Dirichlet series
| L(s) = 1 | + 15·5-s − 24·8-s − 39·11-s + 159·17-s + 159·19-s + 300·23-s − 21·25-s − 81·27-s − 231·29-s − 243·31-s + 1.42e3·37-s − 360·40-s − 69·41-s − 60·43-s + 540·47-s + 429·49-s − 750·53-s − 585·55-s + 1.26e3·59-s + 1.51e3·61-s + 192·64-s − 4.81e3·67-s − 1.37e3·71-s + 672·73-s − 2.30e3·79-s − 2.49e3·83-s + 2.38e3·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.06·8-s − 1.06·11-s + 2.26·17-s + 1.91·19-s + 2.71·23-s − 0.167·25-s − 0.577·27-s − 1.47·29-s − 1.40·31-s + 6.31·37-s − 1.42·40-s − 0.262·41-s − 0.212·43-s + 1.67·47-s + 1.25·49-s − 1.94·53-s − 1.43·55-s + 2.78·59-s + 3.18·61-s + 3/8·64-s − 8.77·67-s − 2.29·71-s + 1.07·73-s − 3.27·79-s − 3.29·83-s + 3.04·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 3^{18} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.94102\times 10^{14}\) |
| Root analytic conductor: | \(2.59349\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 3^{18} \cdot 19^{18} ,\ ( \ : [3/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(13.96263321\) |
| \(L(\frac12)\) | \(\approx\) | \(13.96263321\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T^{3} + p^{6} T^{6} )^{3} \) |
| 3 | \( ( 1 + p^{3} T^{3} + p^{6} T^{6} )^{3} \) | |
| 19 | \( 1 - 159 T + 13686 T^{2} - 788305 T^{3} + 29184318 T^{4} - 505794303 p T^{5} + 3370763402 p^{2} T^{6} - 14723448843 p^{3} T^{7} + 47668417356 p^{4} T^{8} - 93456061057 p^{5} T^{9} + 47668417356 p^{7} T^{10} - 14723448843 p^{9} T^{11} + 3370763402 p^{11} T^{12} - 505794303 p^{13} T^{13} + 29184318 p^{15} T^{14} - 788305 p^{18} T^{15} + 13686 p^{21} T^{16} - 159 p^{24} T^{17} + p^{27} T^{18} \) | |
| good | 5 | \( 1 - 3 p T + 246 T^{2} - 1159 T^{3} + 54357 T^{4} - 810861 T^{5} + 3072918 p T^{6} - 136065291 T^{7} + 2462066769 T^{8} - 27741272414 T^{9} + 468353973072 T^{10} - 954076381371 p T^{11} + 76647368673307 T^{12} - 776925994590561 T^{13} + 11127654868419702 T^{14} - 107814051762344557 T^{15} + 1571072034131558013 T^{16} - 15127262425863165537 T^{17} + \)\(20\!\cdots\!71\)\( T^{18} - 15127262425863165537 p^{3} T^{19} + 1571072034131558013 p^{6} T^{20} - 107814051762344557 p^{9} T^{21} + 11127654868419702 p^{12} T^{22} - 776925994590561 p^{15} T^{23} + 76647368673307 p^{18} T^{24} - 954076381371 p^{22} T^{25} + 468353973072 p^{24} T^{26} - 27741272414 p^{27} T^{27} + 2462066769 p^{30} T^{28} - 136065291 p^{33} T^{29} + 3072918 p^{37} T^{30} - 810861 p^{39} T^{31} + 54357 p^{42} T^{32} - 1159 p^{45} T^{33} + 246 p^{48} T^{34} - 3 p^{52} T^{35} + p^{54} T^{36} \) |
| 7 | \( 1 - 429 T^{2} + 28946 T^{3} + 4584 p T^{4} - 8494734 T^{5} + 497097844 T^{6} - 577887888 T^{7} - 315655380 p^{3} T^{8} + 6443141075020 T^{9} - 2123236466892 p T^{10} - 21622627686936 p^{2} T^{11} + 64306796941752265 T^{12} - 205195797434250012 T^{13} - 7960893448015400784 T^{14} + \)\(53\!\cdots\!24\)\( T^{15} - \)\(21\!\cdots\!09\)\( T^{16} - \)\(49\!\cdots\!82\)\( T^{17} + \)\(36\!\cdots\!41\)\( T^{18} - \)\(49\!\cdots\!82\)\( p^{3} T^{19} - \)\(21\!\cdots\!09\)\( p^{6} T^{20} + \)\(53\!\cdots\!24\)\( p^{9} T^{21} - 7960893448015400784 p^{12} T^{22} - 205195797434250012 p^{15} T^{23} + 64306796941752265 p^{18} T^{24} - 21622627686936 p^{23} T^{25} - 2123236466892 p^{25} T^{26} + 6443141075020 p^{27} T^{27} - 315655380 p^{33} T^{28} - 577887888 p^{33} T^{29} + 497097844 p^{36} T^{30} - 8494734 p^{39} T^{31} + 4584 p^{43} T^{32} + 28946 p^{45} T^{33} - 429 p^{48} T^{34} + p^{54} T^{36} \) | |
