Dirichlet series
| L(s) = 1 | + 3·8-s − 9·9-s − 12·11-s − 6·13-s + 9·23-s − 6·27-s − 6·29-s − 6·31-s − 12·37-s − 21·41-s + 18·43-s − 54·47-s + 12·49-s − 24·53-s − 30·59-s + 48·61-s + 3·64-s − 6·67-s + 30·71-s − 27·72-s + 6·73-s + 30·79-s + 45·81-s + 6·83-s − 36·88-s + 30·89-s − 12·97-s + ⋯ |
| L(s) = 1 | + 1.06·8-s − 3·9-s − 3.61·11-s − 1.66·13-s + 1.87·23-s − 1.15·27-s − 1.11·29-s − 1.07·31-s − 1.97·37-s − 3.27·41-s + 2.74·43-s − 7.87·47-s + 12/7·49-s − 3.29·53-s − 3.90·59-s + 6.14·61-s + 3/8·64-s − 0.733·67-s + 3.56·71-s − 3.18·72-s + 0.702·73-s + 3.37·79-s + 5·81-s + 0.658·83-s − 3.83·88-s + 3.17·89-s − 1.21·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 5^{18} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1813.64\) |
| Root analytic conductor: | \(1.23172\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 5^{18} \cdot 19^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02294995314\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02294995314\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{3} + T^{6} )^{3} \) |
| 5 | \( ( 1 + T^{3} + T^{6} )^{3} \) | |
| 19 | \( 1 - 60 T^{2} - p T^{3} + 1158 T^{4} - 984 T^{5} - 1954 T^{6} + 118830 T^{7} - 98874 T^{8} - 3550597 T^{9} - 98874 p T^{10} + 118830 p^{2} T^{11} - 1954 p^{3} T^{12} - 984 p^{4} T^{13} + 1158 p^{5} T^{14} - p^{7} T^{15} - 60 p^{7} T^{16} + p^{9} T^{18} \) | |
| good | 3 | \( 1 + p^{2} T^{2} + 2 p T^{3} + 4 p^{2} T^{4} + 2 p^{3} T^{5} + 95 T^{6} + 26 p^{2} T^{7} + 28 p^{2} T^{8} + 692 T^{9} + 29 p^{3} T^{10} + 22 p^{4} T^{11} + 2164 T^{12} + 154 p^{3} T^{13} + 535 p^{2} T^{14} + 4586 T^{15} + 1646 p^{2} T^{16} - 1868 p^{2} T^{17} + 52885 T^{18} - 1868 p^{3} T^{19} + 1646 p^{4} T^{20} + 4586 p^{3} T^{21} + 535 p^{6} T^{22} + 154 p^{8} T^{23} + 2164 p^{6} T^{24} + 22 p^{11} T^{25} + 29 p^{11} T^{26} + 692 p^{9} T^{27} + 28 p^{12} T^{28} + 26 p^{13} T^{29} + 95 p^{12} T^{30} + 2 p^{16} T^{31} + 4 p^{16} T^{32} + 2 p^{16} T^{33} + p^{18} T^{34} + p^{18} T^{36} \) |
| 7 | \( 1 - 12 T^{2} + 10 p T^{3} + 75 T^{4} - 654 T^{5} + 2263 T^{6} + 2712 T^{7} - 18924 T^{8} + 58872 T^{9} + 6621 p T^{10} - 418722 T^{11} + 1283514 T^{12} - 21702 T^{13} - 7279950 T^{14} + 27629878 T^{15} - 25985577 T^{16} - 113758110 T^{17} + 555591451 T^{18} - 113758110 p T^{19} - 25985577 p^{2} T^{20} + 27629878 p^{3} T^{21} - 7279950 p^{4} T^{22} - 21702 p^{5} T^{23} + 1283514 p^{6} T^{24} - 418722 p^{7} T^{25} + 6621 p^{9} T^{26} + 58872 p^{9} T^{27} - 18924 p^{10} T^{28} + 2712 p^{11} T^{29} + 2263 p^{12} T^{30} - 654 p^{13} T^{31} + 75 p^{14} T^{32} + 10 p^{16} T^{33} - 12 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( 1 + 12 T + 54 T^{2} + 62 T^{3} - 450 T^{4} - 2088 T^{5} - 189 p T^{6} + 2256 T^{7} - 48258 T^{8} - 358730 T^{9} - 606774 T^{10} + 2461596 T^{11} + 12981418 T^{12} + 16064244 