5} + 1528947555334 T^{6} - 136581705530426 T^{7} + 13017030881246878 T^{8} - 1103267624796787394 T^{9} + 79159339435445479398 T^{10} - \)\(48\!\cdots\!74\)\( T^{11} + \)\(22\!\cdots\!58\)\( T^{12} - \)\(75\!\cdots\!46\)\( T^{13} + \)\(32\!\cdots\!34\)\( T^{14} - \)\(25\!\cdots\!70\)\( T^{15} + \)\(19\!\cdots\!21\)\( T^{16} + \)\(13\!\cdots\!00\)\( T^{17} - \)\(19\!\cdots\!88\)\( T^{18} + \)\(28\!\cdots\!84\)\( T^{19} - \)\(25\!\cdots\!88\)\( T^{20} + \)\(28\!\cdots\!84\)\( p^{2} T^{21} - \)\(19\!\cdots\!88\)\( p^{4} T^{22} + \)\(13\!\cdots\!00\)\( p^{6} T^{23} + \)\(19\!\cdots\!21\)\( p^{8} T^{24} - \)\(25\!\cdots\!70\)\( p^{10} T^{25} + \)\(32\!\cdots\!34\)\( p^{12} T^{26} - \)\(75\!\cdots\!46\)\( p^{14} T^{27} + \)\(22\!\cdots\!58\)\( p^{16} T^{28} - \)\(48\!\cdots\!74\)\( p^{18} T^{29} + 79159339435445479398 p^{20} T^{30} - 1103267624796787394 p^{22} T^{31} + 13017030881246878 p^{24} T^{32} - 136581705530426 p^{26} T^{33} + 1528947555334 p^{28} T^{34} - 19303994522 p^{30} T^{35} + 265025636 p^{32} T^{36} - 3336918 p^{34} T^{37} + 34322 p^{36} T^{38} - 262 p^{38} T^{39} + p^{40} T^{40} \) | |
| 79 | \( ( 1 + 272 T + 64803 T^{2} + 10901116 T^{3} + 1674804545 T^{4} + 213973062332 T^{5} + 25454078712446 T^{6} + 2660636500626116 T^{7} + 261238159878739402 T^{8} + 23023732241540716524 T^{9} + \)\(19\!\cdots\!26\)\( T^{10} + 23023732241540716524 p^{2} T^{11} + 261238159878739402 p^{4} T^{12} + 2660636500626116 p^{6} T^{13} + 25454078712446 p^{8} T^{14} + 213973062332 p^{10} T^{15} + 1674804545 p^{12} T^{16} + 10901116 p^{14} T^{17} + 64803 p^{16} T^{18} + 272 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 83 | \( 1 + 88 T + 3872 T^{2} + 1964606 T^{3} + 92179758 T^{4} - 3092030696 T^{5} + 1300819643394 T^{6} + 17126358983376 T^{7} - 3760632980538325 T^{8} + 1150485821959793726 T^{9} + 34321557832798535700 T^{10} + \)\(36\!\cdots\!06\)\( T^{11} + \)\(91\!\cdots\!40\)\( T^{12} + \)\(96\!\cdots\!16\)\( T^{13} + \)\(28\!\cdots\!06\)\( T^{14} + \)\(24\!\cdots\!14\)\( T^{15} - \)\(11\!\cdots\!43\)\( T^{16} + \)\(25\!\cdots\!06\)\( T^{17} + \)\(12\!\cdots\!58\)\( T^{18} + \)\(70\!\cdots\!18\)\( T^{19} + \)\(25\!\cdots\!78\)\( T^{20} + \)\(70\!\cdots\!18\)\( p^{2} T^{21} + \)\(12\!\cdots\!58\)\( p^{4} T^{22} + \)\(25\!\cdots\!06\)\( p^{6} T^{23} - \)\(11\!\cdots\!43\)\( p^{8} T^{24} + \)\(24\!\cdots\!14\)\( p^{10} T^{25} + \)\(28\!\cdots\!06\)\( p^{12} T^{26} + \)\(96\!\cdots\!16\)\( p^{14} T^{27} + \)\(91\!\cdots\!40\)\( p^{16} T^{28} + \)\(36\!\cdots\!06\)\( p^{18} T^{29} + 34321557832798535700 p^{20} T^{30} + 1150485821959793726 p^{22} T^{31} - 3760632980538325 p^{24} T^{32} + 17126358983376 p^{26} T^{33} + 1300819643394 p^{28} T^{34} - 3092030696 p^{30} T^{35} + 92179758 p^{32} T^{36} + 1964606 p^{34} T^{37} + 3872 p^{36} T^{38} + 88 p^{38} T^{39} + p^{40} T^{40} \) | |
| 89 | \( 1 + 282 T + 39762 T^{2} + 5844690 T^{3} + 789975536 T^{4} + 66179353318 T^{5} + 4331770971294 T^{6} + 223108493185230 T^{7} - 22043015076602474 T^{8} - 5272983131403765994 T^{9} - \)\(53\!\cdots\!42\)\( T^{10} - \)\(53\!\cdots\!26\)\( T^{11} - \)\(30\!\cdots\!66\)\( T^{12} + \)\(50\!\cdots\!18\)\( T^{13} + \)\(20\!\cdots\!54\)\( T^{14} + \)\(32\!\cdots\!30\)\( T^{15} + \)\(32\!\cdots\!33\)\( T^{16} + \)\(15\!\cdots\!52\)\( T^{17} + \)\(20\!\cdots\!32\)\( T^{18} - \)\(72\!\cdots\!20\)\( T^{19} - \)\(12\!\cdots\!00\)\( T^{20} - \)\(72\!\cdots\!20\)\( p^{2} T^{21} + \)\(20\!\cdots\!32\)\( p^{4} T^{22} + \)\(15\!\cdots\!52\)\( p^{6} T^{23} + \)\(32\!