Dirichlet series
| L(s) = 1 | − 24·5-s − 24·7-s − 372·11-s − 224·13-s + 186·16-s − 840·19-s + 288·25-s + 5.06e3·29-s + 1.71e3·31-s + 1.45e3·32-s + 576·35-s − 868·37-s + 6.00e3·41-s − 1.69e3·47-s + 288·49-s + 1.88e4·53-s + 8.92e3·55-s + 6.58e3·59-s − 5.51e3·61-s + 5.37e3·65-s − 4.70e3·67-s − 9.75e3·71-s + 1.36e4·73-s + 8.92e3·77-s − 5.05e3·79-s − 4.46e3·80-s − 2.47e4·83-s + ⋯ |
| L(s) = 1 | − 0.959·5-s − 0.489·7-s − 3.07·11-s − 1.32·13-s + 0.726·16-s − 2.32·19-s + 0.460·25-s + 6.02·29-s + 1.78·31-s + 1.41·32-s + 0.470·35-s − 0.634·37-s + 3.56·41-s − 0.765·47-s + 0.119·49-s + 6.69·53-s + 2.95·55-s + 1.89·59-s − 1.48·61-s + 1.27·65-s − 1.04·67-s − 1.93·71-s + 2.56·73-s + 1.50·77-s − 0.810·79-s − 0.697·80-s − 3.58·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48336\times 10^{21}\) |
| Root analytic conductor: | \(3.47768\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [2]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(11.76089264\) |
| \(L(\frac12)\) | \(\approx\) | \(11.76089264\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 224 T + 40418 T^{2} + 19043840 T^{3} + 4040836317 T^{4} + 56010813280 p T^{5} + 1160139102776 p^{2} T^{6} + 17268286639392 p^{3} T^{7} + 225235277185570 p^{4} T^{8} + 260685079682208 p^{6} T^{9} + 276202748345804 p^{8} T^{10} + 260685079682208 p^{10} T^{11} + 225235277185570 p^{12} T^{12} + 17268286639392 p^{15} T^{13} + 1160139102776 p^{18} T^{14} + 56010813280 p^{21} T^{15} + 4040836317 p^{24} T^{16} + 19043840 p^{28} T^{17} + 40418 p^{32} T^{18} + 224 p^{36} T^{19} + p^{40} T^{20} \) | |
| good | 2 | \( 1 - 93 p T^{4} - 363 p^{2} T^{5} - 615 p^{4} T^{7} + 33217 T^{8} + 50571 p^{2} T^{9} + 131769 p^{3} T^{10} + 380655 p^{3} T^{11} + 1866739 p^{2} T^{12} - 2235591 p^{5} T^{13} - 769239 p^{6} T^{14} - 29045133 p^{6} T^{15} - 290349581 p^{4} T^{16} - 13910541 p^{6} T^{17} + 392155929 p^{7} T^{18} + 834250623 p^{7} T^{19} + 44248716985 p^{6} T^{20} + 834250623 p^{11} T^{21} + 392155929 p^{15} T^{22} - 13910541 p^{18} T^{23} - 290349581 p^{20} T^{24} - 29045133 p^{26} T^{25} - 769239 p^{30} T^{26} - 2235591 p^{33} T^{27} + 1866739 p^{34} T^{28} + 380655 p^{39} T^{29} + 131769 p^{43} T^{30} + 50571 p^{46} T^{31} + 33217 p^{48} T^{32} - 615 p^{56} T^{33} - 363 p^{62} T^{35} - 93 p^{65} T^{36} + p^{80} T^{40} \) |