| 11 | \( 1 + 39 T - 405 p T^{2} - 208918 T^{3} + 698661 p T^{4} + 439822215 T^{5} - 7208606358 T^{6} - 391072719306 T^{7} + 10182435897753 T^{8} + 26978206991482 T^{9} - 20454437241435936 T^{10} + 213179433300931974 T^{11} + 16107736074727328629 T^{12} - 43068796450244829423 p T^{13} + \)\(11\!\cdots\!06\)\( p T^{14} + \)\(11\!\cdots\!84\)\( T^{15} - \)\(30\!\cdots\!08\)\( T^{16} - \)\(87\!\cdots\!29\)\( T^{17} + \)\(31\!\cdots\!99\)\( T^{18} - \)\(87\!\cdots\!29\)\( p^{3} T^{19} - \)\(30\!\cdots\!08\)\( p^{6} T^{20} + \)\(11\!\cdots\!84\)\( p^{9} T^{21} + \)\(11\!\cdots\!06\)\( p^{13} T^{22} - 43068796450244829423 p^{16} T^{23} + 16107736074727328629 p^{18} T^{24} + 213179433300931974 p^{21} T^{25} - 20454437241435936 p^{24} T^{26} + 26978206991482 p^{27} T^{27} + 10182435897753 p^{30} T^{28} - 391072719306 p^{33} T^{29} - 7208606358 p^{36} T^{30} + 439822215 p^{39} T^{31} + 698661 p^{43} T^{32} - 208918 p^{45} T^{33} - 405 p^{49} T^{34} + 39 p^{51} T^{35} + p^{54} T^{36} \) | |
| 13 | \( 1 + 729 T^{2} - 21810 T^{3} + 2616534 T^{4} + 30974220 T^{5} + 3508964553 T^{6} - 4536838836 p T^{7} + 3698850553920 T^{8} - 1892692517983960 T^{9} + 11163691632855753 T^{10} - 381056445735906420 T^{11} + 29206728781881596064 T^{12} - \)\(89\!\cdots\!56\)\( T^{13} + \)\(56\!\cdots\!23\)\( T^{14} + \)\(28\!\cdots\!58\)\( T^{15} + \)\(11\!\cdots\!32\)\( T^{16} + \)\(40\!\cdots\!32\)\( T^{17} + \)\(28\!\cdots\!79\)\( T^{18} + \)\(40\!\cdots\!32\)\( p^{3} T^{19} + \)\(11\!\cdots\!32\)\( p^{6} T^{20} + \)\(28\!\cdots\!58\)\( p^{9} T^{21} + \)\(56\!\cdots\!23\)\( p^{12} T^{22} - \)\(89\!\cdots\!56\)\( p^{15} T^{23} + 29206728781881596064 p^{18} T^{24} - 381056445735906420 p^{21} T^{25} + 11163691632855753 p^{24} T^{26} - 1892692517983960 p^{27} T^{27} + 3698850553920 p^{30} T^{28} - 4536838836 p^{34} T^{29} + 3508964553 p^{36} T^{30} + 30974220 p^{39} T^{31} + 2616534 p^{42} T^{32} - 21810 p^{45} T^{33} + 729 p^{48} T^{34} + p^{54} T^{36} \) | |
| 17 | \( 1 - 159 T + 9756 T^{2} - 20781 T^{3} - 10288665 T^{4} - 47990625 T^{5} - 8052941346 T^{6} + 5245566451671 T^{7} + 641720909787453 T^{8} - 167926359048832170 T^{9} + 12822721320855672378 T^{10} - 48549089721795656469 T^{11} - \)\(43\!\cdots\!35\)\( T^{12} + \)\(19\!\cdots\!13\)\( T^{13} - \)\(84\!\cdots\!74\)\( T^{14} + \)\(14\!\cdots\!55\)\( T^{15} - \)\(84\!\cdots\!97\)\( T^{16} - \)\(46\!\cdots\!59\)\( T^{17} + \)\(74\!\cdots\!43\)\( T^{18} - \)\(46\!\cdots\!59\)\( p^{3} T^{19} - \)\(84\!\cdots\!97\)\( p^{6} T^{20} + \)\(14\!\cdots\!55\)\( p^{9} T^{21} - \)\(84\!\cdots\!74\)\( p^{12} T^{22} + \)\(19\!\cdots\!13\)\( p^{15} T^{23} - \)\(43\!\cdots\!35\)\( p^{18} T^{24} - 48549089721795656469 p^{21} T^{25} + 12822721320855672378 p^{24} T^{26} - 167926359048832170 p^{27} T^{27} + 641720909787453 p^{30} T^{28} + 5245566451671 p^{33} T^{29} - 8052941346 p^{36} T^{30} - 47990625 p^{39} T^{31} - 10288665 p^{42} T^{32} - 20781 p^{45} T^{33} + 9756 p^{48} T^{34} - 159 p^{51} T^{35} + p^{54} T^{36} \) | |
