Dirichlet series
| L(s) = 1 | − 2·2-s + 3·3-s + 2·4-s − 7·5-s − 6·6-s − 4·7-s − 8-s + 3·9-s + 14·10-s − 6·11-s + 6·12-s − 11·13-s + 8·14-s − 21·15-s + 4·16-s − 32·17-s − 6·18-s + 2·19-s − 14·20-s − 12·21-s + 12·22-s − 6·23-s − 3·24-s + 34·25-s + 22·26-s + 27-s − 8·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 4-s − 3.13·5-s − 2.44·6-s − 1.51·7-s − 0.353·8-s + 9-s + 4.42·10-s − 1.80·11-s + 1.73·12-s − 3.05·13-s + 2.13·14-s − 5.42·15-s + 16-s − 7.76·17-s − 1.41·18-s + 0.458·19-s − 3.13·20-s − 2.61·21-s + 2.55·22-s − 1.25·23-s − 0.612·24-s + 34/5·25-s + 4.31·26-s + 0.192·27-s − 1.51·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{18} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00142015\) |
| Root analytic conductor: | \(0.833485\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 3^{18} \cdot 29^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03191987091\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03191987091\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{3} \) |
| 29 | \( 1 + 10 T - 36 T^{2} - 578 T^{3} + 1016 T^{4} + 21063 T^{5} - 22778 T^{6} - 592636 T^{7} + 552441 T^{8} + 19185875 T^{9} + 552441 p T^{10} - 592636 p^{2} T^{11} - 22778 p^{3} T^{12} + 21063 p^{4} T^{13} + 1016 p^{5} T^{14} - 578 p^{6} T^{15} - 36 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 2 | \( 1 + p T + p T^{2} + T^{3} - p^{2} T^{4} - 15 T^{5} - 7 p T^{6} - 11 p T^{7} - 15 p T^{8} + p^{2} T^{9} + 21 p T^{10} + 9 p^{3} T^{11} + 29 p^{3} T^{12} + 163 T^{13} + 27 p T^{14} + 107 T^{15} - 307 T^{16} - 215 p^{2} T^{17} - 451 T^{18} - 215 p^{3} T^{19} - 307 p^{2} T^{20} + 107 p^{3} T^{21} + 27 p^{5} T^{22} + 163 p^{5} T^{23} + 29 p^{9} T^{24} + 9 p^{10} T^{25} + 21 p^{9} T^{26} + p^{11} T^{27} - 15 p^{11} T^{28} - 11 p^{12} T^{29} - 7 p^{13} T^{30} - 15 p^{13} T^{31} - p^{16} T^{32} + p^{15} T^{33} + p^{17} T^{34} + p^{18} T^{35} + p^{18} T^{36} \) |
| 5 | \( 1 + 7 T + 3 p T^{2} - T^{3} - 57 T^{4} - 198 T^{5} - 122 p T^{6} - 727 T^{7} + 2083 T^{8} + 10771 T^{9} + 26002 T^{10} + 7474 p T^{11} - 1549 p T^{12} - 222133 T^{13} - 680461 T^{14} - 964364 T^{15} + 607712 T^{16} + 6539033 T^{17} + 18434481 T^{18} + 6539033 p T^{19} + 607712 p^{2} T^{20} - 964364 p^{3} T^{21} - 680461 p^{4} T^{22} - 222133 p^{5} T^{23} - 1549 p^{7} T^{24} + 7474 p^{8} T^{25} + 26002 p^{8} T^{26} + 10771 p^{9} T^{27} + 2083 p^{10} T^{28} - 727 p^{11} T^{29} - 122 p^{13} T^{30} - 198 p^{13} T^{31} - 57 p^{14} T^{32} - p^{15} T^{33} + 3 p^{17} T^{34} + 7 p^{17} T^{35} + p^{18} T^{36} \) | |