| 5 | \( 1 + 24 T + 288 T^{2} - 96 p^{2} T^{3} - 833082 T^{4} - 29643696 T^{5} - 468641088 T^{6} - 5555597112 T^{7} + 52643903273 p T^{8} + 15277501818096 T^{9} + 353144157386976 T^{10} + 9089448892285392 T^{11} + 808879438174376 p^{3} T^{12} - 3535915739330907024 T^{13} - \)\(15\!\cdots\!68\)\( T^{14} - \)\(56\!\cdots\!92\)\( T^{15} - \)\(14\!\cdots\!38\)\( T^{16} - \)\(10\!\cdots\!64\)\( T^{17} + \)\(11\!\cdots\!48\)\( p^{2} T^{18} + \)\(40\!\cdots\!72\)\( p T^{19} + \)\(76\!\cdots\!04\)\( T^{20} + \)\(40\!\cdots\!72\)\( p^{5} T^{21} + \)\(11\!\cdots\!48\)\( p^{10} T^{22} - \)\(10\!\cdots\!64\)\( p^{12} T^{23} - \)\(14\!\cdots\!38\)\( p^{16} T^{24} - \)\(56\!\cdots\!92\)\( p^{20} T^{25} - \)\(15\!\cdots\!68\)\( p^{24} T^{26} - 3535915739330907024 p^{28} T^{27} + 808879438174376 p^{35} T^{28} + 9089448892285392 p^{36} T^{29} + 353144157386976 p^{40} T^{30} + 15277501818096 p^{44} T^{31} + 52643903273 p^{49} T^{32} - 5555597112 p^{52} T^{33} - 468641088 p^{56} T^{34} - 29643696 p^{60} T^{35} - 833082 p^{64} T^{36} - 96 p^{70} T^{37} + 288 p^{72} T^{38} + 24 p^{76} T^{39} + p^{80} T^{40} \) | |
| 7 | \( 1 + 24 T + 288 T^{2} + 536 p^{3} T^{3} + 9231938 T^{4} - 31515336 p T^{5} + 8946668960 T^{6} + 962989380664 T^{7} + 30794736155901 T^{8} + 259153231497600 T^{9} + 116133173932339456 T^{10} + 2063179916584866048 p T^{11} + \)\(29\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!08\)\( T^{13} + \)\(12\!\cdots\!64\)\( T^{14} + \)\(41\!\cdots\!52\)\( T^{15} - \)\(37\!\cdots\!14\)\( T^{16} - \)\(20\!\cdots\!64\)\( T^{17} + \)\(88\!\cdots\!84\)\( T^{18} + \)\(13\!\cdots\!20\)\( T^{19} - \)\(87\!\cdots\!80\)\( T^{20} + \)\(13\!\cdots\!20\)\( p^{4} T^{21} + \)\(88\!\cdots\!84\)\( p^{8} T^{22} - \)\(20\!\cdots\!64\)\( p^{12} T^{23} - \)\(37\!\cdots\!14\)\( p^{16} T^{24} + \)\(41\!\cdots\!52\)\( p^{20} T^{25} + \)\(12\!\cdots\!64\)\( p^{24} T^{26} - \)\(17\!\cdots\!08\)\( p^{28} T^{27} + \)\(29\!\cdots\!64\)\( p^{32} T^{28} + 2063179916584866048 p^{37} T^{29} + 116133173932339456 p^{40} T^{30} + 259153231497600 p^{44} T^{31} + 30794736155901 p^{48} T^{32} + 962989380664 p^{52} T^{33} + 8946668960 p^{56} T^{34} - 31515336 p^{61} T^{35} + 9231938 p^{64} T^{36} + 536 p^{71} T^{37} + 288 p^{72} T^{38} + 24 p^{76} T^{39} + p^{80} T^{40} \) | |
| 11 | \( 1 + 372 T + 69192 T^{2} + 13024548 T^{3} + 2580635154 T^{4} + 35436392652 p T^{5} + 51265836458568 T^{6} + 7062048338279700 T^{7} + 924124383741120253 T^{8} + \)\(11\!\cdots\!36\)\( T^{9} + \)\(16\!\cdots\!36\)\( T^{10} + \)\(22\!\cdots\!68\)\( T^{11} + \)\(33\!\cdots\!48\)\( T^{12} + \)\(49\!\cdots\!00\)\( T^{13} + \)\(66\!\cdots\!92\)\( T^{14} + \)\(72\!\cdots\!24\)\( p^{2} T^{15} + \)\(11\!\cdots\!02\)\( T^{16} + \)\(12\!\cdots\!44\)\( T^{17} + \)\(15\!\cdots\!84\)\( T^{18} + \)\(17\!\cdots\!28\)\( T^{19} + \)\(20\!\cdots\!00\)\( T^{20} + \)\(17\!\cdots\!28\)\( p^{4} T^{21} + \)\(15\!\cdots\!84\)\( p^{8} T^{22} + \)\(12\!