| 23 | \( 1 - 300 T + 62943 T^{2} - 8608990 T^{3} + 986107872 T^{4} - 70510509174 T^{5} + 2643614354067 T^{6} + 357592268215554 T^{7} - 47850568572314604 T^{8} + 3814550711942403544 T^{9} + 28856204135615952717 T^{10} + \)\(92\!\cdots\!50\)\( T^{11} - \)\(47\!\cdots\!26\)\( T^{12} + \)\(15\!\cdots\!82\)\( T^{13} - \)\(22\!\cdots\!67\)\( T^{14} + \)\(21\!\cdots\!30\)\( T^{15} - \)\(66\!\cdots\!70\)\( T^{16} - \)\(88\!\cdots\!52\)\( T^{17} + \)\(20\!\cdots\!55\)\( T^{18} - \)\(88\!\cdots\!52\)\( p^{3} T^{19} - \)\(66\!\cdots\!70\)\( p^{6} T^{20} + \)\(21\!\cdots\!30\)\( p^{9} T^{21} - \)\(22\!\cdots\!67\)\( p^{12} T^{22} + \)\(15\!\cdots\!82\)\( p^{15} T^{23} - \)\(47\!\cdots\!26\)\( p^{18} T^{24} + \)\(92\!\cdots\!50\)\( p^{21} T^{25} + 28856204135615952717 p^{24} T^{26} + 3814550711942403544 p^{27} T^{27} - 47850568572314604 p^{30} T^{28} + 357592268215554 p^{33} T^{29} + 2643614354067 p^{36} T^{30} - 70510509174 p^{39} T^{31} + 986107872 p^{42} T^{32} - 8608990 p^{45} T^{33} + 62943 p^{48} T^{34} - 300 p^{51} T^{35} + p^{54} T^{36} \) | |
| 29 | \( 1 + 231 T + 21396 T^{2} - 4585340 T^{3} - 54080943 p T^{4} - 210604776033 T^{5} + 6289978071057 T^{6} + 5349206256102483 T^{7} + 796625356471551867 T^{8} + 28703766041288539106 T^{9} - \)\(11\!\cdots\!70\)\( T^{10} - \)\(25\!\cdots\!63\)\( T^{11} - \)\(20\!\cdots\!44\)\( T^{12} + \)\(35\!\cdots\!61\)\( T^{13} + \)\(15\!\cdots\!48\)\( T^{14} + \)\(25\!\cdots\!44\)\( T^{15} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(56\!\cdots\!47\)\( T^{17} - \)\(14\!\cdots\!73\)\( T^{18} - \)\(56\!\cdots\!47\)\( p^{3} T^{19} + \)\(11\!\cdots\!05\)\( p^{6} T^{20} + \)\(25\!\cdots\!44\)\( p^{9} T^{21} + \)\(15\!\cdots\!48\)\( p^{12} T^{22} + \)\(35\!\cdots\!61\)\( p^{15} T^{23} - \)\(20\!\cdots\!44\)\( p^{18} T^{24} - \)\(25\!\cdots\!63\)\( p^{21} T^{25} - \)\(11\!\cdots\!70\)\( p^{24} T^{26} + 28703766041288539106 p^{27} T^{27} + 796625356471551867 p^{30} T^{28} + 5349206256102483 p^{33} T^{29} + 6289978071057 p^{36} T^{30} - 210604776033 p^{39} T^{31} - 54080943 p^{43} T^{32} - 4585340 p^{45} T^{33} + 21396 p^{48} T^{34} + 231 p^{51} T^{35} + p^{54} T^{36} \) | |
| 31 | \( 1 + 243 T - 193590 T^{2} - 40958129 T^{3} + 22842626355 T^{4} + 3925057886406 T^{5} - 63279440740009 p T^{6} - 262174959926675208 T^{7} + \)\(13\!\cdots\!19\)\( T^{8} + \)\(43\!\cdots\!49\)\( p T^{9} - \)\(75\!\cdots\!76\)\( T^{10} - \)\(53\!\cdots\!99\)\( T^{11} + \)\(35\!\cdots\!86\)\( T^{12} + \)\(16\!\cdots\!09\)\( T^{13} - \)\(14\!\cdots\!32\)\( T^{14} - \)\(38\!\cdots\!81\)\( T^{15} + \)\(53\!\cdots\!97\)\( T^{16} + \)\(43\!\cdots\!12\)\( T^{17} - \)\(16\!\cdots\!27\)\( T^{18} + \)\(43\!\cdots\!12\)\( p^{3} T^{19} + \)\(53\!\cdots\!97\)\( p^{6} T^{20} - \)\(38\!\cdots\!81\)\( p^{9} T^{21} - \)\(14\!\cdots\!32\)\( p^{12} T^{22} + \)\(16\!\cdots\!09\)\( p^{15} T^{23} + \)\(35\!\cdots\!86\)\( p^{18} T^{24} - \)\(53\!\cdots\!99\)\( p^{21} T^{25} - \)\(75\!\cdots\!76\)\( p^{24} T^{26} + \)\(43\!\cdots\!49\)\( p^{28} T^{27} + \)\(13\!\cdots\!19\)\( p^{30} T^{28} - 262174959926675208 p^{33} T^{29} - 63279440740009 p^{37} T^{30} + 3925057886406 p^{39} T^{31} + 22842626355 p^{42} T^{32} - 40958129 p^{45} T^{33} - 193590 p^{48} T^{34} + 243 p^{51} T^{35} + p^{54} T^{36} \) | |