T^{13} - 18819054 T^{14} + 44707214 T^{15} + 450215346 T^{16} - 658075128 T^{17} - 7909286623 T^{18} - 658075128 p T^{19} + 450215346 p^{2} T^{20} + 44707214 p^{3} T^{21} - 18819054 p^{4} T^{22} + 16064244 p^{5} T^{23} + 12981418 p^{6} T^{24} + 2461596 p^{7} T^{25} - 606774 p^{8} T^{26} - 358730 p^{9} T^{27} - 48258 p^{10} T^{28} + 2256 p^{11} T^{29} - 189 p^{13} T^{30} - 2088 p^{13} T^{31} - 450 p^{14} T^{32} + 62 p^{15} T^{33} + 54 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 + 6 T - 36 T^{2} - 242 T^{3} + 636 T^{4} + 4656 T^{5} - 4825 T^{6} - 38268 T^{7} + 59610 T^{8} - 1422 p T^{9} - 2969202 T^{10} + 678618 T^{11} + 82433368 T^{12} + 101181822 T^{13} - 1138702488 T^{14} - 173344858 p T^{15} + 8072642520 T^{16} + 15269964780 T^{17} - 43084006715 T^{18} + 15269964780 p T^{19} + 8072642520 p^{2} T^{20} - 173344858 p^{4} T^{21} - 1138702488 p^{4} T^{22} + 101181822 p^{5} T^{23} + 82433368 p^{6} T^{24} + 678618 p^{7} T^{25} - 2969202 p^{8} T^{26} - 1422 p^{10} T^{27} + 59610 p^{10} T^{28} - 38268 p^{11} T^{29} - 4825 p^{12} T^{30} + 4656 p^{13} T^{31} + 636 p^{14} T^{32} - 242 p^{15} T^{33} - 36 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( 1 - 24 T^{2} - 64 T^{3} + 1074 T^{4} + 2202 T^{5} - 22111 T^{6} - 51354 T^{7} + 548262 T^{8} + 1151412 T^{9} - 12351192 T^{10} - 15987900 T^{11} + 905186 p^{2} T^{12} + 274949712 T^{13} - 5313722724 T^{14} - 3803785696 T^{15} + 5699795166 p T^{16} + 12948389538 T^{17} - 1675738565863 T^{18} + 12948389538 p T^{19} + 5699795166 p^{3} T^{20} - 3803785696 p^{3} T^{21} - 5313722724 p^{4} T^{22} + 274949712 p^{5} T^{23} + 905186 p^{8} T^{24} - 15987900 p^{7} T^{25} - 12351192 p^{8} T^{26} + 1151412 p^{9} T^{27} + 548262 p^{10} T^{28} - 51354 p^{11} T^{29} - 22111 p^{12} T^{30} + 2202 p^{13} T^{31} + 1074 p^{14} T^{32} - 64 p^{15} T^{33} - 24 p^{16} T^{34} + p^{18} T^{36} \) | |
| 23 | \( 1 - 9 T + 108 T^{2} - 36 p T^{3} + 6957 T^{4} - 42561 T^{5} + 283665 T^{6} - 1535715 T^{7} + 8689023 T^{8} - 1643364 p T^{9} + 179933274 T^{10} - 551440233 T^{11} + 1527107154 T^{12} + 5123246373 T^{13} - 57449091258 T^{14} + 576147511422 T^{15} - 3356187507297 T^{16} + 20594999640519 T^{17} - 94032262172837 T^{18} + 20594999640519 p T^{19} - 3356187507297 p^{2} T^{20} + 576147511422 p^{3} T^{21} - 57449091258 p^{4} T^{22} + 5123246373 p^{5} T^{23} + 1527107154 p^{6} T^{24} - 551440233 p^{7} T^{25} + 179933274 p^{8} T^{26} - 1643364 p^{10} T^{27} + 8689023 p^{10} T^{28} - 1535715 p^{11} T^{29} + 283665 p^{12} T^{30} - 42561 p^{13} T^{31} + 6957 p^{14} T^{32} - 36 p^{16} T^{33} + 108 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 + 6 T - 24 T^{2} - 344 T^{3} + 204 T^{4} + 4254 T^{5} - 29037 T^{6} - 268554 T^{7} + 1453836 T^{8} + 11124392 T^{9} + 5650410 T^{10} - 262311462 T^{11} - 56982644 T^{12} - 26057454 T^{13} - 17045685414 