\cdots\!44\)\( p^{12} T^{23} + \)\(11\!\cdots\!02\)\( p^{16} T^{24} + \)\(72\!\cdots\!24\)\( p^{22} T^{25} + \)\(66\!\cdots\!92\)\( p^{24} T^{26} + \)\(49\!\cdots\!00\)\( p^{28} T^{27} + \)\(33\!\cdots\!48\)\( p^{32} T^{28} + \)\(22\!\cdots\!68\)\( p^{36} T^{29} + \)\(16\!\cdots\!36\)\( p^{40} T^{30} + \)\(11\!\cdots\!36\)\( p^{44} T^{31} + 924124383741120253 p^{48} T^{32} + 7062048338279700 p^{52} T^{33} + 51265836458568 p^{56} T^{34} + 35436392652 p^{61} T^{35} + 2580635154 p^{64} T^{36} + 13024548 p^{68} T^{37} + 69192 p^{72} T^{38} + 372 p^{76} T^{39} + p^{80} T^{40} \) | |
| 17 | \( 1 - 1073132 T^{2} + 578424730366 T^{4} - 207297364271553420 T^{6} + \)\(55\!\cdots\!73\)\( T^{8} - \)\(11\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!08\)\( T^{12} - \)\(16\!\cdots\!88\)\( p T^{14} + \)\(34\!\cdots\!14\)\( T^{16} - \)\(36\!\cdots\!12\)\( T^{18} + \)\(32\!\cdots\!60\)\( T^{20} - \)\(36\!\cdots\!12\)\( p^{8} T^{22} + \)\(34\!\cdots\!14\)\( p^{16} T^{24} - \)\(16\!\cdots\!88\)\( p^{25} T^{26} + \)\(19\!\cdots\!08\)\( p^{32} T^{28} - \)\(11\!\cdots\!80\)\( p^{40} T^{30} + \)\(55\!\cdots\!73\)\( p^{48} T^{32} - 207297364271553420 p^{56} T^{34} + 578424730366 p^{64} T^{36} - 1073132 p^{72} T^{38} + p^{80} T^{40} \) | |
| 19 | \( 1 + 840 T + 352800 T^{2} + 79161464 T^{3} - 10666670974 T^{4} - 18945290399976 T^{5} - 9017573725040992 T^{6} - 3531371832470727896 T^{7} - \)\(91\!\cdots\!83\)\( T^{8} - \)\(56\!\cdots\!80\)\( T^{9} + \)\(78\!\cdots\!36\)\( T^{10} + \)\(52\!\cdots\!68\)\( T^{11} + \)\(16\!\cdots\!20\)\( T^{12} + \)\(43\!\cdots\!68\)\( T^{13} - \)\(15\!\cdots\!12\)\( T^{14} - \)\(97\!\cdots\!76\)\( T^{15} - \)\(35\!\cdots\!66\)\( T^{16} - \)\(46\!\cdots\!32\)\( T^{17} + \)\(22\!\cdots\!00\)\( T^{18} + \)\(20\!\cdots\!72\)\( T^{19} + \)\(10\!\cdots\!60\)\( T^{20} + \)\(20\!\cdots\!72\)\( p^{4} T^{21} + \)\(22\!\cdots\!00\)\( p^{8} T^{22} - \)\(46\!\cdots\!32\)\( p^{12} T^{23} - \)\(35\!\cdots\!66\)\( p^{16} T^{24} - \)\(97\!\cdots\!76\)\( p^{20} T^{25} - \)\(15\!\cdots\!12\)\( p^{24} T^{26} + \)\(43\!\cdots\!68\)\( p^{28} T^{27} + \)\(16\!\cdots\!20\)\( p^{32} T^{28} + \)\(52\!\cdots\!68\)\( p^{36} T^{29} + \)\(78\!\cdots\!36\)\( p^{40} T^{30} - \)\(56\!\cdots\!80\)\( p^{44} T^{31} - \)\(91\!\cdots\!83\)\( p^{48} T^{32} - 3531371832470727896 p^{52} T^{33} - 9017573725040992 p^{56} T^{34} - 18945290399976 p^{60} T^{35} - 10666670974 p^{64} T^{36} + 79161464 p^{68} T^{37} + 352800 p^{72} T^{38} + 840 p^{76} T^{39} + p^{80} T^{40} \) | |