| 7 | \( 1 + 4 T - 11 T^{2} - 55 T^{3} + 66 T^{4} + 374 T^{5} + 17 p T^{6} - 613 T^{7} - 7208 T^{8} - 12134 T^{9} + 88572 T^{10} + 142848 T^{11} - 10939 p^{2} T^{12} - 791002 T^{13} + 184498 T^{14} + 3122533 T^{15} + 33800345 T^{16} - 9073301 T^{17} - 365501233 T^{18} - 9073301 p T^{19} + 33800345 p^{2} T^{20} + 3122533 p^{3} T^{21} + 184498 p^{4} T^{22} - 791002 p^{5} T^{23} - 10939 p^{8} T^{24} + 142848 p^{7} T^{25} + 88572 p^{8} T^{26} - 12134 p^{9} T^{27} - 7208 p^{10} T^{28} - 613 p^{11} T^{29} + 17 p^{13} T^{30} + 374 p^{13} T^{31} + 66 p^{14} T^{32} - 55 p^{15} T^{33} - 11 p^{16} T^{34} + 4 p^{17} T^{35} + p^{18} T^{36} \) | |
| 11 | \( 1 + 6 T - 26 T^{2} - 169 T^{3} + 460 T^{4} + 1938 T^{5} - 7939 T^{6} - 6352 T^{7} + 93064 T^{8} - 220713 T^{9} - 588046 T^{10} + 5323182 T^{11} + 330459 T^{12} - 60022879 T^{13} + 69159966 T^{14} + 445746144 T^{15} - 1248237636 T^{16} - 1751031277 T^{17} + 14689255023 T^{18} - 1751031277 p T^{19} - 1248237636 p^{2} T^{20} + 445746144 p^{3} T^{21} + 69159966 p^{4} T^{22} - 60022879 p^{5} T^{23} + 330459 p^{6} T^{24} + 5323182 p^{7} T^{25} - 588046 p^{8} T^{26} - 220713 p^{9} T^{27} + 93064 p^{10} T^{28} - 6352 p^{11} T^{29} - 7939 p^{12} T^{30} + 1938 p^{13} T^{31} + 460 p^{14} T^{32} - 169 p^{15} T^{33} - 26 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 + 11 T + 33 T^{2} + 19 T^{3} + 484 T^{4} + 3635 T^{5} + 5989 T^{6} - 7910 T^{7} + 21400 T^{8} + 328794 T^{9} + 368440 T^{10} - 4176168 T^{11} - 17362703 T^{12} - 14359812 T^{13} + 51809026 T^{14} - 472892417 T^{15} - 4553078857 T^{16} - 4550813008 T^{17} + 32473776559 T^{18} - 4550813008 p T^{19} - 4553078857 p^{2} T^{20} - 472892417 p^{3} T^{21} + 51809026 p^{4} T^{22} - 14359812 p^{5} T^{23} - 17362703 p^{6} T^{24} - 4176168 p^{7} T^{25} + 368440 p^{8} T^{26} + 328794 p^{9} T^{27} + 21400 p^{10} T^{28} - 7910 p^{11} T^{29} + 5989 p^{12} T^{30} + 3635 p^{13} T^{31} + 484 p^{14} T^{32} + 19 p^{15} T^{33} + 33 p^{16} T^{34} + 11 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( ( 1 + 16 T + 190 T^{2} + 1544 T^{3} + 11104 T^{4} + 66663 T^{5} + 376150 T^{6} + 1868520 T^{7} + 8820919 T^{8} + 37225411 T^{9} + 8820919 p T^{10} + 1868520 p^{2} T^{11} + 376150 p^{3} T^{12} + 66663 p^{4} T^{13} + 11104 p^{5} T^{14} + 1544 p^{6} T^{15} + 190 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 19 | \( 1 - 2 T - 33 T^{2} - 43 T^{3} + 907 T^{4} + 88 T^{5} - 15761 T^{6} + 6234 T^{7} + 526856 T^{8} - 253857 T^{9} - 10948627 T^{10} - 451388 T^{11} + 217872346 T^{12} - 111041419 T^{13} - 4764832328 T^{14} + 840233774 T^{15} + 92916967495 T^{16} + 27457863777 T^{17} - 1750903420621 T^{18} + 27457863777 p T^{19} + 92916967495 p^{2} T^{20} + 840233774 p^{3} T^{21} - 4764832328 p^{4} T^{22} - 111041419 p^{5} T^{23} + 217872346 p^{6} T^{24} - 451388 p^{7} T^{25} - 10948627 p^{8} T^{26} - 253857 p^{9} T^{27} + 526856 p^{10} T^{28} + 6234 p^{11} T^{29} - 15761 p^{12} T^{30} + 88 p^{13} T^{31} + 907 p^{14} T^{32} - 43 p^{15} T^{33} - 33 