| 37 | \( ( 1 - 711 T + 523818 T^{2} - 231697495 T^{3} + 100677857724 T^{4} - 33060508326603 T^{5} + 10707446897249600 T^{6} - 2840944079323392633 T^{7} + \)\(75\!\cdots\!24\)\( T^{8} - \)\(16\!\cdots\!29\)\( T^{9} + \)\(75\!\cdots\!24\)\( p^{3} T^{10} - 2840944079323392633 p^{6} T^{11} + 10707446897249600 p^{9} T^{12} - 33060508326603 p^{12} T^{13} + 100677857724 p^{15} T^{14} - 231697495 p^{18} T^{15} + 523818 p^{21} T^{16} - 711 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 41 | \( 1 + 69 T - 282606 T^{2} - 24706307 T^{3} + 41963724861 T^{4} + 5841602317572 T^{5} - 4344769047368775 T^{6} - 949581171094251162 T^{7} + \)\(32\!\cdots\!54\)\( T^{8} + \)\(11\!\cdots\!70\)\( T^{9} - \)\(16\!\cdots\!19\)\( T^{10} - \)\(98\!\cdots\!76\)\( T^{11} + \)\(81\!\cdots\!17\)\( T^{12} + \)\(68\!\cdots\!98\)\( T^{13} + \)\(81\!\cdots\!18\)\( T^{14} - \)\(34\!\cdots\!97\)\( T^{15} - \)\(10\!\cdots\!23\)\( T^{16} + \)\(87\!\cdots\!33\)\( T^{17} + \)\(84\!\cdots\!60\)\( T^{18} + \)\(87\!\cdots\!33\)\( p^{3} T^{19} - \)\(10\!\cdots\!23\)\( p^{6} T^{20} - \)\(34\!\cdots\!97\)\( p^{9} T^{21} + \)\(81\!\cdots\!18\)\( p^{12} T^{22} + \)\(68\!\cdots\!98\)\( p^{15} T^{23} + \)\(81\!\cdots\!17\)\( p^{18} T^{24} - \)\(98\!\cdots\!76\)\( p^{21} T^{25} - \)\(16\!\cdots\!19\)\( p^{24} T^{26} + \)\(11\!\cdots\!70\)\( p^{27} T^{27} + \)\(32\!\cdots\!54\)\( p^{30} T^{28} - 949581171094251162 p^{33} T^{29} - 4344769047368775 p^{36} T^{30} + 5841602317572 p^{39} T^{31} + 41963724861 p^{42} T^{32} - 24706307 p^{45} T^{33} - 282606 p^{48} T^{34} + 69 p^{51} T^{35} + p^{54} T^{36} \) | |
| 43 | \( 1 + 60 T - 119160 T^{2} - 432422 T^{3} + 1607959626 T^{4} + 2382921785586 T^{5} + 874109787509771 T^{6} - 311305402510595343 T^{7} - 90843463843545016626 T^{8} + \)\(14\!\cdots\!36\)\( T^{9} + \)\(96\!\cdots\!79\)\( T^{10} + \)\(18\!\cdots\!53\)\( T^{11} - \)\(55\!\cdots\!50\)\( T^{12} - \)\(16\!\cdots\!02\)\( T^{13} - \)\(27\!\cdots\!13\)\( T^{14} + \)\(17\!\cdots\!38\)\( T^{15} + \)\(52\!\cdots\!69\)\( T^{16} - \)\(84\!\cdots\!32\)\( T^{17} - \)\(37\!\cdots\!73\)\( T^{18} - \)\(84\!\cdots\!32\)\( p^{3} T^{19} + \)\(52\!\cdots\!69\)\( p^{6} T^{20} + \)\(17\!\cdots\!38\)\( p^{9} T^{21} - \)\(27\!\cdots\!13\)\( p^{12} T^{22} - \)\(16\!\cdots\!02\)\( p^{15} T^{23} - \)\(55\!\cdots\!50\)\( p^{18} T^{24} + \)\(18\!\cdots\!53\)\( p^{21} T^{25} + \)\(96\!\cdots\!79\)\( p^{24} T^{26} + \)\(14\!\cdots\!36\)\( p^{27} T^{27} - 90843463843545016626 p^{30} T^{28} - 311305402510595343 p^{33} T^{29} + 874109787509771 p^{36} T^{30} + 2382921785586 p^{39} T^{31} + 1607959626 p^{42} T^{32} - 432422 p^{45} T^{33} - 119160 p^{48} T^{34} + 60 p^{51} T^{35} + p^{54} T^{36} \) | |
| 47 | \( 1 - 540 T - 13203 T^{2} + 85929164 T^{3} - 11740175091 T^{4} - 3401419924638 T^{5} - 280339013623620 T^{6} + 390119430136859838 T^{7} + \)\(10\!\cdots\!75\)\( T^{8} - \)\(47\!\cdots\!38\)\( T^{9} + \)\(10\!\cdots\!49\)\( T^{10} - \)\(50\!\cdots\!70\)\( T^{11} - \)\(69\!\cdots\!11\)\( T^{12} + \)\(95\!\cdots\!60\)\( T^{13} + \)\(66\!\cdots\!15\)\( T^{14} - \)\(32\!\cdots\!50\)\( T^{15} - \)\(26\!\cdots\!45\)\( T^{16} - \)\(33\!\cdots\!46\)\( T^{17} + \)\(39\!\cdots\!59\)\( T^{18} - \)\(33\!\cdots\!46\)\( p^{3} T^{19} - \)\(26\!\cdots\!45\)\( p^{6} T^{20} - \)\(32\!\cdots\!50\)\( p^{9} T^{21} + \)\(66\!\cdots\!15\)\( p^{12} T^{22} + \)\(95\!\cdots\!60\)\( p^{15} T^{23} - \)\(69\!\cdots\!11\)\( p^{18} T^{24} - \)\(50\!\cdots\!70\)\( p^{21} T^{25} + \)\(10\!\cdots\!49\)\( p^{24} T^{26} - \)\(47\!\cdots\!38\)\( p^{27} T^{27} + \)\(10\!\cdots\!75\)\( p^{30} T^{28} + 390119430136859838 p^{33} T^{29} - 280339013623620 p^{36} T^{30} - 3401419924638 p^{39} T^{31} - 11740175091 p^{42} T^{32} + 85929164 p^{45} T^{33} - 13203 p^{48} T^{34} - 540 p^{51} T^{35} + p^{54} T^{36} \) | |