T^{14} - 201096319880 T^{15} + 416793662706 T^{16} + 3837023954478 T^{17} + 32867808984761 T^{18} + 3837023954478 p T^{19} + 416793662706 p^{2} T^{20} - 201096319880 p^{3} T^{21} - 17045685414 p^{4} T^{22} - 26057454 p^{5} T^{23} - 56982644 p^{6} T^{24} - 262311462 p^{7} T^{25} + 5650410 p^{8} T^{26} + 11124392 p^{9} T^{27} + 1453836 p^{10} T^{28} - 268554 p^{11} T^{29} - 29037 p^{12} T^{30} + 4254 p^{13} T^{31} + 204 p^{14} T^{32} - 344 p^{15} T^{33} - 24 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 + 6 T - 159 T^{2} - 1038 T^{3} + 12573 T^{4} + 86448 T^{5} - 710236 T^{6} - 4792296 T^{7} + 34348554 T^{8} + 203116372 T^{9} - 1544952546 T^{10} - 7023351564 T^{11} + 65309888142 T^{12} + 200758789224 T^{13} - 2546614645104 T^{14} - 4427810387768 T^{15} + 90547425741591 T^{16} + 49894569974946 T^{17} - 2936538482485081 T^{18} + 49894569974946 p T^{19} + 90547425741591 p^{2} T^{20} - 4427810387768 p^{3} T^{21} - 2546614645104 p^{4} T^{22} + 200758789224 p^{5} T^{23} + 65309888142 p^{6} T^{24} - 7023351564 p^{7} T^{25} - 1544952546 p^{8} T^{26} + 203116372 p^{9} T^{27} + 34348554 p^{10} T^{28} - 4792296 p^{11} T^{29} - 710236 p^{12} T^{30} + 86448 p^{13} T^{31} + 12573 p^{14} T^{32} - 1038 p^{15} T^{33} - 159 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 37 | \( ( 1 + 6 T + 204 T^{2} + 1033 T^{3} + 18417 T^{4} + 77142 T^{5} + 995859 T^{6} + 3507723 T^{7} + 39994371 T^{8} + 130541864 T^{9} + 39994371 p T^{10} + 3507723 p^{2} T^{11} + 995859 p^{3} T^{12} + 77142 p^{4} T^{13} + 18417 p^{5} T^{14} + 1033 p^{6} T^{15} + 204 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 + 21 T + 261 T^{2} + 2687 T^{3} + 26949 T^{4} + 259677 T^{5} + 2308556 T^{6} + 19035192 T^{7} + 148820031 T^{8} + 1123940412 T^{9} + 8183262762 T^{10} + 57307373268 T^{11} + 384410038481 T^{12} + 2513454276111 T^{13} + 16261430887680 T^{14} + 103803929261429 T^{15} + 653838899005800 T^{16} + 4109490177349875 T^{17} + 26141798868223823 T^{18} + 4109490177349875 p T^{19} + 653838899005800 p^{2} T^{20} + 103803929261429 p^{3} T^{21} + 16261430887680 p^{4} T^{22} + 2513454276111 p^{5} T^{23} + 384410038481 p^{6} T^{24} + 57307373268 p^{7} T^{25} + 8183262762 p^{8} T^{26} + 1123940412 p^{9} T^{27} + 148820031 p^{10} T^{28} + 19035192 p^{11} T^{29} + 2308556 p^{12} T^{30} + 259677 p^{13} T^{31} + 26949 p^{14} T^{32} + 2687 p^{15} T^{33} + 261 p^{16} T^{34} + 21 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( 1 - 18 T + 306 T^{2} - 4368 T^{3} + 51156 T^{4} - 13332 p T^{5} + 5710715 T^{6} - 52576668 T^{7} + 460320960 T^{8} - 3723915104 T^{9} + 28676230302 T^{10} - 210514634202 T^{11} + 1458998470506 T^{12} - 9808849689510 T^{13} + 63687334640082 T^{14} - 402141801223016 T^{15} + 2542637607657924 T^{16} - 16130094780727392 T^{17} + 104122492165077899 T^{18} - 16130094780727392 