Dirichlet series
| L(s) = 1 | − 4·2-s − 3-s − 2·4-s − 14·5-s + 4·6-s − 5·7-s + 26·8-s − 22·9-s + 56·10-s − 17·11-s + 2·12-s − 6·13-s + 20·14-s + 14·15-s − 11·16-s − 19·17-s + 88·18-s − 9·19-s + 28·20-s + 5·21-s + 68·22-s − 13·23-s − 26·24-s + 65·25-s + 24·26-s + 17·27-s + 10·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.577·3-s − 4-s − 6.26·5-s + 1.63·6-s − 1.88·7-s + 9.19·8-s − 7.33·9-s + 17.7·10-s − 5.12·11-s + 0.577·12-s − 1.66·13-s + 5.34·14-s + 3.61·15-s − 2.75·16-s − 4.60·17-s + 20.7·18-s − 2.06·19-s + 6.26·20-s + 1.09·21-s + 14.4·22-s − 2.71·23-s − 5.30·24-s + 13·25-s + 4.70·26-s + 3.27·27-s + 1.88·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{17} \cdot 89^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{17} \cdot 89^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(11^{17} \cdot 89^{17}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.52060\times 10^{15}\) |
| Root analytic conductor: | \(2.79595\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(17\) |
| Selberg data: | \((34,\ 11^{17} \cdot 89^{17} ,\ ( \ : [1/2]^{17} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + T )^{17} \) |
| 89 | \( ( 1 + T )^{17} \) | |
| good | 2 | \( 1 + p^{2} T + 9 p T^{2} + 27 p T^{3} + 159 T^{4} + 395 T^{5} + 949 T^{6} + 1029 p T^{7} + 4319 T^{8} + 8423 T^{9} + 15943 T^{10} + 14225 p T^{11} + 3087 p^{4} T^{12} + 81495 T^{13} + 32747 p^{2} T^{14} + 100537 p T^{15} + 300885 T^{16} + 430927 T^{17} + 300885 p T^{18} + 100537 p^{3} T^{19} + 32747 p^{5} T^{20} + 81495 p^{4} T^{21} + 3087 p^{9} T^{22} + 14225 p^{7} T^{23} + 15943 p^{7} T^{24} + 8423 p^{8} T^{25} + 4319 p^{9} T^{26} + 1029 p^{11} T^{27} + 949 p^{11} T^{28} + 395 p^{12} T^{29} + 159 p^{13} T^{30} + 27 p^{15} T^{31} + 9 p^{16} T^{32} + p^{18} T^{33} + p^{17} T^{34} \) |
| 3 | \( 1 + T + 23 T^{2} + 28 T^{3} + 271 T^{4} + 359 T^{5} + 721 p T^{6} + 2872 T^{7} + 12997 T^{8} + 1814 p^{2} T^{9} + 62114 T^{10} + 71059 T^{11} + 82099 p T^{12} + 83965 p T^{13} + 848456 T^{14} + 784310 T^{15} + 891422 p T^{16} + 2345912 T^{17} + 891422 p^{2} T^{18} + 784310 p^{2} T^{19} + 848456 p^{3} T^{20} + 83965 p^{5} T^{21} + 82099 p^{6} T^{22} + 71059 p^{6} T^{23} + 62114 p^{7} T^{24} + 1814 p^{10} T^{25} + 12997 p^{9} T^{26} + 2872 p^{10} T^{27} + 721 p^{12} T^{28} + 359 p^{12} T^{29} + 271 p^{13} T^{30} + 28 p^{14} T^{31} + 23 p^{15} T^{32} + p^{16} T^{33} + p^{17} T^{34} \) | |
| 5 | \( 1 + 14 T + 131 T^{2} + 911 T^{3} + 1059 p T^{4} + 26563 T^{5} + 119199 T^{6} + 97139 p T^{7} + 1825549 T^{8} + 6380571 T^{9} + 20909286 T^{10} + 64541261 T^{11} + 188531547 T^{12} + 522555257 T^{13} + 275603612 p T^{14} + 692447164 p T^{15} + 8299660116 T^{16} + 18991153972 T^{17} + 8299660116 p T^{18} + 692447164 p^{3} T^{19} + 275603612 p^{4} T^{20} + 522555257 p^{4} T^{21} + 188531547 p^{5} T^{22} + 64541261 p^{6} T^{23} + 20909286 p^{7} T^{24} + 6380571 p^{8} T^{25} + 1825549 p^{9} T^{26} + 97139 p^{11} T^{27} + 119199 p^{11} T^{28} + 26563 p^{12} T^{29} + 1059 p^{14} T^{30} + 911 p^{14} T^{31} + 131 p^{15} T^{32} + 14 p^{16} T^{33} + p^{17} T^{34} \) | |
