Dirichlet series
| L(s) = 1 | + 3·3-s + 12·9-s + 9·13-s + 12·17-s − 18·19-s + 21·23-s + 30·27-s − 9·29-s + 6·31-s − 36·37-s + 27·39-s + 6·41-s − 12·43-s + 21·47-s + 30·49-s + 36·51-s + 36·53-s − 54·57-s − 6·61-s + 60·67-s + 63·69-s − 36·71-s − 60·73-s − 3·79-s + 84·81-s − 6·83-s − 27·87-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4·9-s + 2.49·13-s + 2.91·17-s − 4.12·19-s + 4.37·23-s + 5.77·27-s − 1.67·29-s + 1.07·31-s − 5.91·37-s + 4.32·39-s + 0.937·41-s − 1.82·43-s + 3.06·47-s + 30/7·49-s + 5.04·51-s + 4.94·53-s − 7.15·57-s − 0.768·61-s + 7.33·67-s + 7.58·69-s − 4.27·71-s − 7.02·73-s − 0.337·79-s + 28/3·81-s − 0.658·83-s − 2.89·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{36} \cdot 5^{18} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.75436\times 10^{8}\) |
| Root analytic conductor: | \(1.74192\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{36} \cdot 5^{18} \cdot 19^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.670227909\) |
| \(L(\frac12)\) | \(\approx\) | \(4.670227909\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{3} + T^{6} )^{3} \) | |
| 19 | \( 1 + 18 T + 201 T^{2} + 1655 T^{3} + 11247 T^{4} + 65157 T^{5} + 17909 p T^{6} + 1634736 T^{7} + 394893 p T^{8} + 33096611 T^{9} + 394893 p^{2} T^{10} + 1634736 p^{2} T^{11} + 17909 p^{4} T^{12} + 65157 p^{4} T^{13} + 11247 p^{5} T^{14} + 1655 p^{6} T^{15} + 201 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| good | 3 | \( 1 - p T - p T^{2} + 5 p T^{3} - p T^{4} - 10 p T^{5} + 77 T^{6} - 16 p^{2} T^{7} - 13 p^{2} T^{8} + 1088 T^{9} - 344 p T^{10} - 794 p T^{11} + 5710 T^{12} - 394 p^{2} T^{13} - 893 p^{2} T^{14} + 33998 T^{15} - 15053 p T^{16} - 17378 p T^{17} + 239095 T^{18} - 17378 p^{2} T^{19} - 15053 p^{3} T^{20} + 33998 p^{3} T^{21} - 893 p^{6} T^{22} - 394 p^{7} T^{23} + 5710 p^{6} T^{24} - 794 p^{8} T^{25} - 344 p^{9} T^{26} + 1088 p^{9} T^{27} - 13 p^{12} T^{28} - 16 p^{13} T^{29} + 77 p^{12} T^{30} - 10 p^{14} T^{31} - p^{15} T^{32} + 5 p^{16} T^{33} - p^{17} T^{34} - p^{18} T^{35} + p^{18} T^{36} \) |
| 7 | \( 1 - 30 T^{2} - 8 p T^{3} + 426 T^{4} + 1488 T^{5} - 326 p T^{6} - 17655 T^{7} - 10599 T^{8} + 14088 p T^{9} + 194235 T^{10} - 118653 T^{11} - 527565 T^{12} - 324273 T^{13} - 3934452 T^{14} - 11962751 T^{15} + 1304385 p T^{16} + 70964622 T^{17} + 145801525 T^{18} + 70964622 p T^{19} + 1304385 p^{3} T^{20} - 11962751 p^{3} T^{21} - 3934452 p^{4} T^{22} - 324273 p^{5} T^{23} - 527565 p^{6} T^{24} - 118653 p^{7} T^{25} + 194235 p^{8} T^{26} + 14088 p^{10} T^{27} - 10599 p^{10} T^{28} - 17655 p^{11} T^{29} - 326 p^{13} T^{30} + 1488 p^{13} T^{31} + 426 p^{14} T^{32} - 8 p^{16} T^{33} - 30 p^{16} T^{34} + p^{18} T^{36} \) | |