p^{16} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 + 6 T + 13 T^{2} + 198 T^{3} + 1726 T^{4} + 5684 T^{5} + 22852 T^{6} + 237892 T^{7} + 53137 p T^{8} + 1544926 T^{9} + 12854897 T^{10} + 112304298 T^{11} + 15330732 T^{12} - 1458953602 T^{13} + 1302687861 T^{14} - 5767597954 T^{15} - 381323450868 T^{16} - 1440526017186 T^{17} - 596591623839 T^{18} - 1440526017186 p T^{19} - 381323450868 p^{2} T^{20} - 5767597954 p^{3} T^{21} + 1302687861 p^{4} T^{22} - 1458953602 p^{5} T^{23} + 15330732 p^{6} T^{24} + 112304298 p^{7} T^{25} + 12854897 p^{8} T^{26} + 1544926 p^{9} T^{27} + 53137 p^{11} T^{28} + 237892 p^{11} T^{29} + 22852 p^{12} T^{30} + 5684 p^{13} T^{31} + 1726 p^{14} T^{32} + 198 p^{15} T^{33} + 13 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 - 8 T - 88 T^{2} + 1175 T^{3} + 2935 T^{4} - 95354 T^{5} + 83734 T^{6} + 5121343 T^{7} - 15576190 T^{8} - 214533311 T^{9} + 1072347695 T^{10} + 7530307288 T^{11} - 53974915502 T^{12} - 225699026103 T^{13} + 2357070708448 T^{14} + 5305263707573 T^{15} - 90443497568807 T^{16} - 63546541686843 T^{17} + 3016759237501459 T^{18} - 63546541686843 p T^{19} - 90443497568807 p^{2} T^{20} + 5305263707573 p^{3} T^{21} + 2357070708448 p^{4} T^{22} - 225699026103 p^{5} T^{23} - 53974915502 p^{6} T^{24} + 7530307288 p^{7} T^{25} + 1072347695 p^{8} T^{26} - 214533311 p^{9} T^{27} - 15576190 p^{10} T^{28} + 5121343 p^{11} T^{29} + 83734 p^{12} T^{30} - 95354 p^{13} T^{31} + 2935 p^{14} T^{32} + 1175 p^{15} T^{33} - 88 p^{16} T^{34} - 8 p^{17} T^{35} + p^{18} T^{36} \) | |
| 37 | \( 1 - 20 T + 240 T^{2} - 2269 T^{3} + 18773 T^{4} - 118409 T^{5} + 546048 T^{6} - 1505305 T^{7} - 5399881 T^{8} + 140220268 T^{9} - 1319930786 T^{10} + 9027617334 T^{11} - 54653437317 T^{12} + 266050494186 T^{13} - 662544800951 T^{14} - 2763918353713 T^{15} + 1404605773485 p T^{16} - 511474898941224 T^{17} + 3657434064135413 T^{18} - 511474898941224 p T^{19} + 1404605773485 p^{3} T^{20} - 2763918353713 p^{3} T^{21} - 662544800951 p^{4} T^{22} + 266050494186 p^{5} T^{23} - 54653437317 p^{6} T^{24} + 9027617334 p^{7} T^{25} - 1319930786 p^{8} T^{26} + 140220268 p^{9} T^{27} - 5399881 p^{10} T^{28} - 1505305 p^{11} T^{29} + 546048 p^{12} T^{30} - 118409 p^{13} T^{31} + 18773 p^{14} T^{32} - 2269 p^{15} T^{33} + 240 p^{16} T^{34} - 20 p^{17} T^{35} + p^{18} T^{36} \) | |
| 41 | \( ( 1 + 34 T + 788 T^{2} + 12970 T^{3} + 175640 T^{4} + 1971845 T^{5} + 19301782 T^{6} + 164588816 T^{7} + 1253221727 T^{8} + 8463182695 T^{9} + 1253221727 p T^{10} + 164588816 p^{2} T^{11} + 19301782 p^{3} T^{12} + 1971845 p^{4} T^{13} + 175640 p^{5} T^{14} + 12970 p^{6} T^{15} + 788 p^{7} T^{16} + 34 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 43 | \( 1 + 3 T - 45 T^{2} - 679 T^{3} + 3799 T^{4} + 35953 T^{5} - 41985 