| 23 | \( 1 - 2918468 T^{2} + 4149078813694 T^{4} - 3803355018323069220 T^{6} + \)\(25\!\cdots\!01\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(48\!\cdots\!96\)\( T^{12} - \)\(64\!\cdots\!60\)\( p T^{14} + \)\(36\!\cdots\!26\)\( T^{16} - \)\(75\!\cdots\!28\)\( T^{18} + \)\(17\!\cdots\!28\)\( T^{20} - \)\(75\!\cdots\!28\)\( p^{8} T^{22} + \)\(36\!\cdots\!26\)\( p^{16} T^{24} - \)\(64\!\cdots\!60\)\( p^{25} T^{26} + \)\(48\!\cdots\!96\)\( p^{32} T^{28} - \)\(12\!\cdots\!08\)\( p^{40} T^{30} + \)\(25\!\cdots\!01\)\( p^{48} T^{32} - 3803355018323069220 p^{56} T^{34} + 4149078813694 p^{64} T^{36} - 2918468 p^{72} T^{38} + p^{80} T^{40} \) | |
| 29 | \( ( 1 - 2532 T + 7367922 T^{2} - 12299675220 T^{3} + 21600680272045 T^{4} - 28054073244377280 T^{5} + 1289070220239055352 p T^{6} - \)\(40\!\cdots\!08\)\( T^{7} + \)\(43\!\cdots\!46\)\( T^{8} - \)\(39\!\cdots\!96\)\( T^{9} + \)\(36\!\cdots\!60\)\( T^{10} - \)\(39\!\cdots\!96\)\( p^{4} T^{11} + \)\(43\!\cdots\!46\)\( p^{8} T^{12} - \)\(40\!\cdots\!08\)\( p^{12} T^{13} + 1289070220239055352 p^{17} T^{14} - 28054073244377280 p^{20} T^{15} + 21600680272045 p^{24} T^{16} - 12299675220 p^{28} T^{17} + 7367922 p^{32} T^{18} - 2532 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 31 | \( 1 - 1712 T + 1465472 T^{2} - 2078808464 T^{3} + 3126843229954 T^{4} - 1555017976841232 T^{5} + 240611889464060544 T^{6} - \)\(63\!\cdots\!80\)\( T^{7} - \)\(80\!\cdots\!59\)\( T^{8} + \)\(84\!\cdots\!28\)\( p T^{9} - \)\(15\!\cdots\!12\)\( T^{10} + \)\(16\!\cdots\!64\)\( T^{11} - \)\(26\!\cdots\!04\)\( T^{12} + \)\(72\!\cdots\!04\)\( T^{13} + \)\(35\!\cdots\!12\)\( T^{14} + \)\(78\!\cdots\!48\)\( T^{15} + \)\(58\!\cdots\!86\)\( T^{16} - \)\(19\!\cdots\!16\)\( T^{17} + \)\(39\!\cdots\!12\)\( T^{18} - \)\(71\!\cdots\!88\)\( T^{19} + \)\(19\!\cdots\!88\)\( T^{20} - \)\(71\!\cdots\!88\)\( p^{4} T^{21} + \)\(39\!\cdots\!12\)\( p^{8} T^{22} - \)\(19\!\cdots\!16\)\( p^{12} T^{23} + \)\(58\!\cdots\!86\)\( p^{16} T^{24} + \)\(78\!\cdots\!48\)\( p^{20} T^{25} + \)\(35\!\cdots\!12\)\( p^{24} T^{26} + \)\(72\!\cdots\!04\)\( p^{28} T^{27} - \)\(26\!\cdots\!04\)\( p^{32} T^{28} + \)\(16\!\cdots\!64\)\( p^{36} T^{29} - \)\(15\!\cdots\!12\)\( p^{40} T^{30} + \)\(84\!\cdots\!28\)\( p^{45} T^{31} - \)\(80\!\cdots\!59\)\( p^{48} T^{32} - \)\(63\!\cdots\!80\)\( p^{52} T^{33} + 240611889464060544 p^{56} T^{34} - 1555017976841232 p^{60} T^{35} + 3126843229954 p^{64} T^{36} - 2078808464 p^{68} T^{37} + 1465472 p^{72} T^{38} - 1712 p^{76} T^{39} + p^{80} T^{40} \) | |