\cdots\!33\)\( p^{8} T^{24} + \)\(32\!\cdots\!30\)\( p^{10} T^{25} + \)\(20\!\cdots\!54\)\( p^{12} T^{26} + \)\(50\!\cdots\!18\)\( p^{14} T^{27} - \)\(30\!\cdots\!66\)\( p^{16} T^{28} - \)\(53\!\cdots\!26\)\( p^{18} T^{29} - \)\(53\!\cdots\!42\)\( p^{20} T^{30} - 5272983131403765994 p^{22} T^{31} - 22043015076602474 p^{24} T^{32} + 223108493185230 p^{26} T^{33} + 4331770971294 p^{28} T^{34} + 66179353318 p^{30} T^{35} + 789975536 p^{32} T^{36} + 5844690 p^{34} T^{37} + 39762 p^{36} T^{38} + 282 p^{38} T^{39} + p^{40} T^{40} \) | |
| 97 | \( 1 + 6 T + 18 T^{2} + 1140102 T^{3} - 382689754 T^{4} - 15722217538 T^{5} + 562471395546 T^{6} - 509776239928322 T^{7} + 48963707885608813 T^{8} + 6121063923434238992 T^{9} - \)\(20\!\cdots\!20\)\( T^{10} + \)\(95\!\cdots\!72\)\( T^{11} + \)\(13\!\cdots\!52\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(29\!\cdots\!44\)\( T^{14} - \)\(80\!\cdots\!88\)\( T^{15} - \)\(12\!\cdots\!38\)\( p T^{16} + \)\(82\!\cdots\!32\)\( T^{17} - \)\(13\!\cdots\!08\)\( T^{18} - \)\(11\!\cdots\!24\)\( T^{19} + \)\(15\!\cdots\!88\)\( T^{20} - \)\(11\!\cdots\!24\)\( p^{2} T^{21} - \)\(13\!\cdots\!08\)\( p^{4} T^{22} + \)\(82\!\cdots\!32\)\( p^{6} T^{23} - \)\(12\!\cdots\!38\)\( p^{9} T^{24} - \)\(80\!\cdots\!88\)\( p^{10} T^{25} + \)\(29\!\cdots\!44\)\( p^{12} T^{26} - \)\(10\!\cdots\!72\)\( p^{14} T^{27} + \)\(13\!\cdots\!52\)\( p^{16} T^{28} + \)\(95\!\cdots\!72\)\( p^{18} T^{29} - \)\(20\!\cdots\!20\)\( p^{20} T^{30} + 6121063923434238992 p^{22} T^{31} + 48963707885608813 p^{24} T^{32} - 509776239928322 p^{26} T^{33} + 562471395546 p^{28} T^{34} - 15722217538 p^{30} T^{35} - 382689754 p^{32} T^{36} + 1140102 p^{34} T^{37} + 18 p^{36} T^{38} + 6 p^{38} T^{39} + p^{40} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.60653017024747753754564285223, −2.49949935202486345987750397237, −2.46583097693498556916058786010, −2.38366366872194918140332719496, −2.37152608642371083461219613815, −2.11371022255256149409156643549, −1.89710867107929285196925380974, −1.87916488125645270756987883439, −1.84797507047409435703538819857, −1.79944552180775155468987824455, −1.77932886806918941174820196483, −1.63270004100700456447245651250, −1.49141468872933732011125054700, −1.48972972113455250889065362048, −1.38025982275306081052072914034, −1.27748733845391469524254537310, −1.02216605012289798426793329287, −0.867780779102115481343597379699, −0.55371832181074293703098414619, −0.53301678835876286809451361760, −0.49519184746116075000803980663, −0.46528103066676006982649364183, −0.38239122717864297578385889159, −0.13966724667598895075170756568, −0.00697762193101680358893887494, 0.00697762193101680358893887494, 0.13966724667598895075170756568, 0.38239122717864297578385889159, 0.46528103066676006982649364183, 0.49519184746116075000803980663, 0.53301678835876286809451361760, 0.55371832181074293703098414619, 0.867780779102115481343597379699, 1.02216605012289798426793329287, 1.27748733845391469524254537310, 1.38025982275306081052072914034, 1.48972972113455250889065362048, 1.49141468872933732011125054700, 1.63270004100700456447245651250, 1.77932886806918941174820196483, 1.79944552180775155468987824455, 1.84797507047409435703538819857, 1.87916488125645270756987883439, 1.89710867107929285196925380974, 2.11371022255256149409156643549, 2.37152608642371083461219613815, 2.38366366872194918140332719496, 2.46583097693498556916058786010, 2.49949935202486345987750397237, 2.60653017024747753754564285223
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.