T^{6} - 2376026 T^{7} + 4654921 T^{8} + 80879753 T^{9} - 124963966 T^{10} - 2923869730 T^{11} + 12841489201 T^{12} + 78049687599 T^{13} - 1079976447261 T^{14} - 329598473085 T^{15} + 54748168547443 T^{16} + 44776368685846 T^{17} - 3261703262426839 T^{18} + 44776368685846 p T^{19} + 54748168547443 p^{2} T^{20} - 329598473085 p^{3} T^{21} - 1079976447261 p^{4} T^{22} + 78049687599 p^{5} T^{23} + 12841489201 p^{6} T^{24} - 2923869730 p^{7} T^{25} - 124963966 p^{8} T^{26} + 80879753 p^{9} T^{27} + 4654921 p^{10} T^{28} - 2376026 p^{11} T^{29} - 41985 p^{12} T^{30} + 35953 p^{13} T^{31} + 3799 p^{14} T^{32} - 679 p^{15} T^{33} - 45 p^{16} T^{34} + 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 47 | \( 1 - 19 T + 117 T^{2} + 59 T^{3} - 4860 T^{4} + 1239 p T^{5} - 513325 T^{6} + 3488486 T^{7} - 28131034 T^{8} + 128623658 T^{9} + 1140129796 T^{10} - 17807857568 T^{11} + 124898306503 T^{12} - 813503687318 T^{13} + 3483040291028 T^{14} - 99284260941 T^{15} - 110224753873477 T^{16} + 2037787357356288 T^{17} - 20438556318603675 T^{18} + 2037787357356288 p T^{19} - 110224753873477 p^{2} T^{20} - 99284260941 p^{3} T^{21} + 3483040291028 p^{4} T^{22} - 813503687318 p^{5} T^{23} + 124898306503 p^{6} T^{24} - 17807857568 p^{7} T^{25} + 1140129796 p^{8} T^{26} + 128623658 p^{9} T^{27} - 28131034 p^{10} T^{28} + 3488486 p^{11} T^{29} - 513325 p^{12} T^{30} + 1239 p^{14} T^{31} - 4860 p^{14} T^{32} + 59 p^{15} T^{33} + 117 p^{16} T^{34} - 19 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 + T - 138 T^{2} - 503 T^{3} + 10974 T^{4} + 37822 T^{5} - 525687 T^{6} - 1664660 T^{7} + 25250021 T^{8} + 95315685 T^{9} - 995167016 T^{10} - 10282486106 T^{11} + 12539948599 T^{12} + 691255727491 T^{13} + 356576543387 T^{14} - 20781271975832 T^{15} + 56568783466599 T^{16} + 358216320273226 T^{17} - 8202609795566856 T^{18} + 358216320273226 p T^{19} + 56568783466599 p^{2} T^{20} - 20781271975832 p^{3} T^{21} + 356576543387 p^{4} T^{22} + 691255727491 p^{5} T^{23} + 12539948599 p^{6} T^{24} - 10282486106 p^{7} T^{25} - 995167016 p^{8} T^{26} + 95315685 p^{9} T^{27} + 25250021 p^{10} T^{28} - 1664660 p^{11} T^{29} - 525687 p^{12} T^{30} + 37822 p^{13} T^{31} + 10974 p^{14} T^{32} - 503 p^{15} T^{33} - 138 p^{16} T^{34} + p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( ( 1 - 10 T + 263 T^{2} - 28 p T^{3} + 30369 T^{4} - 110304 T^{5} + 2000488 T^{6} - 1128580 T^{7} + 96017121 T^{8} + 142377228 T^{9} + 96017121 p T^{10} - 1128580 p^{2} T^{11} + 2000488 p^{3} T^{12} - 110304 p^{4} T^{13} + 30369 p^{5} T^{14} - 28 p^{7} T^{15} + 263 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 61 | \( 1 - 24 T + 136 T^{2} + 1099 T^{3} - 14763 T^{4} - 26612 T^{5} + 1514090 T^{6} - 5581559 T^{7} - 104473494 T^{8} + 980262005 T^{9} + 