| 7 | \( 1 + 5 T + 71 T^{2} + 296 T^{3} + 2374 T^{4} + 8573 T^{5} + 7313 p T^{6} + 23580 p T^{7} + 818432 T^{8} + 2412124 T^{9} + 10490003 T^{10} + 585749 p^{2} T^{11} + 112701964 T^{12} + 288526451 T^{13} + 1039584627 T^{14} + 2495559947 T^{15} + 8332516145 T^{16} + 18729602046 T^{17} + 8332516145 p T^{18} + 2495559947 p^{2} T^{19} + 1039584627 p^{3} T^{20} + 288526451 p^{4} T^{21} + 112701964 p^{5} T^{22} + 585749 p^{8} T^{23} + 10490003 p^{7} T^{24} + 2412124 p^{8} T^{25} + 818432 p^{9} T^{26} + 23580 p^{11} T^{27} + 7313 p^{12} T^{28} + 8573 p^{12} T^{29} + 2374 p^{13} T^{30} + 296 p^{14} T^{31} + 71 p^{15} T^{32} + 5 p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 + 6 T + 103 T^{2} + 485 T^{3} + 4982 T^{4} + 18940 T^{5} + 156238 T^{6} + 496481 T^{7} + 3759219 T^{8} + 10425559 T^{9} + 76989697 T^{10} + 193405834 T^{11} + 1398215882 T^{12} + 248715652 p T^{13} + 22586902726 T^{14} + 3725783504 p T^{15} + 325699902400 T^{16} + 657672801078 T^{17} + 325699902400 p T^{18} + 3725783504 p^{3} T^{19} + 22586902726 p^{3} T^{20} + 248715652 p^{5} T^{21} + 1398215882 p^{5} T^{22} + 193405834 p^{6} T^{23} + 76989697 p^{7} T^{24} + 10425559 p^{8} T^{25} + 3759219 p^{9} T^{26} + 496481 p^{10} T^{27} + 156238 p^{11} T^{28} + 18940 p^{12} T^{29} + 4982 p^{13} T^{30} + 485 p^{14} T^{31} + 103 p^{15} T^{32} + 6 p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 + 19 T + 343 T^{2} + 4076 T^{3} + 45188 T^{4} + 408261 T^{5} + 3473264 T^{6} + 25911123 T^{7} + 183648279 T^{8} + 1181308181 T^{9} + 7265524250 T^{10} + 41313625251 T^{11} + 13270168655 p T^{12} + 1151006537758 T^{13} + 5655145176742 T^{14} + 26115133630523 T^{15} + 116312995118686 T^{16} + 488203688952424 T^{17} + 116312995118686 p T^{18} + 26115133630523 p^{2} T^{19} + 5655145176742 p^{3} T^{20} + 1151006537758 p^{4} T^{21} + 13270168655 p^{6} T^{22} + 41313625251 p^{6} T^{23} + 7265524250 p^{7} T^{24} + 1181308181 p^{8} T^{25} + 183648279 p^{9} T^{26} + 25911123 p^{10} T^{27} + 3473264 p^{11} T^{28} + 408261 p^{12} T^{29} + 45188 p^{13} T^{30} + 4076 p^{14} T^{31} + 343 p^{15} T^{32} + 19 p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 + 9 T + 218 T^{2} + 1692 T^{3} + 23052 T^{4} + 157436 T^{5} + 1580431 T^{6} + 9650181 T^{7} + 79172996 T^{8} + 437691504 T^{9} + 3093419038 T^{10} + 15631195181 T^{11} + 98111236251 T^{12} + 456110465137 T^{13} + 2590572247076 T^{14} + 11120048784533 T^{15} + 57817100420697 T^{16} + 229293474035064 T^{17} + 57817100420697 p T^{18} + 11120048784533 p^{2} T^{19} + 2590572247076 p^{3} T^{20} + 456110465137 p^{4} T^{21} + 98111236251 p^{5} T^{22} + 15631195181 p^{6} T^{23} + 3093419038 p^{7} T^{24} + 437691504 p^{8} T^{25} + 79172996 p^{9} T^{26} + 9650181 p^{10} T^{27} + 1580431 p^{11} T^{28} + 157436 p^{12} T^{29} + 23052 