p T^{19} + 2542637607657924 p^{2} T^{20} - 402141801223016 p^{3} T^{21} + 63687334640082 p^{4} T^{22} - 9808849689510 p^{5} T^{23} + 1458998470506 p^{6} T^{24} - 210514634202 p^{7} T^{25} + 28676230302 p^{8} T^{26} - 3723915104 p^{9} T^{27} + 460320960 p^{10} T^{28} - 52576668 p^{11} T^{29} + 5710715 p^{12} T^{30} - 13332 p^{14} T^{31} + 51156 p^{14} T^{32} - 4368 p^{15} T^{33} + 306 p^{16} T^{34} - 18 p^{17} T^{35} + p^{18} T^{36} \) | |
| 47 | \( 1 + 54 T + 1356 T^{2} + 21164 T^{3} + 232746 T^{4} + 1953786 T^{5} + 13295993 T^{6} + 75707046 T^{7} + 350139042 T^{8} + 1073820612 T^{9} - 26712330 p T^{10} - 57214466106 T^{11} - 635365349440 T^{12} - 5490286514262 T^{13} - 43450280385858 T^{14} - 326571706439356 T^{15} - 2274093835734924 T^{16} - 14766282030974478 T^{17} - 97309728069323233 T^{18} - 14766282030974478 p T^{19} - 2274093835734924 p^{2} T^{20} - 326571706439356 p^{3} T^{21} - 43450280385858 p^{4} T^{22} - 5490286514262 p^{5} T^{23} - 635365349440 p^{6} T^{24} - 57214466106 p^{7} T^{25} - 26712330 p^{9} T^{26} + 1073820612 p^{9} T^{27} + 350139042 p^{10} T^{28} + 75707046 p^{11} T^{29} + 13295993 p^{12} T^{30} + 1953786 p^{13} T^{31} + 232746 p^{14} T^{32} + 21164 p^{15} T^{33} + 1356 p^{16} T^{34} + 54 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 + 24 T + 291 T^{2} + 2383 T^{3} + 14523 T^{4} + 17103 T^{5} - 921154 T^{6} - 12834273 T^{7} - 107324901 T^{8} - 637312101 T^{9} - 947859297 T^{10} + 33291341556 T^{11} + 471395548016 T^{12} + 4004364380088 T^{13} + 24356897170467 T^{14} + 66857970224473 T^{15} - 623920153086945 T^{16} - 10507010802745809 T^{17} - 90856515557112514 T^{18} - 10507010802745809 p T^{19} - 623920153086945 p^{2} T^{20} + 66857970224473 p^{3} T^{21} + 24356897170467 p^{4} T^{22} + 4004364380088 p^{5} T^{23} + 471395548016 p^{6} T^{24} + 33291341556 p^{7} T^{25} - 947859297 p^{8} T^{26} - 637312101 p^{9} T^{27} - 107324901 p^{10} T^{28} - 12834273 p^{11} T^{29} - 921154 p^{12} T^{30} + 17103 p^{13} T^{31} + 14523 p^{14} T^{32} + 2383 p^{15} T^{33} + 291 p^{16} T^{34} + 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( 1 + 30 T + 573 T^{2} + 9628 T^{3} + 129378 T^{4} + 1518252 T^{5} + 16533623 T^{6} + 158715534 T^{7} + 1418462370 T^{8} + 11976736128 T^{9} + 92121439677 T^{10} + 692457310398 T^{11} + 5086477233968 T^{12} + 36574293273468 T^{13} + 279528859009461 T^{14} + 2166682169159116 T^{15} + 17100207477031410 T^{16} + 136473122266472478 T^{17} + 1054927121059171445 T^{18} + 136473122266472478 p T^{19} + 17100207477031410 p^{2} T^{20} + 2166682169159116 p^{3} T^{21} + 279528859009461 p^{4} T^{22} + 36574293273468 p^{5} T^{23} + 5086477233968 p^{6} T^{24} + 692457310398 p^{7} T^{25} + 92121439677 p^{8} T^{26} + 11976736128 p^{9} T^{27} + 1418462370 p^{10} T^{28} + 158715534 p^{11} T^{29} + 16533623 p^{12} T^{30} + 