Dirichlet series
| L(s) = 1 | − 3·7-s − 3·8-s − 6·9-s + 3·11-s + 9·13-s − 6·17-s + 9·19-s + 21·23-s + 27-s + 6·29-s − 30·31-s + 6·37-s + 15·41-s − 12·43-s + 24·47-s + 21·49-s − 12·53-s + 9·56-s − 15·59-s + 72·61-s + 18·63-s + 3·64-s + 18·67-s + 18·72-s − 66·73-s − 9·77-s − 12·79-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.06·8-s − 2·9-s + 0.904·11-s + 2.49·13-s − 1.45·17-s + 2.06·19-s + 4.37·23-s + 0.192·27-s + 1.11·29-s − 5.38·31-s + 0.986·37-s + 2.34·41-s − 1.82·43-s + 3.50·47-s + 3·49-s − 1.64·53-s + 1.20·56-s − 1.95·59-s + 9.21·61-s + 2.26·63-s + 3/8·64-s + 2.19·67-s + 2.12·72-s − 7.72·73-s − 1.02·77-s − 1.35·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 5^{18} \cdot 37^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.94183\times 10^{8}\) |
| Root analytic conductor: | \(1.71885\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 5^{18} \cdot 37^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3959483208\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3959483208\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T^{3} + T^{6} )^{3} \) | |
| 37 | \( 1 - 6 T + 51 T^{2} - 72 T^{3} - 1014 T^{4} + 21426 T^{5} - 58572 T^{6} + 334884 T^{7} + 3037587 T^{8} - 20967095 T^{9} + 3037587 p T^{10} + 334884 p^{2} T^{11} - 58572 p^{3} T^{12} + 21426 p^{4} T^{13} - 1014 p^{5} T^{14} - 72 p^{6} T^{15} + 51 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 3 | \( 1 + 2 p T^{2} - T^{3} + 2 p T^{4} - 2 p T^{5} - 43 T^{6} + 4 p T^{7} - 5 p^{3} T^{8} + 296 T^{9} - 13 p^{2} T^{10} + 347 p T^{11} - 727 T^{12} + 95 p T^{13} - 833 p T^{14} - 5284 T^{15} + 4690 p T^{16} - 1739 p^{2} T^{17} + 91369 T^{18} - 1739 p^{3} T^{19} + 4690 p^{3} T^{20} - 5284 p^{3} T^{21} - 833 p^{5} T^{22} + 95 p^{6} T^{23} - 727 p^{6} T^{24} + 347 p^{8} T^{25} - 13 p^{10} T^{26} + 296 p^{9} T^{27} - 5 p^{13} T^{28} + 4 p^{12} T^{29} - 43 p^{12} T^{30} - 2 p^{14} T^{31} + 2 p^{15} T^{32} - p^{15} T^{33} + 2 p^{17} T^{34} + p^{18} T^{36} \) |
| 7 | \( 1 + 3 T - 12 T^{2} - 52 T^{3} - 12 T^{4} + 264 T^{5} + 634 T^{6} - 807 T^{7} - 3270 T^{8} - 2180 T^{9} - 1767 p T^{10} + 89499 T^{11} + 430981 T^{12} + 92244 T^{13} - 2019036 T^{14} - 5245783 T^{15} + 3992172 T^{16} + 13478016 T^{17} - 48956869 T^{18} + 13478016 p T^{19} + 3992172 p^{2} T^{20} - 5245783 p^{3} T^{21} - 2019036 p^{4} T^{22} + 92244 p^{5} T^{23} + 430981 p^{6} T^{24} + 89499 p^{7} T^{25} - 1767 p^{9} T^{26} - 2180 p^{9} T^{27} - 3270 p^{10} T^{28} - 807 p^{11} T^{29} + 634 p^{12} T^{30} + 264 p^{13} T^{31} - 12 p^{14} T^{32} - 52 p^{15} T^{33} - 12 p^{16} T^{34} + 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 11 | \( 1 - 3 T - 36 T^{2} - 25 T^{3} + 1047 T^{4} + 2532 T^{5} - 12621 T^{6} - 76689 T^{7} + 17634 T^{8} + 1051201 T^{9} + 2549484 T^{10} - 6762375 T^{11} - 43754234 T^{12} - 3648006 p T^{13} + 356344608 T^{14} + 94504948 p T^{15} - 8744247 T^{16} - 6607381173 T^{17} - 17805778051 T^{18} - 6607381173 p T^{19} - 8744247 p^{2} T^{20} + 94504948 p^{4} T^{21} + 356344608 p^{4} T^{22} - 3648006 p^{6} T^{23} - 43754234 p^{6} T^{24} - 6762375 p^{7} T^{25} + 2549484 p^{8} T^{26} + 1051201 