| 53 | \( 1 + 750 T + 296715 T^{2} + 25260959 T^{3} + 49791640173 T^{4} + 48476673186933 T^{5} + 24443633972110776 T^{6} + 5321550099318843423 T^{7} + \)\(26\!\cdots\!89\)\( T^{8} + \)\(21\!\cdots\!49\)\( T^{9} + \)\(10\!\cdots\!53\)\( T^{10} + \)\(27\!\cdots\!20\)\( T^{11} + \)\(88\!\cdots\!40\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{13} + \)\(33\!\cdots\!05\)\( T^{14} + \)\(10\!\cdots\!63\)\( T^{15} + \)\(26\!\cdots\!31\)\( T^{16} + \)\(16\!\cdots\!41\)\( T^{17} + \)\(80\!\cdots\!22\)\( T^{18} + \)\(16\!\cdots\!41\)\( p^{3} T^{19} + \)\(26\!\cdots\!31\)\( p^{6} T^{20} + \)\(10\!\cdots\!63\)\( p^{9} T^{21} + \)\(33\!\cdots\!05\)\( p^{12} T^{22} + \)\(67\!\cdots\!10\)\( p^{15} T^{23} + \)\(88\!\cdots\!40\)\( p^{18} T^{24} + \)\(27\!\cdots\!20\)\( p^{21} T^{25} + \)\(10\!\cdots\!53\)\( p^{24} T^{26} + \)\(21\!\cdots\!49\)\( p^{27} T^{27} + \)\(26\!\cdots\!89\)\( p^{30} T^{28} + 5321550099318843423 p^{33} T^{29} + 24443633972110776 p^{36} T^{30} + 48476673186933 p^{39} T^{31} + 49791640173 p^{42} T^{32} + 25260959 p^{45} T^{33} + 296715 p^{48} T^{34} + 750 p^{51} T^{35} + p^{54} T^{36} \) | |
| 59 | \( 1 - 1263 T + 567306 T^{2} - 109995417 T^{3} - 93850306389 T^{4} + 179900810006973 T^{5} - 116382458908768884 T^{6} + 35406951659458348389 T^{7} + \)\(17\!\cdots\!91\)\( T^{8} - \)\(12\!\cdots\!36\)\( T^{9} + \)\(91\!\cdots\!80\)\( T^{10} - \)\(33\!\cdots\!25\)\( T^{11} + \)\(32\!\cdots\!55\)\( T^{12} + \)\(59\!\cdots\!15\)\( T^{13} - \)\(46\!\cdots\!86\)\( T^{14} + \)\(15\!\cdots\!51\)\( T^{15} - \)\(11\!\cdots\!57\)\( T^{16} - \)\(27\!\cdots\!71\)\( T^{17} + \)\(20\!\cdots\!29\)\( T^{18} - \)\(27\!\cdots\!71\)\( p^{3} T^{19} - \)\(11\!\cdots\!57\)\( p^{6} T^{20} + \)\(15\!\cdots\!51\)\( p^{9} T^{21} - \)\(46\!\cdots\!86\)\( p^{12} T^{22} + \)\(59\!\cdots\!15\)\( p^{15} T^{23} + \)\(32\!\cdots\!55\)\( p^{18} T^{24} - \)\(33\!\cdots\!25\)\( p^{21} T^{25} + \)\(91\!\cdots\!80\)\( p^{24} T^{26} - \)\(12\!\cdots\!36\)\( p^{27} T^{27} + \)\(17\!\cdots\!91\)\( p^{30} T^{28} + 35406951659458348389 p^{33} T^{29} - 116382458908768884 p^{36} T^{30} + 179900810006973 p^{39} T^{31} - 93850306389 p^{42} T^{32} - 109995417 p^{45} T^{33} + 567306 p^{48} T^{34} - 1263 p^{51} T^{35} + p^{54} T^{36} \) | |
| 61 | \( 1 - 1518 T + 1199022 T^{2} - 574394181 T^{3} + 153750836442 T^{4} + 23017406948772 T^{5} - 56863188027087987 T^{6} + 41656558932007911912 T^{7} - \)\(16\!\cdots\!27\)\( T^{8} + \)\(28\!\cdots\!40\)\( T^{9} + \)\(37\!\cdots\!63\)\( T^{10} - \)\(22\!\cdots\!55\)\( T^{11} + \)\(70\!\cdots\!76\)\( T^{12} - \)\(13\!\cdots\!19\)\( T^{13} - \)\(13\!\cdots\!70\)\( T^{14} + \)\(99\!\cdots\!50\)\( T^{15} - \)\(39\!\cdots\!14\)\( T^{16} + \)\(52\!\cdots\!87\)\( T^{17} + \)\(14\!\cdots\!68\)\( T^{18} + \)\(52\!\cdots\!87\)\( p^{3} T^{19} - \)\(39\!\cdots\!14\)\( p^{6} T^{20} + \)\(99\!\cdots\!50\)\( p^{9} T^{21} - \)\(13\!\cdots\!70\)\( p^{12} T^{22} - \)\(13\!\cdots\!19\)\( p^{15} T^{23} + \)\(70\!\cdots\!76\)\( p^{18} T^{24} - \)\(22\!\cdots\!55\)\( p^{21} T^{25} + \)\(37\!\cdots\!63\)\( p^{24} T^{26} + \)\(28\!\cdots\!40\)\( p^{27} T^{27} - \)\(16\!\cdots\!27\)\( p^{30} T^{28} + 41656558932007911912 p^{33} T^{29} - 56863188027087987 p^{36} T^{30} + 23017406948772 p^{39} T^{31} + 153750836442 p^{42} T^{32} - 574394181 p^{45} T^{33} + 1199022 p^{48} T^{34} - 1518 p^{51} T^{35} + p^{54} T^{36} \) | |