| 11 | \( 1 - 63 T^{2} + 4 p T^{3} + 2091 T^{4} - 2613 T^{5} - 46116 T^{6} + 82578 T^{7} + 738579 T^{8} - 1787012 T^{9} - 8742693 T^{10} + 29039016 T^{11} + 71264527 T^{12} - 360619665 T^{13} - 242084916 T^{14} + 3208864817 T^{15} - 3249789753 T^{16} - 13633003335 T^{17} + 63006122477 T^{18} - 13633003335 p T^{19} - 3249789753 p^{2} T^{20} + 3208864817 p^{3} T^{21} - 242084916 p^{4} T^{22} - 360619665 p^{5} T^{23} + 71264527 p^{6} T^{24} + 29039016 p^{7} T^{25} - 8742693 p^{8} T^{26} - 1787012 p^{9} T^{27} + 738579 p^{10} T^{28} + 82578 p^{11} T^{29} - 46116 p^{12} T^{30} - 2613 p^{13} T^{31} + 2091 p^{14} T^{32} + 4 p^{16} T^{33} - 63 p^{16} T^{34} + p^{18} T^{36} \) | |
| 13 | \( 1 - 9 T + 42 T^{2} - 56 T^{3} - 483 T^{4} + 2544 T^{5} - 883 T^{6} - 57462 T^{7} + 337455 T^{8} - 810543 T^{9} - 4653 p^{2} T^{10} + 14240355 T^{11} - 40507358 T^{12} - 64499622 T^{13} + 1006056672 T^{14} - 4087813045 T^{15} + 5931552477 T^{16} + 21393484542 T^{17} - 147891204575 T^{18} + 21393484542 p T^{19} + 5931552477 p^{2} T^{20} - 4087813045 p^{3} T^{21} + 1006056672 p^{4} T^{22} - 64499622 p^{5} T^{23} - 40507358 p^{6} T^{24} + 14240355 p^{7} T^{25} - 4653 p^{10} T^{26} - 810543 p^{9} T^{27} + 337455 p^{10} T^{28} - 57462 p^{11} T^{29} - 883 p^{12} T^{30} + 2544 p^{13} T^{31} - 483 p^{14} T^{32} - 56 p^{15} T^{33} + 42 p^{16} T^{34} - 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( 1 - 12 T + 48 T^{2} - 64 T^{3} + 168 T^{4} - 3246 T^{5} + 23771 T^{6} - 79803 T^{7} + 104925 T^{8} + 162366 T^{9} - 864318 T^{10} - 6851928 T^{11} + 64025579 T^{12} - 91887513 T^{13} - 1135726431 T^{14} + 8248244426 T^{15} - 24746600640 T^{16} + 42677717274 T^{17} - 113060408317 T^{18} + 42677717274 p T^{19} - 24746600640 p^{2} T^{20} + 8248244426 p^{3} T^{21} - 1135726431 p^{4} T^{22} - 91887513 p^{5} T^{23} + 64025579 p^{6} T^{24} - 6851928 p^{7} T^{25} - 864318 p^{8} T^{26} + 162366 p^{9} T^{27} + 104925 p^{10} T^{28} - 79803 p^{11} T^{29} + 23771 p^{12} T^{30} - 3246 p^{13} T^{31} + 168 p^{14} T^{32} - 64 p^{15} T^{33} + 48 p^{16} T^{34} - 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 23 | \( 1 - 21 T + 198 T^{2} - 1233 T^{3} + 6453 T^{4} - 36237 T^{5} + 241698 T^{6} - 1511247 T^{7} + 8090838 T^{8} - 43861221 T^{9} + 264728340 T^{10} - 1562024865 T^{11} + 8122560891 T^{12} - 38968468131 T^{13} + 198605895516 T^{14} - 1089829424997 T^{15} + 5564386576566 T^{16} - 25408579416960 T^{17} + 116501933790259 T^{18} - 25408579416960 p T^{19} + 5564386576566 p^{2} T^{20} - 1089829424997 p^{3} T^{21} + 198605895516 p^{4} T^{22} - 38968468131 p^{5} T^{23} + 8122560891 p^{6} T^{24} - 1562024865 p^{7} T^{25} + 264728340 p^{8} T^{26} - 43861221 p^{9} T^{27} + 8090838 p^{10} T^{28} - 1511247 p^{11} T^{29} + 241698 p^{12} T^{30} - 36237 p^{13} T^{31} + 6453 p^{14} T^{32} - 1233 p^{15} T^{33} + 198 p^{16} T^{34} - 21 p^{17} T^{35} + p^{18} T^{36} \) | |