p^{9} T^{27} + 17634 p^{10} T^{28} - 76689 p^{11} T^{29} - 12621 p^{12} T^{30} + 2532 p^{13} T^{31} + 1047 p^{14} T^{32} - 25 p^{15} T^{33} - 36 p^{16} T^{34} - 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 13 | \( 1 - 9 T + 12 T^{2} + 166 T^{3} - 1230 T^{4} + 4614 T^{5} - 3586 T^{6} - 57171 T^{7} + 373608 T^{8} - 1558104 T^{9} + 4153581 T^{10} - 78261 T^{11} - 70793459 T^{12} + 462702072 T^{13} - 1658739612 T^{14} + 2755040033 T^{15} + 6523900026 T^{16} - 73404622692 T^{17} + 335990961751 T^{18} - 73404622692 p T^{19} + 6523900026 p^{2} T^{20} + 2755040033 p^{3} T^{21} - 1658739612 p^{4} T^{22} + 462702072 p^{5} T^{23} - 70793459 p^{6} T^{24} - 78261 p^{7} T^{25} + 4153581 p^{8} T^{26} - 1558104 p^{9} T^{27} + 373608 p^{10} T^{28} - 57171 p^{11} T^{29} - 3586 p^{12} T^{30} + 4614 p^{13} T^{31} - 1230 p^{14} T^{32} + 166 p^{15} T^{33} + 12 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( 1 + 6 T + 21 T^{2} + 67 T^{3} + 471 T^{4} + 2472 T^{5} + 1956 T^{6} - 20721 T^{7} - 60363 T^{8} + 299219 T^{9} + 1228398 T^{10} - 3229794 T^{11} + 4508926 T^{12} + 180022257 T^{13} + 761803446 T^{14} - 1371694325 T^{15} - 19219017192 T^{16} - 47819790435 T^{17} - 133953713731 T^{18} - 47819790435 p T^{19} - 19219017192 p^{2} T^{20} - 1371694325 p^{3} T^{21} + 761803446 p^{4} T^{22} + 180022257 p^{5} T^{23} + 4508926 p^{6} T^{24} - 3229794 p^{7} T^{25} + 1228398 p^{8} T^{26} + 299219 p^{9} T^{27} - 60363 p^{10} T^{28} - 20721 p^{11} T^{29} + 1956 p^{12} T^{30} + 2472 p^{13} T^{31} + 471 p^{14} T^{32} + 67 p^{15} T^{33} + 21 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 19 | \( 1 - 9 T + 36 T^{2} + 79 T^{3} - 1080 T^{4} + 4590 T^{5} + 2813 T^{6} - 76959 T^{7} + 552861 T^{8} - 72717 p T^{9} + 854811 T^{10} + 41423544 T^{11} - 178661933 T^{12} + 452431251 T^{13} + 2417721534 T^{14} - 12270665413 T^{15} + 40661413173 T^{16} + 162506449242 T^{17} - 848463623213 T^{18} + 162506449242 p T^{19} + 40661413173 p^{2} T^{20} - 12270665413 p^{3} T^{21} + 2417721534 p^{4} T^{22} + 452431251 p^{5} T^{23} - 178661933 p^{6} T^{24} + 41423544 p^{7} T^{25} + 854811 p^{8} T^{26} - 72717 p^{10} T^{27} + 552861 p^{10} T^{28} - 76959 p^{11} T^{29} + 2813 p^{12} T^{30} + 4590 p^{13} T^{31} - 1080 p^{14} T^{32} + 79 p^{15} T^{33} + 36 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 - 21 T + 147 T^{2} - 58 T^{3} - 4884 T^{4} + 33573 T^{5} - 149331 T^{6} + 373095 T^{7} + 3271059 T^{8} - 40794359 T^{9} + 172891893 T^{10} - 203994831 T^{11} - 1776540860 T^{12} + 20856284523 T^{13} - 134826657936 T^{14} + 381844459358 T^{15} + 34422745242 p T^{16} - 11682991682382 T^{17} + 63697897064549 T^{18} - 11682991682382 p T^{19} + 34422745242 p^{3} T^{20} + 381844459358 p^{3} T^{21} - 134826657936 p^{4} T^{22} + 20856284523 p^{5} T^{23} - 1776540860 p^{6} T^{24} - 203994831 p^{7} T^{25} + 172891893 p^{8} T^{26} - 40794359 p^{9} T^{27} + 3271059 p^{10} T^{28} + 373095 p^{11} T^{29} - 149331 p^{12} T^{30} + 33573 p^{13} T^{31} - 4884 p^{14} T^{32} - 58 p^{15} T^{33} + 147 p^{16} T^{34} - 21 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 - 6 T - 93 T^{2} - 22 