p^{13} T^{30} + 1692 p^{14} T^{31} + 218 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 23 | \( 1 + 13 T + 292 T^{2} + 3166 T^{3} + 41486 T^{4} + 384772 T^{5} + 3814472 T^{6} + 30898033 T^{7} + 254449171 T^{8} + 1829533855 T^{9} + 13079362000 T^{10} + 84476388867 T^{11} + 536841389389 T^{12} + 136519698428 p T^{13} + 17979779624001 T^{14} + 95701530349769 T^{15} + 497574883860290 T^{16} + 2415235149405822 T^{17} + 497574883860290 p T^{18} + 95701530349769 p^{2} T^{19} + 17979779624001 p^{3} T^{20} + 136519698428 p^{5} T^{21} + 536841389389 p^{5} T^{22} + 84476388867 p^{6} T^{23} + 13079362000 p^{7} T^{24} + 1829533855 p^{8} T^{25} + 254449171 p^{9} T^{26} + 30898033 p^{10} T^{27} + 3814472 p^{11} T^{28} + 384772 p^{12} T^{29} + 41486 p^{13} T^{30} + 3166 p^{14} T^{31} + 292 p^{15} T^{32} + 13 p^{16} T^{33} + p^{17} T^{34} \) | |
| 29 | \( 1 + 54 T + 1597 T^{2} + 33489 T^{3} + 552664 T^{4} + 7581944 T^{5} + 89433037 T^{6} + 927615248 T^{7} + 8597297255 T^{8} + 72069137675 T^{9} + 551846035714 T^{10} + 3893954518753 T^{11} + 25544386421542 T^{12} + 157319110670964 T^{13} + 920122539314289 T^{14} + 5178825251044550 T^{15} + 28432156586748857 T^{16} + 153888495139765366 T^{17} + 28432156586748857 p T^{18} + 5178825251044550 p^{2} T^{19} + 920122539314289 p^{3} T^{20} + 157319110670964 p^{4} T^{21} + 25544386421542 p^{5} T^{22} + 3893954518753 p^{6} T^{23} + 551846035714 p^{7} T^{24} + 72069137675 p^{8} T^{25} + 8597297255 p^{9} T^{26} + 927615248 p^{10} T^{27} + 89433037 p^{11} T^{28} + 7581944 p^{12} T^{29} + 552664 p^{13} T^{30} + 33489 p^{14} T^{31} + 1597 p^{15} T^{32} + 54 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 - 8 T + 262 T^{2} - 1920 T^{3} + 33337 T^{4} - 226684 T^{5} + 2798609 T^{6} - 17831342 T^{7} + 178014336 T^{8} - 1069895919 T^{9} + 9290156674 T^{10} - 52837059105 T^{11} + 415768519689 T^{12} - 2239112511063 T^{13} + 16309614026676 T^{14} - 83155569046504 T^{15} + 567108617542226 T^{16} - 2734694609306666 T^{17} + 567108617542226 p T^{18} - 83155569046504 p^{2} T^{19} + 16309614026676 p^{3} T^{20} - 2239112511063 p^{4} T^{21} + 415768519689 p^{5} T^{22} - 52837059105 p^{6} T^{23} + 9290156674 p^{7} T^{24} - 1069895919 p^{8} T^{25} + 178014336 p^{9} T^{26} - 17831342 p^{10} T^{27} + 2798609 p^{11} T^{28} - 226684 p^{12} T^{29} + 33337 p^{13} T^{30} - 1920 p^{14} T^{31} + 262 p^{15} T^{32} - 8 p^{16} T^{33} + p^{17} T^{34} \) | |
| 37 | \( 1 + 298 T^{2} + 261 T^{3} + 44799 T^{4} + 70375 T^{5} + 4608976 T^{6} + 9664407 T^{7} + 365527206 T^{8} + 24664339 p T^{9} + 23713347868 T^{10} + 66070514256 T^{11} + 1304426356714 T^{12} + 3851767956715 T^{13} + 62212010974931 T^{14} + 185660907062828 T^{15} + 2605327625974507 T^{16} + 7494225130142366 T^{17} + 2605327625974507 p T^{18} + 185660907062828 p^{2} T^{19} + 62212010974931 p^{3} T^{20} + 3851767956715 p^{4} T^{21} + 1304426356714 p^{5} T^{22} + 66070514256 p^{6} T^{23} + 23713347868 p^{7} T^{24} + 24664339 p^{9} T^{25} + 365527206 p^{9} T^{26} + 9664407 p^{10} T^{27} + 4608976 p^{11} T^{28} + 70375 p^{12} T^{29} + 44799 p^{13} T^{30} + 261 p^{14} T^{31} + 298 p^{15} T^{32} + p^{17} T^{34} \) | |