936281331 T^{10} - 49133146846 T^{11} + 54221391570 T^{12} + 3713982961879 T^{13} - 19803841436352 T^{14} - 185670567383233 T^{15} + 2005909869108029 T^{16} + 1472327064254111 T^{17} - 96825153288065495 T^{18} + 1472327064254111 p T^{19} + 2005909869108029 p^{2} T^{20} - 185670567383233 p^{3} T^{21} - 19803841436352 p^{4} T^{22} + 3713982961879 p^{5} T^{23} + 54221391570 p^{6} T^{24} - 49133146846 p^{7} T^{25} + 936281331 p^{8} T^{26} + 980262005 p^{9} T^{27} - 104473494 p^{10} T^{28} - 5581559 p^{11} T^{29} + 1514090 p^{12} T^{30} - 26612 p^{13} T^{31} - 14763 p^{14} T^{32} + 1099 p^{15} T^{33} + 136 p^{16} T^{34} - 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 + 14 T - 32 T^{2} - 682 T^{3} + 5796 T^{4} + 60182 T^{5} + 497107 T^{6} + 2601082 T^{7} - 46571706 T^{8} - 89197296 T^{9} + 2836640474 T^{10} - 30159412 p T^{11} + 17069214395 T^{12} - 670507002066 T^{13} - 16280748321974 T^{14} + 27121309354516 T^{15} - 177525510497970 T^{16} - 6590984722833208 T^{17} - 10231191562009335 T^{18} - 6590984722833208 p T^{19} - 177525510497970 p^{2} T^{20} + 27121309354516 p^{3} T^{21} - 16280748321974 p^{4} T^{22} - 670507002066 p^{5} T^{23} + 17069214395 p^{6} T^{24} - 30159412 p^{8} T^{25} + 2836640474 p^{8} T^{26} - 89197296 p^{9} T^{27} - 46571706 p^{10} T^{28} + 2601082 p^{11} T^{29} + 497107 p^{12} T^{30} + 60182 p^{13} T^{31} + 5796 p^{14} T^{32} - 682 p^{15} T^{33} - 32 p^{16} T^{34} + 14 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 + 28 T + 134 T^{2} - 5058 T^{3} - 92230 T^{4} - 295310 T^{5} + 9519255 T^{6} + 138484508 T^{7} + 374996946 T^{8} - 9994700294 T^{9} - 127252903162 T^{10} - 320416895110 T^{11} + 6864434010851 T^{12} + 77716662201882 T^{13} + 144679893780110 T^{14} - 3606790368380910 T^{15} - 29546900140126224 T^{16} + 49009998494139752 T^{17} + 1914476521633020253 T^{18} + 49009998494139752 p T^{19} - 29546900140126224 p^{2} T^{20} - 3606790368380910 p^{3} T^{21} + 144679893780110 p^{4} T^{22} + 77716662201882 p^{5} T^{23} + 6864434010851 p^{6} T^{24} - 320416895110 p^{7} T^{25} - 127252903162 p^{8} T^{26} - 9994700294 p^{9} T^{27} + 374996946 p^{10} T^{28} + 138484508 p^{11} T^{29} + 9519255 p^{12} T^{30} - 295310 p^{13} T^{31} - 92230 p^{14} T^{32} - 5058 p^{15} T^{33} + 134 p^{16} T^{34} + 28 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( 1 - 43 T + 888 T^{2} - 13358 T^{3} + 184502 T^{4} - 2351213 T^{5} + 25832040 T^{6} - 252404694 T^{7} + 2354167993 T^{8} - 20333166464 T^{9} + 148975109506 T^{10} - 914346534322 T^{11} + 4423731769883 T^{12} + 1268511191925 T^{13} - 426612869319200 T^{14} + 6611832777671515 T^{15} - 74648373280790749 T^{16} + 776591982937366054 T^{17} - 7215964915516431616 T^{18} + 776591982937366054 p T^{19} - 74648373280790749 p^{2} T^{20} + 