T^{3} + 8928 T^{4} + 19962 T^{5} - 357687 T^{6} - 2615880 T^{7} + 8777298 T^{8} + 121083154 T^{9} + 162540579 T^{10} - 125814699 p T^{11} - 15322934321 T^{12} + 40616311527 T^{13} + 503002817118 T^{14} + 486455873336 T^{15} - 233154904308 p T^{16} - 16068676818150 T^{17} + 60830950400159 T^{18} - 16068676818150 p T^{19} - 233154904308 p^{3} T^{20} + 486455873336 p^{3} T^{21} + 503002817118 p^{4} T^{22} + 40616311527 p^{5} T^{23} - 15322934321 p^{6} T^{24} - 125814699 p^{8} T^{25} + 162540579 p^{8} T^{26} + 121083154 p^{9} T^{27} + 8777298 p^{10} T^{28} - 2615880 p^{11} T^{29} - 357687 p^{12} T^{30} + 19962 p^{13} T^{31} + 8928 p^{14} T^{32} - 22 p^{15} T^{33} - 93 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( ( 1 + 15 T + 252 T^{2} + 2460 T^{3} + 23970 T^{4} + 171897 T^{5} + 1238013 T^{6} + 7174422 T^{7} + 44195340 T^{8} + 233297407 T^{9} + 44195340 p T^{10} + 7174422 p^{2} T^{11} + 1238013 p^{3} T^{12} + 171897 p^{4} T^{13} + 23970 p^{5} T^{14} + 2460 p^{6} T^{15} + 252 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 - 15 T + 153 T^{2} - 232 T^{3} - 4623 T^{4} + 68268 T^{5} + 11820 T^{6} - 3532119 T^{7} + 47652870 T^{8} - 161780975 T^{9} - 179925090 T^{10} + 12789184647 T^{11} - 50336816723 T^{12} - 40970414304 T^{13} + 5042044193889 T^{14} - 30998889586672 T^{15} + 120048690321942 T^{16} + 783383570104482 T^{17} - 6706826807596909 T^{18} + 783383570104482 p T^{19} + 120048690321942 p^{2} T^{20} - 30998889586672 p^{3} T^{21} + 5042044193889 p^{4} T^{22} - 40970414304 p^{5} T^{23} - 50336816723 p^{6} T^{24} + 12789184647 p^{7} T^{25} - 179925090 p^{8} T^{26} - 161780975 p^{9} T^{27} + 47652870 p^{10} T^{28} - 3532119 p^{11} T^{29} + 11820 p^{12} T^{30} + 68268 p^{13} T^{31} - 4623 p^{14} T^{32} - 232 p^{15} T^{33} + 153 p^{16} T^{34} - 15 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( ( 1 + 6 T + 270 T^{2} + 1477 T^{3} + 36318 T^{4} + 179541 T^{5} + 3114747 T^{6} + 13635519 T^{7} + 186120102 T^{8} + 704128613 T^{9} + 186120102 p T^{10} + 13635519 p^{2} T^{11} + 3114747 p^{3} T^{12} + 179541 p^{4} T^{13} + 36318 p^{5} T^{14} + 1477 p^{6} T^{15} + 270 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 47 | \( 1 - 24 T + 84 T^{2} + 2196 T^{3} - 15942 T^{4} - 84894 T^{5} + 567504 T^{6} + 6701250 T^{7} - 13495236 T^{8} - 522017154 T^{9} + 729234816 T^{10} + 22425256182 T^{11} + 31163479632 T^{12} - 971203166376 T^{13} - 4900324356960 T^{14} + 46773786037500 T^{15} + 4404141502950 p T^{16} - 1004893793330856 T^{17} - 7414065670195541 T^{18} - 1004893793330856 p T^{19} + 4404141502950 p^{3} T^{20} + 46773786037500 p^{3} T^{21} - 4900324356960 p^{4} T^{22} - 971203166376 p^{5} T^{23} + 31163479632 p^{6} T^{24} + 22425256182 p^{7} T^{25} + 729234816 p^{8} T^{26} - 522017154 p^{9} T^{27} - 13495236 p^{10} T^{28} + 6701250 p^{11} T^{29} + 567504 p^{12} T^{30} - 84894 p^{13} T^{31} - 15942 p^{14} T^{32} + 2196 p^{15} T^{33} + 84 p^{16} T^{34} - 24 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 + 12 T + 12 T^{2} - 216 T^{3} + 3156 T^{4} + 54768 T^{5} + 127908 T^{6} - 704931 T^{7} - 1991472 T^{8} + 27778086 T^{9} + 619155501 