1518252 p^{13} T^{31} + 129378 p^{14} T^{32} + 9628 p^{15} T^{33} + 573 p^{16} T^{34} + 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 - 48 T + 990 T^{2} - 194 p T^{3} + 102156 T^{4} - 849936 T^{5} + 7812013 T^{6} - 74784588 T^{7} + 718864722 T^{8} - 5950590774 T^{9} + 33514831692 T^{10} - 78543630756 T^{11} - 590421131742 T^{12} + 9214097558616 T^{13} - 84845300690442 T^{14} + 926479189322134 T^{15} - 12142887187358028 T^{16} + 136084494979145028 T^{17} - 1194796974753478619 T^{18} + 136084494979145028 p T^{19} - 12142887187358028 p^{2} T^{20} + 926479189322134 p^{3} T^{21} - 84845300690442 p^{4} T^{22} + 9214097558616 p^{5} T^{23} - 590421131742 p^{6} T^{24} - 78543630756 p^{7} T^{25} + 33514831692 p^{8} T^{26} - 5950590774 p^{9} T^{27} + 718864722 p^{10} T^{28} - 74784588 p^{11} T^{29} + 7812013 p^{12} T^{30} - 849936 p^{13} T^{31} + 102156 p^{14} T^{32} - 194 p^{16} T^{33} + 990 p^{16} T^{34} - 48 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 + 6 T + 207 T^{2} + 604 T^{3} + 18474 T^{4} - 71760 T^{5} + 575527 T^{6} - 17582442 T^{7} + 2393316 T^{8} - 1417520940 T^{9} + 6029448003 T^{10} - 41994622458 T^{11} + 1065916553352 T^{12} - 196973581320 T^{13} + 75584766090021 T^{14} - 304384220102468 T^{15} + 1907060825765040 T^{16} - 50135883483912234 T^{17} - 19239827276472947 T^{18} - 50135883483912234 p T^{19} + 1907060825765040 p^{2} T^{20} - 304384220102468 p^{3} T^{21} + 75584766090021 p^{4} T^{22} - 196973581320 p^{5} T^{23} + 1065916553352 p^{6} T^{24} - 41994622458 p^{7} T^{25} + 6029448003 p^{8} T^{26} - 1417520940 p^{9} T^{27} + 2393316 p^{10} T^{28} - 17582442 p^{11} T^{29} + 575527 p^{12} T^{30} - 71760 p^{13} T^{31} + 18474 p^{14} T^{32} + 604 p^{15} T^{33} + 207 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 - 30 T + 354 T^{2} - 1838 T^{3} - 3468 T^{4} + 186348 T^{5} - 2275285 T^{6} + 11480964 T^{7} + 66252870 T^{8} - 1614689322 T^{9} + 13065077136 T^{10} - 34254930342 T^{11} - 526955336614 T^{12} + 8959610373186 T^{13} - 76062700065846 T^{14} + 350522695238686 T^{15} + 507080153779524 T^{16} - 25294237688086500 T^{17} + 266752723846557611 T^{18} - 25294237688086500 p T^{19} + 507080153779524 p^{2} T^{20} + 350522695238686 p^{3} T^{21} - 76062700065846 p^{4} T^{22} + 8959610373186 p^{5} T^{23} - 526955336614 p^{6} T^{24} - 34254930342 p^{7} T^{25} + 13065077136 p^{8} T^{26} - 1614689322 p^{9} T^{27} + 66252870 p^{10} T^{28} + 11480964 p^{11} T^{29} - 2275285 p^{12} T^{30} + 186348 p^{13} T^{31} - 3468 p^{14} T^{32} - 1838 p^{15} T^{33} + 354 p^{16} T^{34} - 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( 1 - 6 T - 102 T^{2} + 2042 T^{3} - 9603 T^{4} - 171738 T^{5} + 2378032 T^{6} - 5038980 T^{7} - 147932841 T^{8} + 2198560186 T^{9} - 5688934680 T^{10} - 131985171978 T^{11} + 1836527358910 T^{12} - 7218235686708 T^{13} - 