| 41 | \( 1 + 16 T + 453 T^{2} + 5075 T^{3} + 87568 T^{4} + 783194 T^{5} + 10699895 T^{6} + 81535800 T^{7} + 969930534 T^{8} + 6504463195 T^{9} + 70488412354 T^{10} + 424682463093 T^{11} + 4301086931145 T^{12} + 23672151743334 T^{13} + 226818876306268 T^{14} + 1155990665203640 T^{15} + 10506448109111062 T^{16} + 1223383029796138 p T^{17} + 10506448109111062 p T^{18} + 1155990665203640 p^{2} T^{19} + 226818876306268 p^{3} T^{20} + 23672151743334 p^{4} T^{21} + 4301086931145 p^{5} T^{22} + 424682463093 p^{6} T^{23} + 70488412354 p^{7} T^{24} + 6504463195 p^{8} T^{25} + 969930534 p^{9} T^{26} + 81535800 p^{10} T^{27} + 10699895 p^{11} T^{28} + 783194 p^{12} T^{29} + 87568 p^{13} T^{30} + 5075 p^{14} T^{31} + 453 p^{15} T^{32} + 16 p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 + 21 T + 635 T^{2} + 10354 T^{3} + 186562 T^{4} + 2507745 T^{5} + 34337626 T^{6} + 394999643 T^{7} + 4485566135 T^{8} + 45243138327 T^{9} + 444574794806 T^{10} + 3995892739557 T^{11} + 34791142624833 T^{12} + 281650392889266 T^{13} + 2203443284018490 T^{14} + 16171871050662483 T^{15} + 114596696498424058 T^{16} + 764855812390396882 T^{17} + 114596696498424058 p T^{18} + 16171871050662483 p^{2} T^{19} + 2203443284018490 p^{3} T^{20} + 281650392889266 p^{4} T^{21} + 34791142624833 p^{5} T^{22} + 3995892739557 p^{6} T^{23} + 444574794806 p^{7} T^{24} + 45243138327 p^{8} T^{25} + 4485566135 p^{9} T^{26} + 394999643 p^{10} T^{27} + 34337626 p^{11} T^{28} + 2507745 p^{12} T^{29} + 186562 p^{13} T^{30} + 10354 p^{14} T^{31} + 635 p^{15} T^{32} + 21 p^{16} T^{33} + p^{17} T^{34} \) | |
| 47 | \( 1 + 3 T + 250 T^{2} + 371 T^{3} + 33334 T^{4} + 22946 T^{5} + 3343522 T^{6} + 1459384 T^{7} + 279281463 T^{8} + 93945994 T^{9} + 19977389091 T^{10} + 4963694657 T^{11} + 1261694984906 T^{12} + 261531500924 T^{13} + 71819731153486 T^{14} + 13794657637652 T^{15} + 3705902147518917 T^{16} + 662753529486762 T^{17} + 3705902147518917 p T^{18} + 13794657637652 p^{2} T^{19} + 71819731153486 p^{3} T^{20} + 261531500924 p^{4} T^{21} + 1261694984906 p^{5} T^{22} + 4963694657 p^{6} T^{23} + 19977389091 p^{7} T^{24} + 93945994 p^{8} T^{25} + 279281463 p^{9} T^{26} + 1459384 p^{10} T^{27} + 3343522 p^{11} T^{28} + 22946 p^{12} T^{29} + 33334 p^{13} T^{30} + 371 p^{14} T^{31} + 250 p^{15} T^{32} + 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 + 43 T + 1410 T^{2} + 33358 T^{3} + 675584 T^{4} + 11618788 T^{5} + 179363735 T^{6} + 2483009941 T^{7} + 31645436621 T^{8} + 371340320094 T^{9} + 4073339090926 T^{10} + 41763626670105 T^{11} + 404191124601225 T^{12} + 69623198130199 p T^{13} + 31995550140310973 T^{14} + 263126417689346049 T^{15} + 2062360498416540901 T^{16} + 15370732770012071402 T^{17} + 