| 29 | \( 1 + 9 T + 147 T^{2} + 1375 T^{3} + 396 p T^{4} + 100296 T^{5} + 676689 T^{6} + 4895667 T^{7} + 31889931 T^{8} + 193602848 T^{9} + 1226994852 T^{10} + 7039485624 T^{11} + 42093387622 T^{12} + 244712457144 T^{13} + 1397291787837 T^{14} + 8075635171144 T^{15} + 1556325539814 p T^{16} + 251270657250657 T^{17} + 1366372141482263 T^{18} + 251270657250657 p T^{19} + 1556325539814 p^{3} T^{20} + 8075635171144 p^{3} T^{21} + 1397291787837 p^{4} T^{22} + 244712457144 p^{5} T^{23} + 42093387622 p^{6} T^{24} + 7039485624 p^{7} T^{25} + 1226994852 p^{8} T^{26} + 193602848 p^{9} T^{27} + 31889931 p^{10} T^{28} + 4895667 p^{11} T^{29} + 676689 p^{12} T^{30} + 100296 p^{13} T^{31} + 396 p^{15} T^{32} + 1375 p^{15} T^{33} + 147 p^{16} T^{34} + 9 p^{17} T^{35} + p^{18} T^{36} \) | |
| 31 | \( 1 - 6 T - 102 T^{2} + 1224 T^{3} + 2580 T^{4} - 96879 T^{5} + 298022 T^{6} + 4022799 T^{7} - 32446188 T^{8} - 44362004 T^{9} + 1613208291 T^{10} - 4954266645 T^{11} - 41465477817 T^{12} + 359148907419 T^{13} - 32162367582 T^{14} - 12032492894270 T^{15} + 52760500526631 T^{16} + 163570141427940 T^{17} - 2355860837392087 T^{18} + 163570141427940 p T^{19} + 52760500526631 p^{2} T^{20} - 12032492894270 p^{3} T^{21} - 32162367582 p^{4} T^{22} + 359148907419 p^{5} T^{23} - 41465477817 p^{6} T^{24} - 4954266645 p^{7} T^{25} + 1613208291 p^{8} T^{26} - 44362004 p^{9} T^{27} - 32446188 p^{10} T^{28} + 4022799 p^{11} T^{29} + 298022 p^{12} T^{30} - 96879 p^{13} T^{31} + 2580 p^{14} T^{32} + 1224 p^{15} T^{33} - 102 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 37 | \( ( 1 + 18 T + 318 T^{2} + 3496 T^{3} + 37998 T^{4} + 316977 T^{5} + 2653443 T^{6} + 18306267 T^{7} + 129024912 T^{8} + 772889291 T^{9} + 129024912 p T^{10} + 18306267 p^{2} T^{11} + 2653443 p^{3} T^{12} + 316977 p^{4} T^{13} + 37998 p^{5} T^{14} + 3496 p^{6} T^{15} + 318 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 - 6 T + 63 T^{2} + 23 T^{3} - 654 T^{4} + 15651 T^{5} + 31250 T^{6} - 110562 T^{7} + 4981365 T^{8} - 17571753 T^{9} + 176508270 T^{10} - 772481352 T^{11} + 3158166590 T^{12} - 1998446358 T^{13} - 208652873376 T^{14} - 716906113312 T^{15} + 798313099356 T^{16} - 118250183353971 T^{17} + 437498682442919 T^{18} - 118250183353971 p T^{19} + 798313099356 p^{2} T^{20} - 716906113312 p^{3} T^{21} - 208652873376 p^{4} T^{22} - 1998446358 p^{5} T^{23} + 3158166590 p^{6} T^{24} - 772481352 p^{7} T^{25} + 176508270 p^{8} T^{26} - 17571753 p^{9} T^{27} + 4981365 p^{10} T^{28} - 110562 p^{11} T^{29} + 31250 p^{12} T^{30} + 15651 p^{13} T^{31} - 654 p^{14} T^{32} + 23 p^{15} T^{33} + 63 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( 1 + 12 T + 240 T^{2} + 2298 T^{3} + 28938 T^{4} + 272442 T^{5} + 2686682 T^{6} + 24657588 T^{7} + 207114888 T^{8} + 1816166398 T^{9} + 14051883372 T^{10} + 115347470916 T^{11} + 855547670352 T^{12} + 6492780744624 T^{13} + 46986553320060 T^{14} + 331376684874568 T^{15} + 2338731689634252 T^{16} + 360599735924808 p T^{17} + 105603186903554615 T^{18} + 360599735924808 p^{2} T^{19} + 2338731689634252 p^{2} T^{20} + 331376684874568 p^{3} T^{21} + 46986553320060 p^{4} T^{22} + 6492780744624 p^{5} T^{23} + 855547670352 p^{6} T^{24} + 115347470916 p^{7} T^{25} + 14051883372 p^{8} T^{26} + 1816166398 p^{9} T^{27} + 207114888 p^{10} T^{28} + 24657588 p^{11} T^{29} + 2686682 p^{12} T^{30} + 272442 p^{13} T^{31} + 28938 p^{14} T^{32} + 2298 p^{15} T^{33} + 240 p^{16} T^{34} + 12 p^{17} T^{35} + p^{18} T^{36} \) | |