| 37 | \( 1 + 868 T + 376712 T^{2} - 1656617492 T^{3} - 4033498332218 T^{4} - 9142538078588916 T^{5} - 5044065071088486840 T^{6} - \)\(34\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} + \)\(25\!\cdots\!08\)\( T^{9} + \)\(43\!\cdots\!48\)\( T^{10} + \)\(29\!\cdots\!60\)\( T^{11} - \)\(17\!\cdots\!88\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} - \)\(61\!\cdots\!24\)\( T^{14} - \)\(43\!\cdots\!72\)\( T^{15} + \)\(13\!\cdots\!14\)\( T^{16} + \)\(10\!\cdots\!00\)\( T^{17} - \)\(95\!\cdots\!56\)\( p^{2} T^{18} + \)\(15\!\cdots\!44\)\( p T^{19} - \)\(14\!\cdots\!44\)\( T^{20} + \)\(15\!\cdots\!44\)\( p^{5} T^{21} - \)\(95\!\cdots\!56\)\( p^{10} T^{22} + \)\(10\!\cdots\!00\)\( p^{12} T^{23} + \)\(13\!\cdots\!14\)\( p^{16} T^{24} - \)\(43\!\cdots\!72\)\( p^{20} T^{25} - \)\(61\!\cdots\!24\)\( p^{24} T^{26} - \)\(52\!\cdots\!48\)\( p^{28} T^{27} - \)\(17\!\cdots\!88\)\( p^{32} T^{28} + \)\(29\!\cdots\!60\)\( p^{36} T^{29} + \)\(43\!\cdots\!48\)\( p^{40} T^{30} + \)\(25\!\cdots\!08\)\( p^{44} T^{31} + \)\(15\!\cdots\!61\)\( p^{48} T^{32} - \)\(34\!\cdots\!84\)\( p^{52} T^{33} - 5044065071088486840 p^{56} T^{34} - 9142538078588916 p^{60} T^{35} - 4033498332218 p^{64} T^{36} - 1656617492 p^{68} T^{37} + 376712 p^{72} T^{38} + 868 p^{76} T^{39} + p^{80} T^{40} \) | |
| 41 | \( 1 - 6000 T + 18000000 T^{2} - 37340125080 T^{3} + 56899759216998 T^{4} - 77888429365321320 T^{5} + \)\(14\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!36\)\( T^{7} + \)\(66\!\cdots\!13\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{9} + \)\(23\!\cdots\!80\)\( T^{10} - \)\(41\!\cdots\!28\)\( T^{11} + \)\(74\!\cdots\!52\)\( T^{12} - \)\(33\!\cdots\!64\)\( p T^{13} + \)\(25\!\cdots\!68\)\( T^{14} - \)\(43\!\cdots\!16\)\( T^{15} + \)\(67\!\cdots\!50\)\( T^{16} - \)\(10\!\cdots\!76\)\( T^{17} + \)\(18\!\cdots\!92\)\( T^{18} - \)\(36\!\cdots\!64\)\( T^{19} + \)\(66\!\cdots\!16\)\( T^{20} - \)\(36\!\cdots\!64\)\( p^{4} T^{21} + \)\(18\!\cdots\!92\)\( p^{8} T^{22} - \)\(10\!\cdots\!76\)\( p^{12} T^{23} + \)\(67\!\cdots\!50\)\( p^{16} T^{24} - \)\(43\!\cdots\!16\)\( p^{20} T^{25} + \)\(25\!\cdots\!68\)\( p^{24} T^{26} - \)\(33\!\cdots\!64\)\( p^{29} T^{27} + \)\(74\!\cdots\!52\)\( p^{32} T^{28} - \)\(41\!\cdots\!28\)\( p^{36} T^{29} + \)\(23\!\cdots\!80\)\( p^{40} T^{30} - \)\(12\!\cdots\!16\)\( p^{44} T^{31} + \)\(66\!\cdots\!13\)\( p^{48} T^{32} - \)\(30\!\cdots\!36\)\( p^{52} T^{33} + \)\(14\!\cdots\!00\)\( p^{56} T^{34} - 77888429365321320 p^{60} T^{35} + 56899759216998 p^{64} T^{36} - 37340125080 p^{68} T^{37} + 18000000 p^{72} T^{38} - 6000 p^{76} T^{39} + p^{80} T^{40} \) | |