2062360498416540901 p T^{18} + 263126417689346049 p^{2} T^{19} + 31995550140310973 p^{3} T^{20} + 69623198130199 p^{5} T^{21} + 404191124601225 p^{5} T^{22} + 41763626670105 p^{6} T^{23} + 4073339090926 p^{7} T^{24} + 371340320094 p^{8} T^{25} + 31645436621 p^{9} T^{26} + 2483009941 p^{10} T^{27} + 179363735 p^{11} T^{28} + 11618788 p^{12} T^{29} + 675584 p^{13} T^{30} + 33358 p^{14} T^{31} + 1410 p^{15} T^{32} + 43 p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 + T + 573 T^{2} + 174 T^{3} + 167621 T^{4} - 32256 T^{5} + 33135989 T^{6} - 17821500 T^{7} + 4935444973 T^{8} - 3788250046 T^{9} + 585489001793 T^{10} - 529803173225 T^{11} + 57114809849379 T^{12} - 55216064833358 T^{13} + 4669793434702739 T^{14} - 4514267039153637 T^{15} + 323526005919517784 T^{16} - 296271322575565554 T^{17} + 323526005919517784 p T^{18} - 4514267039153637 p^{2} T^{19} + 4669793434702739 p^{3} T^{20} - 55216064833358 p^{4} T^{21} + 57114809849379 p^{5} T^{22} - 529803173225 p^{6} T^{23} + 585489001793 p^{7} T^{24} - 3788250046 p^{8} T^{25} + 4935444973 p^{9} T^{26} - 17821500 p^{10} T^{27} + 33135989 p^{11} T^{28} - 32256 p^{12} T^{29} + 167621 p^{13} T^{30} + 174 p^{14} T^{31} + 573 p^{15} T^{32} + p^{16} T^{33} + p^{17} T^{34} \) | |
| 61 | \( 1 + 29 T + 821 T^{2} + 14307 T^{3} + 239969 T^{4} + 3062552 T^{5} + 38388388 T^{6} + 393572404 T^{7} + 4099525066 T^{8} + 36297546274 T^{9} + 342727213341 T^{10} + 2830417088439 T^{11} + 26052407944211 T^{12} + 211801904971628 T^{13} + 1928246262464348 T^{14} + 15309505059159386 T^{15} + 133234611245159719 T^{16} + 999350777497285962 T^{17} + 133234611245159719 p T^{18} + 15309505059159386 p^{2} T^{19} + 1928246262464348 p^{3} T^{20} + 211801904971628 p^{4} T^{21} + 26052407944211 p^{5} T^{22} + 2830417088439 p^{6} T^{23} + 342727213341 p^{7} T^{24} + 36297546274 p^{8} T^{25} + 4099525066 p^{9} T^{26} + 393572404 p^{10} T^{27} + 38388388 p^{11} T^{28} + 3062552 p^{12} T^{29} + 239969 p^{13} T^{30} + 14307 p^{14} T^{31} + 821 p^{15} T^{32} + 29 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 - 17 T + 630 T^{2} - 8620 T^{3} + 193250 T^{4} - 2319116 T^{5} + 40095189 T^{6} - 436178439 T^{7} + 6326275459 T^{8} - 63305386082 T^{9} + 804051004408 T^{10} - 7464905018994 T^{11} + 85157790538372 T^{12} - 736615213115310 T^{13} + 7668098026717573 T^{14} - 61877817993864171 T^{15} + 593986018360584894 T^{16} - 4466165774957263462 T^{17} + 593986018360584894 p T^{18} - 61877817993864171 p^{2} T^{19} + 7668098026717573 p^{3} T^{20} - 736615213115310 p^{4} T^{21} + 85157790538372 p^{5} T^{22} - 7464905018994 p^{6} T^{23} + 804051004408 p^{7} T^{24} - 63305386082 p^{8} T^{25} + 6326275459 p^{9} T^{26} - 436178439 p^{10} T^{27} + 40095189 p^{11} T^{28} - 2319116 p^{12} T^{29} + 193250 p^{13} T^{30} - 8620 p^{14} T^{31} + 630 p^{15} T^{32} - 17 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 - 6 T + 546 T^{2} - 1898 T^{3} + 144480 T^{4} - 264866 