96820347723174 T^{14} + 1377160644756914 T^{15} - 5181865698234243 T^{16} - 50668215199958292 T^{17} + 899838020415018344 T^{18} - 50668215199958292 p T^{19} - 5181865698234243 p^{2} T^{20} + 1377160644756914 p^{3} T^{21} - 96820347723174 p^{4} T^{22} - 7218235686708 p^{5} T^{23} + 1836527358910 p^{6} T^{24} - 131985171978 p^{7} T^{25} - 5688934680 p^{8} T^{26} + 2198560186 p^{9} T^{27} - 147932841 p^{10} T^{28} - 5038980 p^{11} T^{29} + 2378032 p^{12} T^{30} - 171738 p^{13} T^{31} - 9603 p^{14} T^{32} + 2042 p^{15} T^{33} - 102 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( 1 - 30 T + 264 T^{2} + 1470 T^{3} - 33240 T^{4} - 125544 T^{5} + 4596407 T^{6} + 2918700 T^{7} - 547571964 T^{8} + 1650169966 T^{9} + 44453440776 T^{10} - 263232926274 T^{11} - 2577280237590 T^{12} + 21794452087590 T^{13} + 134824908924708 T^{14} - 1426521127815674 T^{15} - 5608834503876108 T^{16} + 71852151965555244 T^{17} - 184335855665903137 T^{18} + 71852151965555244 p T^{19} - 5608834503876108 p^{2} T^{20} - 1426521127815674 p^{3} T^{21} + 134824908924708 p^{4} T^{22} + 21794452087590 p^{5} T^{23} - 2577280237590 p^{6} T^{24} - 263232926274 p^{7} T^{25} + 44453440776 p^{8} T^{26} + 1650169966 p^{9} T^{27} - 547571964 p^{10} T^{28} + 2918700 p^{11} T^{29} + 4596407 p^{12} T^{30} - 125544 p^{13} T^{31} - 33240 p^{14} T^{32} + 1470 p^{15} T^{33} + 264 p^{16} T^{34} - 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 - 6 T - 351 T^{2} + 938 T^{3} + 66000 T^{4} + 19278 T^{5} - 7510003 T^{6} - 22770942 T^{7} + 518209080 T^{8} + 3170544342 T^{9} - 20200055973 T^{10} - 214378348578 T^{11} + 686253917522 T^{12} + 6540119014050 T^{13} - 102222630078993 T^{14} + 4482135785546 T^{15} + 15453053980709838 T^{16} - 5322951962235078 T^{17} - 1505233343118807703 T^{18} - 5322951962235078 p T^{19} + 15453053980709838 p^{2} T^{20} + 4482135785546 p^{3} T^{21} - 102222630078993 p^{4} T^{22} + 6540119014050 p^{5} T^{23} + 686253917522 p^{6} T^{24} - 214378348578 p^{7} T^{25} - 20200055973 p^{8} T^{26} + 3170544342 p^{9} T^{27} + 518209080 p^{10} T^{28} - 22770942 p^{11} T^{29} - 7510003 p^{12} T^{30} + 19278 p^{13} T^{31} + 66000 p^{14} T^{32} + 938 p^{15} T^{33} - 351 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 - 30 T + 435 T^{2} - 5573 T^{3} + 66813 T^{4} - 619113 T^{5} + 5310960 T^{6} - 55668477 T^{7} + 586345215 T^{8} - 5231458759 T^{9} + 48080173737 T^{10} - 477944806500 T^{11} + 4263974250112 T^{12} - 43577777566608 T^{13} + 557400285055035 T^{14} - 6415842996111161 T^{15} + 64740225185507541 T^{16} - 658238065888057515 T^{17} + 6544436150121443150 T^{18} - 658238065888057515 p T^{19} + 64740225185507541 p^{2} T^{20} - 6415842996111161 p^{3} T^{21} + 557400285055035 p^{4} T^{22} - 43577777566608 p^{5} T^{23} + 4263974250112 p^{6} T^{24} - 477944806500 p^{7} T^{25} + 48080173737 