| 43 | \( 1 - 33918068 T^{2} + 596219655783742 T^{4} - \)\(71\!\cdots\!08\)\( T^{6} + \)\(66\!\cdots\!05\)\( T^{8} - \)\(49\!\cdots\!68\)\( T^{10} + \)\(30\!\cdots\!00\)\( T^{12} - \)\(16\!\cdots\!36\)\( T^{14} + \)\(77\!\cdots\!94\)\( T^{16} - \)\(31\!\cdots\!92\)\( T^{18} + \)\(11\!\cdots\!04\)\( T^{20} - \)\(31\!\cdots\!92\)\( p^{8} T^{22} + \)\(77\!\cdots\!94\)\( p^{16} T^{24} - \)\(16\!\cdots\!36\)\( p^{24} T^{26} + \)\(30\!\cdots\!00\)\( p^{32} T^{28} - \)\(49\!\cdots\!68\)\( p^{40} T^{30} + \)\(66\!\cdots\!05\)\( p^{48} T^{32} - \)\(71\!\cdots\!08\)\( p^{56} T^{34} + 596219655783742 p^{64} T^{36} - 33918068 p^{72} T^{38} + p^{80} T^{40} \) | |
| 47 | \( 1 + 36 p T + 648 p^{2} T^{2} + 26276943948 T^{3} + 97293874867602 T^{4} + 108348578537962092 T^{5} + \)\(38\!\cdots\!52\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(24\!\cdots\!73\)\( T^{8} + \)\(42\!\cdots\!44\)\( T^{9} + \)\(20\!\cdots\!80\)\( T^{10} + \)\(25\!\cdots\!48\)\( T^{11} - \)\(14\!\cdots\!20\)\( T^{12} + \)\(12\!\cdots\!08\)\( T^{13} + \)\(12\!\cdots\!72\)\( T^{14} - \)\(11\!\cdots\!08\)\( T^{15} - \)\(12\!\cdots\!58\)\( T^{16} - \)\(93\!\cdots\!20\)\( T^{17} - \)\(17\!\cdots\!68\)\( T^{18} - \)\(37\!\cdots\!44\)\( T^{19} - \)\(25\!\cdots\!76\)\( T^{20} - \)\(37\!\cdots\!44\)\( p^{4} T^{21} - \)\(17\!\cdots\!68\)\( p^{8} T^{22} - \)\(93\!\cdots\!20\)\( p^{12} T^{23} - \)\(12\!\cdots\!58\)\( p^{16} T^{24} - \)\(11\!\cdots\!08\)\( p^{20} T^{25} + \)\(12\!\cdots\!72\)\( p^{24} T^{26} + \)\(12\!\cdots\!08\)\( p^{28} T^{27} - \)\(14\!\cdots\!20\)\( p^{32} T^{28} + \)\(25\!\cdots\!48\)\( p^{36} T^{29} + \)\(20\!\cdots\!80\)\( p^{40} T^{30} + \)\(42\!\cdots\!44\)\( p^{44} T^{31} + \)\(24\!\cdots\!73\)\( p^{48} T^{32} + \)\(18\!\cdots\!80\)\( p^{52} T^{33} + \)\(38\!\cdots\!52\)\( p^{56} T^{34} + 108348578537962092 p^{60} T^{35} + 97293874867602 p^{64} T^{36} + 26276943948 p^{68} T^{37} + 648 p^{74} T^{38} + 36 p^{77} T^{39} + p^{80} T^{40} \) | |
| 53 | \( ( 1 - 9408 T + 57674550 T^{2} - 284517722688 T^{3} + 1271484713910637 T^{4} - 5142906192780910080 T^{5} + \)\(18\!\cdots\!76\)\( T^{6} - \)\(64\!\cdots\!72\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} - \)\(63\!\cdots\!08\)\( T^{9} + \)\(18\!\cdots\!68\)\( T^{10} - \)\(63\!\cdots\!08\)\( p^{4} T^{11} + \)\(20\!\cdots\!70\)\( p^{8} T^{12} - \)\(64\!\cdots\!72\)\( p^{12} T^{13} + \)\(18\!\cdots\!76\)\( p^{16} T^{14} - 5142906192780910080 p^{20} T^{15} + 1271484713910637 p^{24} T^{16} - 284517722688 p^{28} T^{17} + 57674550 p^{32} T^{18} - 9408 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 59 | \( 1 - 6588 T + 21700872 T^{2} + 36543986484 T^{3} - 236467147283598 T^{4} - 511519892404192620 T^{5} + \)\(91\!\cdots\!44\)\( T^{6} - \)\(20\!