T^{5} + 26009536 T^{6} - 22506673 T^{7} + 3628005204 T^{8} - 1472288549 T^{9} + 416317804274 T^{10} - 127328340622 T^{11} + 40862963717576 T^{12} - 17063281343400 T^{13} + 3533831580868640 T^{14} - 2007114425513095 T^{15} + 275137061789603223 T^{16} - 169429507489640374 T^{17} + 275137061789603223 p T^{18} - 2007114425513095 p^{2} T^{19} + 3533831580868640 p^{3} T^{20} - 17063281343400 p^{4} T^{21} + 40862963717576 p^{5} T^{22} - 127328340622 p^{6} T^{23} + 416317804274 p^{7} T^{24} - 1472288549 p^{8} T^{25} + 3628005204 p^{9} T^{26} - 22506673 p^{10} T^{27} + 26009536 p^{11} T^{28} - 264866 p^{12} T^{29} + 144480 p^{13} T^{30} - 1898 p^{14} T^{31} + 546 p^{15} T^{32} - 6 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 - 3 T + 557 T^{2} - 1135 T^{3} + 158966 T^{4} - 217565 T^{5} + 31339097 T^{6} - 28650991 T^{7} + 4818772203 T^{8} - 2971920715 T^{9} + 613588544346 T^{10} - 260469446367 T^{11} + 66722919831829 T^{12} - 20120878779478 T^{13} + 6300805446837420 T^{14} - 1441042001995694 T^{15} + 521821363969691001 T^{16} - 103051164910028512 T^{17} + 521821363969691001 p T^{18} - 1441042001995694 p^{2} T^{19} + 6300805446837420 p^{3} T^{20} - 20120878779478 p^{4} T^{21} + 66722919831829 p^{5} T^{22} - 260469446367 p^{6} T^{23} + 613588544346 p^{7} T^{24} - 2971920715 p^{8} T^{25} + 4818772203 p^{9} T^{26} - 28650991 p^{10} T^{27} + 31339097 p^{11} T^{28} - 217565 p^{12} T^{29} + 158966 p^{13} T^{30} - 1135 p^{14} T^{31} + 557 p^{15} T^{32} - 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 79 | \( 1 + 43 T + 1674 T^{2} + 44031 T^{3} + 1053439 T^{4} + 20775458 T^{5} + 379431166 T^{6} + 6109624000 T^{7} + 92399242824 T^{8} + 1272539884208 T^{9} + 16633967321658 T^{10} + 201698435386535 T^{11} + 2339289840922100 T^{12} + 25456832642577312 T^{13} + 266420685450293861 T^{14} + 2634255279873683226 T^{15} + 25132154079465975515 T^{16} + \)\(22\!\cdots\!22\)\( T^{17} + 25132154079465975515 p T^{18} + 2634255279873683226 p^{2} T^{19} + 266420685450293861 p^{3} T^{20} + 25456832642577312 p^{4} T^{21} + 2339289840922100 p^{5} T^{22} + 201698435386535 p^{6} T^{23} + 16633967321658 p^{7} T^{24} + 1272539884208 p^{8} T^{25} + 92399242824 p^{9} T^{26} + 6109624000 p^{10} T^{27} + 379431166 p^{11} T^{28} + 20775458 p^{12} T^{29} + 1053439 p^{13} T^{30} + 44031 p^{14} T^{31} + 1674 p^{15} T^{32} + 43 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 + 14 T + 861 T^{2} + 10388 T^{3} + 348353 T^{4} + 3661745 T^{5} + 88771858 T^{6} + 816082902 T^{7} + 16124020083 T^{8} + 129450668997 T^{9} + 2247339629326 T^{10} + 15703285151793 T^{11} + 254512313957605 T^{12} + 1551730701609035 T^{13} + 24743470287825060 T^{14} + 134749296813963041 T^{15} + 2180502358260350123 T^{16} + 11195021344544404466 T^{17} + 2180502358260350123 p T^{18} + 134749296813963041 p^{2} T^{19} + 24743470287825060 p^{3} T^{20} + 1551730701609035 p^{4} T^{21} + 254512313957605 p^{5} T^{22} + 15703285151793 p^{6} T^{23} + 2247339629326 p^{7} T^{24} + 129450668997 p^{8} T^{25} + 16124020083 p^{9} T^{26} + 816082902 p^{10} T^{27} + 88771858 p^{11} T^{28} + 3661745 p^{12} T^{29} + 348353 p^{13} T^{30} + 10388 p^{14} T^{31} + 861 p^{15} T^{32} + 14 p^{16} T^{33} + p^{17} T^{34} \) | |