| 67 | \( 1 + 4812 T + 12562071 T^{2} + 23151535670 T^{3} + 33620592779052 T^{4} + 40836666860395428 T^{5} + 43088607529172003083 T^{6} + \)\(40\!\cdots\!48\)\( T^{7} + \)\(34\!\cdots\!26\)\( T^{8} + \)\(27\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!65\)\( T^{10} + \)\(13\!\cdots\!20\)\( T^{11} + \)\(87\!\cdots\!24\)\( T^{12} + \)\(53\!\cdots\!88\)\( T^{13} + \)\(31\!\cdots\!99\)\( T^{14} + \)\(18\!\cdots\!98\)\( T^{15} + \)\(10\!\cdots\!28\)\( T^{16} + \)\(56\!\cdots\!52\)\( T^{17} + \)\(30\!\cdots\!09\)\( T^{18} + \)\(56\!\cdots\!52\)\( p^{3} T^{19} + \)\(10\!\cdots\!28\)\( p^{6} T^{20} + \)\(18\!\cdots\!98\)\( p^{9} T^{21} + \)\(31\!\cdots\!99\)\( p^{12} T^{22} + \)\(53\!\cdots\!88\)\( p^{15} T^{23} + \)\(87\!\cdots\!24\)\( p^{18} T^{24} + \)\(13\!\cdots\!20\)\( p^{21} T^{25} + \)\(19\!\cdots\!65\)\( p^{24} T^{26} + \)\(27\!\cdots\!24\)\( p^{27} T^{27} + \)\(34\!\cdots\!26\)\( p^{30} T^{28} + \)\(40\!\cdots\!48\)\( p^{33} T^{29} + 43088607529172003083 p^{36} T^{30} + 40836666860395428 p^{39} T^{31} + 33620592779052 p^{42} T^{32} + 23151535670 p^{45} T^{33} + 12562071 p^{48} T^{34} + 4812 p^{51} T^{35} + p^{54} T^{36} \) | |
| 71 | \( 1 + 1371 T + 911451 T^{2} + 155192177 T^{3} + 103001344404 T^{4} + 315950229948324 T^{5} + 342061759907953557 T^{6} + \)\(16\!\cdots\!52\)\( T^{7} + \)\(43\!\cdots\!48\)\( T^{8} + \)\(37\!\cdots\!53\)\( T^{9} + \)\(56\!\cdots\!09\)\( T^{10} + \)\(47\!\cdots\!23\)\( T^{11} + \)\(20\!\cdots\!40\)\( T^{12} + \)\(69\!\cdots\!05\)\( T^{13} + \)\(58\!\cdots\!05\)\( T^{14} + \)\(66\!\cdots\!49\)\( T^{15} + \)\(44\!\cdots\!78\)\( T^{16} + \)\(17\!\cdots\!74\)\( T^{17} + \)\(65\!\cdots\!79\)\( T^{18} + \)\(17\!\cdots\!74\)\( p^{3} T^{19} + \)\(44\!\cdots\!78\)\( p^{6} T^{20} + \)\(66\!\cdots\!49\)\( p^{9} T^{21} + \)\(58\!\cdots\!05\)\( p^{12} T^{22} + \)\(69\!\cdots\!05\)\( p^{15} T^{23} + \)\(20\!\cdots\!40\)\( p^{18} T^{24} + \)\(47\!\cdots\!23\)\( p^{21} T^{25} + \)\(56\!\cdots\!09\)\( p^{24} T^{26} + \)\(37\!\cdots\!53\)\( p^{27} T^{27} + \)\(43\!\cdots\!48\)\( p^{30} T^{28} + \)\(16\!\cdots\!52\)\( p^{33} T^{29} + 342061759907953557 p^{36} T^{30} + 315950229948324 p^{39} T^{31} + 103001344404 p^{42} T^{32} + 155192177 p^{45} T^{33} + 911451 p^{48} T^{34} + 1371 p^{51} T^{35} + p^{54} T^{36} \) | |
| 73 | \( 1 - 672 T + 799719 T^{2} - 327880981 T^{3} + 7229399607 p T^{4} - 197575919695170 T^{5} + 241522998540986914 T^{6} - \)\(11\!\cdots\!50\)\( T^{7} + \)\(15\!\cdots\!81\)\( T^{8} - \)\(61\!\cdots\!96\)\( T^{9} + \)\(64\!\cdots\!08\)\( T^{10} - \)\(30\!\cdots\!04\)\( T^{11} + \)\(31\!\cdots\!11\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{13} + \)\(13\!\cdots\!76\)\( T^{14} - \)\(61\!\cdots\!91\)\( T^{15} + \)\(58\!\cdots\!60\)\( T^{16} - \)\(31\!\cdots\!00\)\( p T^{17} + \)\(22\!\cdots\!67\)\( T^{18} - \)\(31\!\cdots\!00\)\( p^{4} T^{19} + \)\(58\!\cdots\!60\)\( p^{6} T^{20} - \)\(61\!\cdots\!91\)\( p^{9} T^{21} + \)\(13\!\cdots\!76\)\( p^{12} T^{22} - \)\(13\!\cdots\!80\)\( p^{15} T^{23} + \)\(31\!\cdots\!11\)\( p^{18} T^{24} - \)\(30\!\cdots\!04\)\( p^{21} T^{25} + \)\(64\!\cdots\!08\)\( p^{24} T^{26} - \)\(61\!\cdots\!96\)\( p^{27} T^{27} + \)\(15\!\cdots\!81\)\( p^{30} T^{28} - \)\(11\!\cdots\!50\)\( p^{33} T^{29} + 241522998540986914 p^{36} T^{30} - 197575919695170 p^{39} T^{31} + 7229399607 p^{43} T^{32} - 327880981 p^{45} T^{33} + 799719 p^{48} T^{34} - 672 p^{51} T^{35} + p^{54} T^{36} \) | |