T^{10} + 1983609282 T^{11} - 28401902973 T^{12} - 392661460122 T^{13} - 277255990338 T^{14} + 3005739796023 T^{15} - 117095939378184 T^{16} - 982292556325290 T^{17} - 6819971615075297 T^{18} - 982292556325290 p T^{19} - 117095939378184 p^{2} T^{20} + 3005739796023 p^{3} T^{21} - 277255990338 p^{4} T^{22} - 392661460122 p^{5} T^{23} - 28401902973 p^{6} T^{24} + 1983609282 p^{7} T^{25} + 619155501 p^{8} T^{26} + 27778086 p^{9} T^{27} - 1991472 p^{10} T^{28} - 704931 p^{11} T^{29} + 127908 p^{12} T^{30} + 54768 p^{13} T^{31} + 3156 p^{14} T^{32} - 216 p^{15} T^{33} + 12 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( 1 + 15 T - 6 T^{2} - 850 T^{3} + 4005 T^{4} + 48948 T^{5} - 741201 T^{6} - 5774862 T^{7} + 38583429 T^{8} + 235038739 T^{9} - 2720290053 T^{10} - 5184840615 T^{11} + 241516031038 T^{12} + 738715247250 T^{13} - 8224378284120 T^{14} - 437906681023 T^{15} + 502384188728853 T^{16} - 1186659444732948 T^{17} - 43276887657686053 T^{18} - 1186659444732948 p T^{19} + 502384188728853 p^{2} T^{20} - 437906681023 p^{3} T^{21} - 8224378284120 p^{4} T^{22} + 738715247250 p^{5} T^{23} + 241516031038 p^{6} T^{24} - 5184840615 p^{7} T^{25} - 2720290053 p^{8} T^{26} + 235038739 p^{9} T^{27} + 38583429 p^{10} T^{28} - 5774862 p^{11} T^{29} - 741201 p^{12} T^{30} + 48948 p^{13} T^{31} + 4005 p^{14} T^{32} - 850 p^{15} T^{33} - 6 p^{16} T^{34} + 15 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 - 72 T + 2550 T^{2} - 59220 T^{3} + 1021719 T^{4} - 14181366 T^{5} + 168466182 T^{6} - 1795562832 T^{7} + 17658931554 T^{8} - 162240142228 T^{9} + 1407709464399 T^{10} - 11770515369801 T^{11} + 97103741026686 T^{12} - 799802183508687 T^{13} + 6565995414999621 T^{14} - 53586629707880118 T^{15} + 435186947641027473 T^{16} - 3504373396609340358 T^{17} + 27702888597423976035 T^{18} - 3504373396609340358 p T^{19} + 435186947641027473 p^{2} T^{20} - 53586629707880118 p^{3} T^{21} + 6565995414999621 p^{4} T^{22} - 799802183508687 p^{5} T^{23} + 97103741026686 p^{6} T^{24} - 11770515369801 p^{7} T^{25} + 1407709464399 p^{8} T^{26} - 162240142228 p^{9} T^{27} + 17658931554 p^{10} T^{28} - 1795562832 p^{11} T^{29} + 168466182 p^{12} T^{30} - 14181366 p^{13} T^{31} + 1021719 p^{14} T^{32} - 59220 p^{15} T^{33} + 2550 p^{16} T^{34} - 72 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 18 T + 192 T^{2} - 540 T^{3} - 5523 T^{4} + 127680 T^{5} - 820128 T^{6} + 2594364 T^{7} + 16298172 T^{8} - 2847760 T^{9} - 487012113 T^{10} - 13633286211 T^{11} + 331234706436 T^{12} - 3466309693995 T^{13} + 13555097981685 T^{14} + 31100075619522 T^{15} - 1401735594295341 T^{16} + 8192086551278730 T^{17} - 70513083004996977 T^{18} + 8192086551278730 p T^{19} - 1401735594295341 p^{2} T^{20} + 31100075619522 p^{3} T^{21} + 13555097981685 p^{4} T^{22} - 3466309693995 p^{5} T^{23} + 331234706436 p^{6} T^{24} - 13633286211 p^{7} T^{25} - 487012113 p^{8} T^{26} - 2847760 p^{9} T^{27} + 16298172 p^{10} T^{28} + 2594364 p^{11} T^{29} - 820128 p^{12} T^{30} + 127680 p^{13} T^{31} - 5523 p^{14} T^{32} - 540 p^{15} T^{33} + 192 p^{16} T^{34} - 18 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 + 282 T^{2} - 698 T^{3} + 38391 T^{4} - 163962 T^{5} + 4227456 T^{6} - 17933328 T^{7} + 6225666 p T^{8} - 1832878336 T^{9} + 39812119341 T^{10} - 202518432735 T^{11} + 3390554673808 T^{12} - 17405077139859 T^{13} + 295787653580721 T^{14} - 1328627030120510 T^{15} + 23131648822791471 T^{16} - 109363899869193642 T^{17} + 1644589961272972187 T^{18} - 109363899869193642 p T^{19} + 23131648822791471 p^{2} T^{20} - 1328627030120510 p^{3} T^{21} + 295787653580721 p^{4} T^{22} - 17405077139859 p^{5} T^{23} + 3390554673808 p^{6} T^{24} - 202518432735 p^{7} T^{25} + 39812119341 p^{8} T^{26} - 1832878336 p^{9} T^{27} + 6225666 p^{11} T^{28} - 17933328 p^{11} T^{29} + 4227456 p^{12} T^{30} - 163962 p^{13} T^{31} + 38391 p^{14} T^{32} - 698 p^{15} T^{33} + 282 p^{16} T^{34} + p^{18} T^{36} \) | |
| 73 | \( ( 1 + 33 T + 894 T^{2} + 16396 T^{3} + 265473 T^{4} + 3537900 T^{5} + 43247235 T^{6} + 463319817 T^{7} + 4607852349 T^{8} + 40851616841 T^{9} + 4607852349 p T^{10} + 463319817 p^{2} T^{11} + 43247235 p^{3} T^{12} + 3537900 p^{4} T^{13} + 265473 p^{5} T^{14} + 16396 p^{6} T^{15} + 894 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 79 | \( 1 + 12 T + 63 T^{2} + 3646 T^{3} + 32022 T^{4} + 123039 T^{5} + 6913283 T^{6} + 45456741 T^{7} + 122087364 T^{8} + 9231606054 T^{9} + 43304432736 T^{10} + 88478769645 T^{11} + 9864126235408 T^{12} + 30787160014584 T^{13} + 76383448082055 T^{14} + 8801155292966369 T^{15} + 16568424419533479 T^{16} + 90172021313534856 T^{17} + 6687687216879900979 T^{18} + 90172021313534856 p T^{19} + 16568424419533479 p^{2} T^{20} + 8801155292966369 p^{3} T^{21} + 76383448082055 p^{4} T^{22} + 30787160014584 p^{5} T^{23} + 9864126235408 p^{6} T^{24} + 88478769645 p^{7} T^{25} + 43304432736 p^{8} T^{26} + 9231606054 p^{9} T^{27} + 122087364 p^{10} T^{28} + 45456741 p^{11} T^{29} + 6913283 p^{12} T^{30} + 123039 p^{13} T^{31} + 32022 p^{14} T^{32} + 3646 p^{15} T^{33} + 63 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 - 9 T + 327 T^{2} - 2759 T^{3} + 40443 T^{4} - 454608 T^{5} + 3012447 T^{6} - 60412050 T^{7} + 386280327 T^{8} - 6748744090 T^{9} + 68684796600 T^{10} - 582695472804 T^{11} + 8006732912806 T^{12} - 46268810094558 T^{13} + 650070072216759 T^{14} - 5242306056282014 T^{15} + 48849906553528659 T^{16} - 621769677832591962 T^{17} + 3957906529903260167 T^{18} - 621769677832591962 p T^{19} + 48849906553528659 p^{2} T^{20} - 5242306056282014 p^{3} T^{21} + 650070072216759 p^{4} T^{22} - 46268810094558 p^{5} T^{23} + 8006732912806 p^{6} T^{24} - 582695472804 p^{7} T^{25} + 68684796600 p^{8} T^{26} - 6748744090 p^{9} T^{27} + 386280327 p^{10} T^{28} - 60412050 p^{11} T^{29} + 3012447 p^{12} T^{30} - 454608 p^{13} T^{31} + 40443 p^{14} T^{32} - 2759 p^{15} T^{33} + 327 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 + 96 T^{2} - 40 T^{3} - 864 T^{4} - 136548 T^{5} - 220215 T^{6} - 22550967 T^{7} - 59899209 T^{8} - 144029306 T^{9} + 4373014254 T^{10} + 132576060000 T^{11} + 1983728286061 T^{12} + 14675897304639 T^{13} + 45180712403919 T^{14} + 988920612928418 T^{15} - 15104337396974736 T^{16} - 98446719257486130 