| 97 | \( 1 + 6 T + 785 T^{2} + 2157 T^{3} + 282976 T^{4} - 146931 T^{5} + 65810209 T^{6} - 238188177 T^{7} + 11977138846 T^{8} - 70289526848 T^{9} + 1906810005802 T^{10} - 13036264590507 T^{11} + 269361522781905 T^{12} - 1912864281550806 T^{13} + 33037970035002121 T^{14} - 239951503371819260 T^{15} + 3551451585999558523 T^{16} - 25496154379997197544 T^{17} + 3551451585999558523 p T^{18} - 239951503371819260 p^{2} T^{19} + 33037970035002121 p^{3} T^{20} - 1912864281550806 p^{4} T^{21} + 269361522781905 p^{5} T^{22} - 13036264590507 p^{6} T^{23} + 1906810005802 p^{7} T^{24} - 70289526848 p^{8} T^{25} + 11977138846 p^{9} T^{26} - 238188177 p^{10} T^{27} + 65810209 p^{11} T^{28} - 146931 p^{12} T^{29} + 282976 p^{13} T^{30} + 2157 p^{14} T^{31} + 785 p^{15} T^{32} + 6 p^{16} T^{33} + p^{17} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.13691537591248562451340020621, −3.13635287222479499484115453953, −2.95112956551799471191762512947, −2.92757900294846629724248011950, −2.76110831947460695524313044099, −2.69298713370244354322240698524, −2.62586521096373983729024079600, −2.61810725272068101817571409839, −2.60932402845929368038269620457, −2.58401049580822942450052604221, −2.51502803476804423994382214277, −2.21689419520670023050913781284, −2.18695533321708683082968726250, −2.16925224921182410398330362179, −2.12356832381138694364153435024, −1.95375984045294468791995430809, −1.94260556843987469411530916037, −1.81671180477138578904023806933, −1.80463074723794028553052443640, −1.74174006529741697204208740003, −1.46136262264299101195519446880, −1.42489331723267534000838889264, −1.26897266377966690911988066006, −1.18192008689225712647386281333, −1.06073626165509197498515269888, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.06073626165509197498515269888, 1.18192008689225712647386281333, 1.26897266377966690911988066006, 1.42489331723267534000838889264, 1.46136262264299101195519446880, 1.74174006529741697204208740003, 1.80463074723794028553052443640, 1.81671180477138578904023806933, 1.94260556843987469411530916037, 1.95375984045294468791995430809, 2.12356832381138694364153435024, 2.16925224921182410398330362179, 2.18695533321708683082968726250, 2.21689419520670023050913781284, 2.51502803476804423994382214277, 2.58401049580822942450052604221, 2.60932402845929368038269620457, 2.61810725272068101817571409839, 2.62586521096373983729024079600, 2.69298713370244354322240698524, 2.76110831947460695524313044099, 2.92757900294846629724248011950, 2.95112956551799471191762512947, 3.13635287222479499484115453953, 3.13691537591248562451340020621
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.