| 79 | \( 1 + 2301 T + 2451522 T^{2} + 1874372391 T^{3} + 1439661659037 T^{4} + 1041903621624846 T^{5} + 444749154061822749 T^{6} + 58223242123863571104 T^{7} - \)\(23\!\cdots\!68\)\( T^{8} - \)\(23\!\cdots\!66\)\( T^{9} - \)\(19\!\cdots\!45\)\( T^{10} - \)\(89\!\cdots\!34\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} + \)\(79\!\cdots\!96\)\( T^{13} + \)\(12\!\cdots\!50\)\( T^{14} + \)\(48\!\cdots\!59\)\( T^{15} - \)\(22\!\cdots\!93\)\( p T^{16} + \)\(15\!\cdots\!67\)\( T^{17} + \)\(32\!\cdots\!36\)\( T^{18} + \)\(15\!\cdots\!67\)\( p^{3} T^{19} - \)\(22\!\cdots\!93\)\( p^{7} T^{20} + \)\(48\!\cdots\!59\)\( p^{9} T^{21} + \)\(12\!\cdots\!50\)\( p^{12} T^{22} + \)\(79\!\cdots\!96\)\( p^{15} T^{23} + \)\(22\!\cdots\!21\)\( p^{18} T^{24} - \)\(89\!\cdots\!34\)\( p^{21} T^{25} - \)\(19\!\cdots\!45\)\( p^{24} T^{26} - \)\(23\!\cdots\!66\)\( p^{27} T^{27} - \)\(23\!\cdots\!68\)\( p^{30} T^{28} + 58223242123863571104 p^{33} T^{29} + 444749154061822749 p^{36} T^{30} + 1041903621624846 p^{39} T^{31} + 1439661659037 p^{42} T^{32} + 1874372391 p^{45} T^{33} + 2451522 p^{48} T^{34} + 2301 p^{51} T^{35} + p^{54} T^{36} \) | |
| 83 | \( 1 + 30 p T + 1065456 T^{2} - 23567652 T^{3} + 3721342093155 T^{4} + 3830694895808646 T^{5} - 935573143773760872 T^{6} + \)\(12\!\cdots\!42\)\( T^{7} + \)\(48\!\cdots\!33\)\( T^{8} + \)\(36\!\cdots\!42\)\( T^{9} - \)\(97\!\cdots\!82\)\( T^{10} + \)\(30\!\cdots\!48\)\( T^{11} + \)\(16\!\cdots\!24\)\( T^{12} - \)\(10\!\cdots\!78\)\( T^{13} + \)\(10\!\cdots\!05\)\( T^{14} + \)\(14\!\cdots\!98\)\( T^{15} - \)\(37\!\cdots\!10\)\( T^{16} + \)\(61\!\cdots\!08\)\( T^{17} + \)\(79\!\cdots\!55\)\( T^{18} + \)\(61\!\cdots\!08\)\( p^{3} T^{19} - \)\(37\!\cdots\!10\)\( p^{6} T^{20} + \)\(14\!\cdots\!98\)\( p^{9} T^{21} + \)\(10\!\cdots\!05\)\( p^{12} T^{22} - \)\(10\!\cdots\!78\)\( p^{15} T^{23} + \)\(16\!\cdots\!24\)\( p^{18} T^{24} + \)\(30\!\cdots\!48\)\( p^{21} T^{25} - \)\(97\!\cdots\!82\)\( p^{24} T^{26} + \)\(36\!\cdots\!42\)\( p^{27} T^{27} + \)\(48\!\cdots\!33\)\( p^{30} T^{28} + \)\(12\!\cdots\!42\)\( p^{33} T^{29} - 935573143773760872 p^{36} T^{30} + 3830694895808646 p^{39} T^{31} + 3721342093155 p^{42} T^{32} - 23567652 p^{45} T^{33} + 1065456 p^{48} T^{34} + 30 p^{52} T^{35} + p^{54} T^{36} \) | |
| 89 | \( 1 - 7203 T + 26611794 T^{2} - 66959457507 T^{3} + 128800423734633 T^{4} - 201822923906812263 T^{5} + \)\(26\!\cdots\!94\)\( T^{6} - \)\(31\!\cdots\!17\)\( T^{7} + \)\(32\!\cdots\!35\)\( T^{8} - \)\(30\!\cdots\!74\)\( T^{9} + \)\(25\!\cdots\!14\)\( T^{10} - \)\(18\!\cdots\!73\)\( T^{11} + \)\(11\!\cdots\!33\)\( T^{12} - \)\(39\!\cdots\!71\)\( T^{13} - \)\(23\!\cdots\!30\)\( T^{14} + \)\(67\!\cdots\!25\)\( T^{15} - \)\(10\!\cdots\!95\)\( p T^{16} + \)\(92\!\cdots\!19\)\( T^{17} - \)\(82\!\cdots\!29\)\( T^{18} + \)\(92\!\cdots\!19\)\( p^{3} T^{19} - \)\(10\!\cdots\!95\)\( p^{7} T^{20} + \)\(67\!\cdots\!25\)\( p^{9} T^{21} - \)\(23\!\cdots\!30\)\( p^{12} T^{22} - \)\(39\!\cdots\!71\)\( p^{15} T^{23} + \)\(11\!\cdots\!33\)\( p^{18} T^{24} - \)\(18\!\cdots\!73\)\( p^{21} T^{25} + \)\(25\!\cdots\!14\)\( p^{24} T^{26} - \)\(30\!\cdots\!74\)\( p^{27} T^{27} + \)\(32\!\cdots\!35\)\( p^{30} T^{28} - \)\(31\!\cdots\!17\)\( p^{33} T^{29} + \)\(26\!\cdots\!94\)\( p^{36} T^{30} - 201822923906812263 p^{39} T^{31} + 128800423734633 p^{42} T^{32} - 66959457507 p^{45} T^{33} + 26611794 p^{48} T^{34} - 7203 p^{51} T^{35} + p^{54} T^{36} \) | |