T^{17} - 1748580601717637017 T^{18} - 98446719257486130 p T^{19} - 15104337396974736 p^{2} T^{20} + 988920612928418 p^{3} T^{21} + 45180712403919 p^{4} T^{22} + 14675897304639 p^{5} T^{23} + 1983728286061 p^{6} T^{24} + 132576060000 p^{7} T^{25} + 4373014254 p^{8} T^{26} - 144029306 p^{9} T^{27} - 59899209 p^{10} T^{28} - 22550967 p^{11} T^{29} - 220215 p^{12} T^{30} - 136548 p^{13} T^{31} - 864 p^{14} T^{32} - 40 p^{15} T^{33} + 96 p^{16} T^{34} + p^{18} T^{36} \) | |
| 97 | \( 1 + 3 T - 615 T^{2} - 2210 T^{3} + 192165 T^{4} + 734115 T^{5} - 42511738 T^{6} - 147852240 T^{7} + 7691856702 T^{8} + 21264360507 T^{9} - 1213086172254 T^{10} - 2458367210181 T^{11} + 169932015117391 T^{12} + 238145223790752 T^{13} - 21219082297023282 T^{14} - 17796014686502032 T^{15} + 2378333475203776203 T^{16} + 6802959619318272 p T^{17} - \)\(24\!\cdots\!65\)\( T^{18} + 6802959619318272 p^{2} T^{19} + 2378333475203776203 p^{2} T^{20} - 17796014686502032 p^{3} T^{21} - 21219082297023282 p^{4} T^{22} + 238145223790752 p^{5} T^{23} + 169932015117391 p^{6} T^{24} - 2458367210181 p^{7} T^{25} - 1213086172254 p^{8} T^{26} + 21264360507 p^{9} T^{27} + 7691856702 p^{10} T^{28} - 147852240 p^{11} T^{29} - 42511738 p^{12} T^{30} + 734115 p^{13} T^{31} + 192165 p^{14} T^{32} - 2210 p^{15} T^{33} - 615 p^{16} T^{34} + 3 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−2.88074294796785923322252694529, −2.82761269349839925751937545859, −2.80871948888994716559451215949, −2.79707600479895560895477872381, −2.69719722064179375024510589183, −2.62349553087136314056512855262, −2.58768679108223922831209188929, −2.35841535030215127547625688334, −2.35024297097190756004068839500, −2.14634990313551078252900100615, −1.98415295208640506950617574637, −1.97178130974407670438710996757, −1.95561494730454181526005157334, −1.80135731958188192620457014763, −1.76091555215556404909670053875, −1.75445784910735608109642365896, −1.31288798725764633736014040707, −1.20107262393097438248240051846, −1.13153058792138832002442735000, −1.08270332256196387737485026513, −0.891067138656458317683489155134, −0.76987525189443189769713118913, −0.73898235793506164080505871052, −0.63192166766973306414403667321, −0.05756900099449319465007910553, 0.05756900099449319465007910553, 0.63192166766973306414403667321, 0.73898235793506164080505871052, 0.76987525189443189769713118913, 0.891067138656458317683489155134, 1.08270332256196387737485026513, 1.13153058792138832002442735000, 1.20107262393097438248240051846, 1.31288798725764633736014040707, 1.75445784910735608109642365896, 1.76091555215556404909670053875, 1.80135731958188192620457014763, 1.95561494730454181526005157334, 1.97178130974407670438710996757, 1.98415295208640506950617574637, 2.14634990313551078252900100615, 2.35024297097190756004068839500, 2.35841535030215127547625688334, 2.58768679108223922831209188929, 2.62349553087136314056512855262, 2.69719722064179375024510589183, 2.79707600479895560895477872381, 2.80871948888994716559451215949, 2.82761269349839925751937545859, 2.88074294796785923322252694529
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.