6611832777671515 p^{3} T^{21} - 426612869319200 p^{4} T^{22} + 1268511191925 p^{5} T^{23} + 4423731769883 p^{6} T^{24} - 914346534322 p^{7} T^{25} + 148975109506 p^{8} T^{26} - 20333166464 p^{9} T^{27} + 2354167993 p^{10} T^{28} - 252404694 p^{11} T^{29} + 25832040 p^{12} T^{30} - 2351213 p^{13} T^{31} + 184502 p^{14} T^{32} - 13358 p^{15} T^{33} + 888 p^{16} T^{34} - 43 p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( 1 - 9 T - 82 T^{2} + 1665 T^{3} + 6709 T^{4} - 152438 T^{5} - 404348 T^{6} + 22461545 T^{7} - 7157894 T^{8} - 1870541243 T^{9} + 8126850221 T^{10} + 158962023876 T^{11} - 597143483446 T^{12} - 147971140983 p T^{13} + 119565726907184 T^{14} + 729473769567002 T^{15} - 10369364506544525 T^{16} - 2028724400936535 T^{17} + 902286961401370031 T^{18} - 2028724400936535 p T^{19} - 10369364506544525 p^{2} T^{20} + 729473769567002 p^{3} T^{21} + 119565726907184 p^{4} T^{22} - 147971140983 p^{6} T^{23} - 597143483446 p^{6} T^{24} + 158962023876 p^{7} T^{25} + 8126850221 p^{8} T^{26} - 1870541243 p^{9} T^{27} - 7157894 p^{10} T^{28} + 22461545 p^{11} T^{29} - 404348 p^{12} T^{30} - 152438 p^{13} T^{31} + 6709 p^{14} T^{32} + 1665 p^{15} T^{33} - 82 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 - 8 T - 234 T^{2} + 2464 T^{3} + 21170 T^{4} - 394800 T^{5} + 600379 T^{6} + 37896208 T^{7} - 446353614 T^{8} - 1105697210 T^{9} + 61509840758 T^{10} - 252712240386 T^{11} - 4304401003813 T^{12} + 47969845142298 T^{13} + 8400474449334 T^{14} - 4216300370766766 T^{15} + 35323954454545452 T^{16} + 147533091486972446 T^{17} - 4237337014813148067 T^{18} + 147533091486972446 p T^{19} + 35323954454545452 p^{2} T^{20} - 4216300370766766 p^{3} T^{21} + 8400474449334 p^{4} T^{22} + 47969845142298 p^{5} T^{23} - 4304401003813 p^{6} T^{24} - 252712240386 p^{7} T^{25} + 61509840758 p^{8} T^{26} - 1105697210 p^{9} T^{27} - 446353614 p^{10} T^{28} + 37896208 p^{11} T^{29} + 600379 p^{12} T^{30} - 394800 p^{13} T^{31} + 21170 p^{14} T^{32} + 2464 p^{15} T^{33} - 234 p^{16} T^{34} - 8 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 + 6 T - 156 T^{2} - 694 T^{3} - 3404 T^{4} - 25746 T^{5} + 3035191 T^{6} + 13236946 T^{7} - 278777232 T^{8} - 1416799932 T^{9} - 11405650892 T^{10} + 7940957862 T^{11} + 4209032266621 T^{12} + 13140194419338 T^{13} - 249945404179812 T^{14} - 1607993216703362 T^{15} - 17507012849570242 T^{16} + 71140627461568478 T^{17} + 3552231900008687859 T^{18} + 71140627461568478 p T^{19} - 17507012849570242 p^{2} T^{20} - 1607993216703362 p^{3} T^{21} - 249945404179812 p^{4} T^{22} + 13140194419338 p^{5} T^{23} + 4209032266621 p^{6} T^{24} + 7940957862 p^{7} T^{25} - 11405650892 p^{8} T^{26} - 1416799932 p^{9} T^{27} - 278777232 p^{10} T^{28} + 13236946 p^{11} T^{29} + 3035191 p^{12} T^{30} - 25746 