| 97 | \( 1 + 1977 T + 3381279 T^{2} + 3025512820 T^{3} + 1531358340507 T^{4} - 889766373728709 T^{5} - 2063983446093421240 T^{6} - \)\(16\!\cdots\!13\)\( T^{7} + \)\(37\!\cdots\!61\)\( T^{8} + \)\(19\!\cdots\!36\)\( T^{9} + \)\(16\!\cdots\!19\)\( T^{10} + \)\(22\!\cdots\!25\)\( T^{11} - \)\(17\!\cdots\!36\)\( T^{12} - \)\(20\!\cdots\!75\)\( T^{13} - \)\(14\!\cdots\!79\)\( T^{14} - \)\(57\!\cdots\!40\)\( T^{15} - \)\(17\!\cdots\!99\)\( T^{16} - \)\(36\!\cdots\!73\)\( T^{17} - \)\(54\!\cdots\!70\)\( T^{18} - \)\(36\!\cdots\!73\)\( p^{3} T^{19} - \)\(17\!\cdots\!99\)\( p^{6} T^{20} - \)\(57\!\cdots\!40\)\( p^{9} T^{21} - \)\(14\!\cdots\!79\)\( p^{12} T^{22} - \)\(20\!\cdots\!75\)\( p^{15} T^{23} - \)\(17\!\cdots\!36\)\( p^{18} T^{24} + \)\(22\!\cdots\!25\)\( p^{21} T^{25} + \)\(16\!\cdots\!19\)\( p^{24} T^{26} + \)\(19\!\cdots\!36\)\( p^{27} T^{27} + \)\(37\!\cdots\!61\)\( p^{30} T^{28} - \)\(16\!\cdots\!13\)\( p^{33} T^{29} - 2063983446093421240 p^{36} T^{30} - 889766373728709 p^{39} T^{31} + 1531358340507 p^{42} T^{32} + 3025512820 p^{45} T^{33} + 3381279 p^{48} T^{34} + 1977 p^{51} T^{35} + p^{54} T^{36} \) | |
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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
−3.10094780579570374650311705907, −3.06690997837225555073830969793, −3.02820642080051744315865180988, −3.00578414075128094742081324263, −2.90619004106848331819660701051, −2.86574225652129771468589691937, −2.52943835461818877484820711040, −2.48433641606285855416172344735, −2.43141039578484915116405688816, −2.22718965673757040396347731520, −2.14542162868018825491951059258, −2.13853427033224568603723245691, −2.08230210386692083102245317186, −1.75286864516471748463411595389, −1.54462573773181508691420225049, −1.46649940577381504447608441330, −1.43138395662897402780244960913, −1.22822249945304653490391551145, −1.03818969529017731655765797494, −0.952877513198563474717349534625, −0.877245660074129718237863029423, −0.76954420461540573444207596393, −0.43903945143011044817395276315, −0.32679119148142718067125749983, −0.17177206140208725491467195673, 0.17177206140208725491467195673, 0.32679119148142718067125749983, 0.43903945143011044817395276315, 0.76954420461540573444207596393, 0.877245660074129718237863029423, 0.952877513198563474717349534625, 1.03818969529017731655765797494, 1.22822249945304653490391551145, 1.43138395662897402780244960913, 1.46649940577381504447608441330, 1.54462573773181508691420225049, 1.75286864516471748463411595389, 2.08230210386692083102245317186, 2.13853427033224568603723245691, 2.14542162868018825491951059258, 2.22718965673757040396347731520, 2.43141039578484915116405688816, 2.48433641606285855416172344735, 2.52943835461818877484820711040, 2.86574225652129771468589691937, 2.90619004106848331819660701051, 3.00578414075128094742081324263, 3.02820642080051744315865180988, 3.06690997837225555073830969793, 3.10094780579570374650311705907
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.