p^{8} T^{26} - 5231458759 p^{9} T^{27} + 586345215 p^{10} T^{28} - 55668477 p^{11} T^{29} + 5310960 p^{12} T^{30} - 619113 p^{13} T^{31} + 66813 p^{14} T^{32} - 5573 p^{15} T^{33} + 435 p^{16} T^{34} - 30 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( 1 + 12 T - 117 T^{2} - 1660 T^{3} + 8916 T^{4} + 104304 T^{5} - 1154855 T^{6} - 10276506 T^{7} + 59355276 T^{8} + 1410458938 T^{9} + 6572269509 T^{10} - 160841835810 T^{11} - 1215678077420 T^{12} + 15064013595612 T^{13} + 51850114649097 T^{14} - 2029920582684316 T^{15} - 5564153308798890 T^{16} + 129689943182999256 T^{17} + 1217791787568262043 T^{18} + 129689943182999256 p T^{19} - 5564153308798890 p^{2} T^{20} - 2029920582684316 p^{3} T^{21} + 51850114649097 p^{4} T^{22} + 15064013595612 p^{5} T^{23} - 1215678077420 p^{6} T^{24} - 160841835810 p^{7} T^{25} + 6572269509 p^{8} T^{26} + 1410458938 p^{9} T^{27} + 59355276 p^{10} T^{28} - 10276506 p^{11} T^{29} - 1154855 p^{12} T^{30} + 104304 p^{13} T^{31} + 8916 p^{14} T^{32} - 1660 p^{15} T^{33} - 117 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−3.40924458962546139149572900868, −3.38702973498213077824883939747, −3.09797616203591873879154278088, −3.06420170532237962167805788610, −3.04637925271467669850723554427, −3.02385895421583769238087299505, −2.95371432639306828263201144443, −2.85413822781681200940261663491, −2.66691206729761745339915015627, −2.56746325981405006283886389191, −2.49896957647338591792666511834, −2.49133464522052084807702608092, −2.41477678417114575671769099627, −2.21554340582056243304813703371, −2.07792696166784797609453768899, −2.01332324628375114090870676913, −1.93479151852949990431148058137, −1.68757290958890336619381577458, −1.62181826236487701499298767948, −1.49522190374989653156568944878, −1.45513449057481048142848604328, −1.27946738305926572915466329657, −0.61985090747894710948866347094, −0.46809461467384583807723188460, −0.06374269368462409012132998672, 0.06374269368462409012132998672, 0.46809461467384583807723188460, 0.61985090747894710948866347094, 1.27946738305926572915466329657, 1.45513449057481048142848604328, 1.49522190374989653156568944878, 1.62181826236487701499298767948, 1.68757290958890336619381577458, 1.93479151852949990431148058137, 2.01332324628375114090870676913, 2.07792696166784797609453768899, 2.21554340582056243304813703371, 2.41477678417114575671769099627, 2.49133464522052084807702608092, 2.49896957647338591792666511834, 2.56746325981405006283886389191, 2.66691206729761745339915015627, 2.85413822781681200940261663491, 2.95371432639306828263201144443, 3.02385895421583769238087299505, 3.04637925271467669850723554427, 3.06420170532237962167805788610, 3.09797616203591873879154278088, 3.38702973498213077824883939747, 3.40924458962546139149572900868
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.