| 47 | \( 1 - 21 T + 249 T^{2} - 1705 T^{3} + 7407 T^{4} + 12618 T^{5} - 583447 T^{6} + 7350318 T^{7} - 51581931 T^{8} + 208104852 T^{9} + 505734876 T^{10} - 14991808218 T^{11} + 156479774492 T^{12} - 1078804467438 T^{13} + 5418486666579 T^{14} - 8439344112262 T^{15} - 137601149581965 T^{16} + 1906230965505180 T^{17} - 14765452353581053 T^{18} + 1906230965505180 p T^{19} - 137601149581965 p^{2} T^{20} - 8439344112262 p^{3} T^{21} + 5418486666579 p^{4} T^{22} - 1078804467438 p^{5} T^{23} + 156479774492 p^{6} T^{24} - 14991808218 p^{7} T^{25} + 505734876 p^{8} T^{26} + 208104852 p^{9} T^{27} - 51581931 p^{10} T^{28} + 7350318 p^{11} T^{29} - 583447 p^{12} T^{30} + 12618 p^{13} T^{31} + 7407 p^{14} T^{32} - 1705 p^{15} T^{33} + 249 p^{16} T^{34} - 21 p^{17} T^{35} + p^{18} T^{36} \) | |
| 53 | \( 1 - 36 T + 543 T^{2} - 3827 T^{3} + 3564 T^{4} + 132189 T^{5} - 558184 T^{6} - 5599533 T^{7} + 71923494 T^{8} - 293109864 T^{9} - 52105539 T^{10} - 3345404754 T^{11} + 197001941273 T^{12} - 2257514751168 T^{13} + 9904683359790 T^{14} + 324660293363 p T^{15} - 267429424087362 T^{16} - 1446626793759954 T^{17} + 27456313834467749 T^{18} - 1446626793759954 p T^{19} - 267429424087362 p^{2} T^{20} + 324660293363 p^{4} T^{21} + 9904683359790 p^{4} T^{22} - 2257514751168 p^{5} T^{23} + 197001941273 p^{6} T^{24} - 3345404754 p^{7} T^{25} - 52105539 p^{8} T^{26} - 293109864 p^{9} T^{27} + 71923494 p^{10} T^{28} - 5599533 p^{11} T^{29} - 558184 p^{12} T^{30} + 132189 p^{13} T^{31} + 3564 p^{14} T^{32} - 3827 p^{15} T^{33} + 543 p^{16} T^{34} - 36 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( 1 - 39 T^{2} + 502 T^{3} + 1674 T^{4} - 24981 T^{5} + 169391 T^{6} - 1145193 T^{7} - 9708582 T^{8} + 1133214 T^{9} - 305490948 T^{10} - 3374588697 T^{11} - 34666849786 T^{12} + 12488280876 T^{13} + 3641009611275 T^{14} - 30343829924873 T^{15} + 182538114798531 T^{16} + 1816194158967570 T^{17} - 8149118718931753 T^{18} + 1816194158967570 p T^{19} + 182538114798531 p^{2} T^{20} - 30343829924873 p^{3} T^{21} + 3641009611275 p^{4} T^{22} + 12488280876 p^{5} T^{23} - 34666849786 p^{6} T^{24} - 3374588697 p^{7} T^{25} - 305490948 p^{8} T^{26} + 1133214 p^{9} T^{27} - 9708582 p^{10} T^{28} - 1145193 p^{11} T^{29} + 169391 p^{12} T^{30} - 24981 p^{13} T^{31} + 1674 p^{14} T^{32} + 502 p^{15} T^{33} - 39 p^{16} T^{34} + p^{18} T^{36} \) | |