p^{13} T^{31} - 3404 p^{14} T^{32} - 694 p^{15} T^{33} - 156 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( 1 + 81 T + 3185 T^{2} + 82471 T^{3} + 1613811 T^{4} + 25949863 T^{5} + 362302042 T^{6} + 4544285207 T^{7} + 52100768296 T^{8} + 547505111492 T^{9} + 5193536282736 T^{10} + 42479546681972 T^{11} + 263323717596387 T^{12} + 516943190734146 T^{13} - 18045629276362598 T^{14} - 399707182302942667 T^{15} - 5752758554039954081 T^{16} - 68393514389764220053 T^{17} - \)\(71\!\cdots\!78\)\( T^{18} - 68393514389764220053 p T^{19} - 5752758554039954081 p^{2} T^{20} - 399707182302942667 p^{3} T^{21} - 18045629276362598 p^{4} T^{22} + 516943190734146 p^{5} T^{23} + 263323717596387 p^{6} T^{24} + 42479546681972 p^{7} T^{25} + 5193536282736 p^{8} T^{26} + 547505111492 p^{9} T^{27} + 52100768296 p^{10} T^{28} + 4544285207 p^{11} T^{29} + 362302042 p^{12} T^{30} + 25949863 p^{13} T^{31} + 1613811 p^{14} T^{32} + 82471 p^{15} T^{33} + 3185 p^{16} T^{34} + 81 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−4.27790468720207800230934456506, −4.21712110329007367991069909983, −3.77783770503437198700932372242, −3.68821588661257684820173630756, −3.66740160253107730561586249301, −3.64903650112333742707224865203, −3.62433848994998155292623325916, −3.59486991487819349566625928273, −3.52307780115957818715897757023, −3.39878355710959014789584512321, −3.18949509739808649516602472934, −3.02280751128944763674251069583, −2.86944808053454088208169353824, −2.69930477157737553204957959417, −2.59294275511740168070333641521, −2.54215679615060119698579005786, −2.42845099258208896765894669285, −2.42521494632738723457307893262, −2.32574764180977286338758692199, −2.09730961705240444748345141327, −1.93482905611807858650630086516, −1.91113329301265944928779355277, −1.80351932704876151773841839122, −0.840602669496891280351210769258, −0.33663499338349850951616216263, 0.33663499338349850951616216263, 0.840602669496891280351210769258, 1.80351932704876151773841839122, 1.91113329301265944928779355277, 1.93482905611807858650630086516, 2.09730961705240444748345141327, 2.32574764180977286338758692199, 2.42521494632738723457307893262, 2.42845099258208896765894669285, 2.54215679615060119698579005786, 2.59294275511740168070333641521, 2.69930477157737553204957959417, 2.86944808053454088208169353824, 3.02280751128944763674251069583, 3.18949509739808649516602472934, 3.39878355710959014789584512321, 3.52307780115957818715897757023, 3.59486991487819349566625928273, 3.62433848994998155292623325916, 3.64903650112333742707224865203, 3.66740160253107730561586249301, 3.68821588661257684820173630756, 3.77783770503437198700932372242, 4.21712110329007367991069909983, 4.27790468720207800230934456506
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.