| 61 | \( 1 + 6 T - 54 T^{2} - 242 T^{3} - 2442 T^{4} + 12975 T^{5} + 465655 T^{6} - 864537 T^{7} - 16904925 T^{8} + 13210122 T^{9} + 256152303 T^{10} + 11672530380 T^{11} + 29107771053 T^{12} - 648205911990 T^{13} + 5762205798 p T^{14} + 11413908552391 T^{15} + 162253277830059 T^{16} + 810700869771165 T^{17} - 14245135841588795 T^{18} + 810700869771165 p T^{19} + 162253277830059 p^{2} T^{20} + 11413908552391 p^{3} T^{21} + 5762205798 p^{5} T^{22} - 648205911990 p^{5} T^{23} + 29107771053 p^{6} T^{24} + 11672530380 p^{7} T^{25} + 256152303 p^{8} T^{26} + 13210122 p^{9} T^{27} - 16904925 p^{10} T^{28} - 864537 p^{11} T^{29} + 465655 p^{12} T^{30} + 12975 p^{13} T^{31} - 2442 p^{14} T^{32} - 242 p^{15} T^{33} - 54 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 - 60 T + 1875 T^{2} - 40109 T^{3} + 663714 T^{4} - 9185874 T^{5} + 112819378 T^{6} - 1280016735 T^{7} + 13673851101 T^{8} - 138063505173 T^{9} + 1319828951241 T^{10} - 12024607105899 T^{11} + 105417784903380 T^{12} - 895545057170898 T^{13} + 7405956957401286 T^{14} - 60041614406582225 T^{15} + 482906069978833632 T^{16} - 3897120329706299907 T^{17} + 31738431431319501145 T^{18} - 3897120329706299907 p T^{19} + 482906069978833632 p^{2} T^{20} - 60041614406582225 p^{3} T^{21} + 7405956957401286 p^{4} T^{22} - 895545057170898 p^{5} T^{23} + 105417784903380 p^{6} T^{24} - 12024607105899 p^{7} T^{25} + 1319828951241 p^{8} T^{26} - 138063505173 p^{9} T^{27} + 13673851101 p^{10} T^{28} - 1280016735 p^{11} T^{29} + 112819378 p^{12} T^{30} - 9185874 p^{13} T^{31} + 663714 p^{14} T^{32} - 40109 p^{15} T^{33} + 1875 p^{16} T^{34} - 60 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 + 36 T + 1047 T^{2} + 290 p T^{3} + 364878 T^{4} + 74442 p T^{5} + 72110120 T^{6} + 855581955 T^{7} + 9786481959 T^{8} + 99679810449 T^{9} + 991596533109 T^{10} + 8798566545159 T^{11} + 77078189271716 T^{12} + 595569207806793 T^{13} + 4664346665674401 T^{14} + 31783142899498435 T^{15} + 237955496611910838 T^{16} + 1588252243306229463 T^{17} + 13768085977761059687 T^{18} + 1588252243306229463 p T^{19} + 237955496611910838 p^{2} T^{20} + 31783142899498435 p^{3} T^{21} + 4664346665674401 p^{4} T^{22} + 595569207806793 p^{5} T^{23} + 77078189271716 p^{6} T^{24} + 8798566545159 p^{7} T^{25} + 991596533109 p^{8} T^{26} + 99679810449 p^{9} T^{27} + 9786481959 p^{10} T^{28} + 855581955 p^{11} T^{29} + 72110120 p^{12} T^{30} + 74442 p^{14} T^{31} + 364878 p^{14} T^{32} + 290 p^{16} T^{33} + 1047 p^{16} T^{34} + 36 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( 1 + 60 T + 1857 T^{2} + 40055 T^{3} + 696276 T^{4} + 10642896 T^{5} + 149545138 T^{6} + 1961277810 T^{7} + 24197042775 T^{8} + 283726628176 T^{9} + 3191093369445 T^{10} + 34560583853568 T^{11} + 360828195679945 T^{12} + 3638116815101478 T^{13} + 486492735531978 p T^{14} + 336146384240489390 T^{15} + 3086843752984411524 T^{16} + 27520566736904334933 T^{17} + \)\(23\!\cdots\!55\)\( T^{18} + 27520566736904334933 p T^{19} + 3086843752984411524 p^{2} T^{20} + 336146384240489390 p^{3} T^{21} + 486492735531978 p^{5} T^{22} + 3638116815101478 p^{5} T^{23} + 360828195679945 p^{6} T^{24} + 34560583853568 p^{7} T^{25} + 3191093369445 p^{8} T^{26} + 283726628176 p^{9} T^{27} + 24197042775 p^{10} T^{28} + 1961277810 p^{11} T^{29} + 149545138 p^{12} T^{30} + 10642896 p^{13} T^{31} + 696276 p^{14} T^{32} + 40055 p^{15} T^{33} + 1857 p^{16} T^{34} + 60 p^{17} T^{35} + p^{18} T^{36} \) | |
| 79 | \( 1 + 3 T + 285 T^{2} + 6 T^{3} + 42825 T^{4} - 104895 T^{5} + 4394561 T^{6} - 18321768 T^{7} + 406868271 T^{8} - 1576413395 T^{9} + 32753225115 T^{10} - 1489951341 p T^{11} + 2135542378668 T^{12} - 10299490089726 T^{13} + 102272756006754 T^{14} - 849297579908639 T^{15} + 5639845015570206 T^{16} - 57147948660727863 T^{17} + 384086746071217007 T^{18} - 57147948660727863 p T^{19} + 5639845015570206 p^{2} T^{20} - 849297579908639 p^{3} T^{21} + 102272756006754 p^{4} T^{22} - 10299490089726 p^{5} T^{23} + 2135542378668 p^{6} T^{24} - 1489951341 p^{8} T^{25} + 32753225115 p^{8} T^{26} - 1576413395 p^{9} T^{27} + 406868271 p^{10} T^{28} - 18321768 p^{11} T^{29} + 4394561 p^{12} T^{30} - 104895 p^{13} T^{31} + 42825 p^{14} T^{32} + 6 p^{15} T^{33} + 285 p^{16} T^{34} + 3 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 + 6 T - 450 T^{2} - 1438 T^{3} + 107829 T^{4} + 79356 T^{5} - 17956465 T^{6} + 16932225 T^{7} + 2346048624 T^{8} - 3801802206 T^{9} - 262551556683 T^{10} + 355232275185 T^{11} + 27308841915125 T^{12} - 16356492164568 T^{13} - 2766870430250763 T^{14} + 147218248317878 T^{15} + 268020431522043147 T^{16} + 11558963037606894 T^{17} - 23601723692520203929 T^{18} + 11558963037606894 p T^{19} + 268020431522043147 p^{2} T^{20} + 147218248317878 p^{3} T^{21} - 2766870430250763 p^{4} T^{22} - 16356492164568 p^{5} T^{23} + 27308841915125 p^{6} T^{24} + 355232275185 p^{7} T^{25} - 262551556683 p^{8} T^{26} - 3801802206 p^{9} T^{27} + 2346048624 p^{10} T^{28} + 16932225 p^{11} T^{29} - 17956465 p^{12} T^{30} + 79356 p^{13} T^{31} + 107829 p^{14} T^{32} - 1438 p^{15} T^{33} - 450 p^{16} T^{34} + 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 - 6 T + 129 T^{2} - 425 T^{3} + 36435 T^{4} - 80826 T^{5} + 3187968 T^{6} + 7776624 T^{7} + 559639026 T^{8} + 1319431964 T^{9} + 42670244739 T^{10} + 304616056461 T^{11} + 6833784047149 T^{12} + 29454146327049 T^{13} + 574876437279921 T^{14} + 3152275611695548 T^{15} + 77537024704662525 T^{16} + 261373670294989197 T^{17} + 5873925100914373667 T^{18} + 261373670294989197 p T^{19} + 77537024704662525 p^{2} T^{20} + 3152275611695548 p^{3} T^{21} + 574876437279921 p^{4} T^{22} + 29454146327049 p^{5} T^{23} + 6833784047149 p^{6} T^{24} + 304616056461 p^{7} T^{25} + 42670244739 p^{8} T^{26} + 1319431964 p^{9} T^{27} + 559639026 p^{10} T^{28} + 7776624 p^{11} T^{29} + 3187968 p^{12} T^{30} - 80826 p^{13} T^{31} + 36435 p^{14} T^{32} - 425 p^{15} T^{33} + 129 p^{16} T^{34} - 6 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( 1 + 57 T + 2133 T^{2} + 59351 T^{3} + 1372848 T^{4} + 27319278 T^{5} + 483744040 T^{6} + 7741385523 T^{7} + 113686459008 T^{8} + 1545432433909 T^{9} + 19617556193682 T^{10} + 233965708809759 T^{11} + 2640171634129633 T^{12} + 28376797380784203 T^{13} + 293059091369117238 T^{14} + 2936717413948183280 T^{15} + 297999183092675013 p T^{16} + \)\(28\!\cdots\!92\)\( T^{17} + \)\(27\!\cdots\!09\)\( T^{18} + \)\(28\!\cdots\!92\)\( p T^{19} + 297999183092675013 p^{3} T^{20} + 2936717413948183280 p^{3} T^{21} + 293059091369117238 p^{4} T^{22} + 28376797380784203 p^{5} T^{23} + 2640171634129633 p^{6} T^{24} + 233965708809759 p^{7} T^{25} + 19617556193682 p^{8} T^{26} + 1545432433909 p^{9} T^{27} + 113686459008 p^{10} T^{28} + 7741385523 p^{11} T^{29} + 483744040 p^{12} T^{30} + 27319278 p^{13} T^{31} + 1372848 p^{14} T^{32} + 59351 p^{15} T^{33} + 2133 p^{16} T^{34} + 57 p^{17} T^{35} + p^{18} T^{36} \) | |
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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
−2.88398827360393141804576004386, −2.83059485781771187390290782161, −2.79711242316252060685475987649, −2.71952908166086898882401784720, −2.68152783562314864104926926031, −2.66134902135247703153820156007, −2.41650122820220261354097770447, −2.29266454530154745020323890118, −2.20985224191035459686939047281, −2.13154991309907048355371829764, −2.10338935061628216512316786486, −2.01827482783724259057924323816, −1.79644525367869500679002965322, −1.71061334172644079882253470308, −1.62071034802330350571774624376, −1.60302020002330042788057809140, −1.54881373653986461206402587647, −1.47465051502578330026075542080, −1.22040151997546246337850177570, −0.996814919982386616523436531897, −0.973920049023747323381824895310, −0.955094985757206389908485166138, −0.808720931616163485852469047015, −0.78806876298540525708994465675, −0.090784322932838528747600465566, 0.090784322932838528747600465566, 0.78806876298540525708994465675, 0.808720931616163485852469047015, 0.955094985757206389908485166138, 0.973920049023747323381824895310, 0.996814919982386616523436531897, 1.22040151997546246337850177570, 1.47465051502578330026075542080, 1.54881373653986461206402587647, 1.60302020002330042788057809140, 1.62071034802330350571774624376, 1.71061334172644079882253470308, 1.79644525367869500679002965322, 2.01827482783724259057924323816, 2.10338935061628216512316786486, 2.13154991309907048355371829764, 2.20985224191035459686939047281, 2.29266454530154745020323890118, 2.41650122820220261354097770447, 2.66134902135247703153820156007, 2.68152783562314864104926926031, 2.71952908166086898882401784720, 2.79711242316252060685475987649, 2.83059485781771187390290782161, 2.88398827360393141804576004386
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.