\cdots\!28\)\( T^{7} - \)\(60\!\cdots\!67\)\( T^{8} + \)\(41\!\cdots\!32\)\( T^{9} + \)\(16\!\cdots\!68\)\( T^{10} - \)\(61\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!96\)\( T^{12} + \)\(40\!\cdots\!20\)\( T^{13} - \)\(24\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!76\)\( T^{15} + \)\(18\!\cdots\!82\)\( T^{16} - \)\(40\!\cdots\!72\)\( T^{17} - \)\(62\!\cdots\!56\)\( T^{18} + \)\(52\!\cdots\!36\)\( T^{19} - \)\(14\!\cdots\!84\)\( T^{20} + \)\(52\!\cdots\!36\)\( p^{4} T^{21} - \)\(62\!\cdots\!56\)\( p^{8} T^{22} - \)\(40\!\cdots\!72\)\( p^{12} T^{23} + \)\(18\!\cdots\!82\)\( p^{16} T^{24} + \)\(21\!\cdots\!76\)\( p^{20} T^{25} - \)\(24\!\cdots\!80\)\( p^{24} T^{26} + \)\(40\!\cdots\!20\)\( p^{28} T^{27} + \)\(15\!\cdots\!96\)\( p^{32} T^{28} - \)\(61\!\cdots\!08\)\( p^{36} T^{29} + \)\(16\!\cdots\!68\)\( p^{40} T^{30} + \)\(41\!\cdots\!32\)\( p^{44} T^{31} - \)\(60\!\cdots\!67\)\( p^{48} T^{32} - \)\(20\!\cdots\!28\)\( p^{52} T^{33} + \)\(91\!\cdots\!44\)\( p^{56} T^{34} - 511519892404192620 p^{60} T^{35} - 236467147283598 p^{64} T^{36} + 36543986484 p^{68} T^{37} + 21700872 p^{72} T^{38} - 6588 p^{76} T^{39} + p^{80} T^{40} \) | |
| 61 | \( ( 1 + 2756 T + 60622382 T^{2} + 29237585060 T^{3} + 1301988253782189 T^{4} - 2936083304258463728 T^{5} + \)\(18\!\cdots\!68\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!22\)\( T^{8} - \)\(69\!\cdots\!20\)\( T^{9} + \)\(77\!\cdots\!76\)\( T^{10} - \)\(69\!\cdots\!20\)\( p^{4} T^{11} + \)\(41\!\cdots\!22\)\( p^{8} T^{12} - \)\(75\!\cdots\!00\)\( p^{12} T^{13} + \)\(18\!\cdots\!68\)\( p^{16} T^{14} - 2936083304258463728 p^{20} T^{15} + 1301988253782189 p^{24} T^{16} + 29237585060 p^{28} T^{17} + 60622382 p^{32} T^{18} + 2756 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 67 | \( 1 + 4704 T + 11063808 T^{2} + 75339719264 T^{3} + 474170561130434 T^{4} - 1249918289696962656 T^{5} - \)\(82\!\cdots\!48\)\( T^{6} - \)\(67\!\cdots\!96\)\( T^{7} - \)\(43\!\cdots\!11\)\( T^{8} - \)\(19\!\cdots\!76\)\( T^{9} - \)\(47\!\cdots\!60\)\( T^{10} - \)\(32\!\cdots\!64\)\( T^{11} + \)\(21\!\cdots\!20\)\( T^{12} + \)\(44\!\cdots\!84\)\( T^{13} + \)\(27\!\cdots\!56\)\( T^{14} + \)\(11\!\cdots\!32\)\( T^{15} + \)\(62\!\cdots\!14\)\( T^{16} + \)\(23\!\cdots\!68\)\( T^{17} + \)\(65\!\cdots\!32\)\( T^{18} - \)\(18\!\cdots\!60\)\( T^{19} - \)\(17\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!60\)\( p^{4} T^{21} + \)\(65\!\cdots\!32\)\( p^{8} T^{22} + \)\(23\!\cdots\!68\)\( p^{12} T^{23} + \)\(62\!\cdots\!14\)\( p^{16} T^{24} + \)\(11\!\cdots\!32\)\( p^{20} T^{25} + \)\(27\!\cdots\!56\)\( p^{24} T^{26} + \)\(44\!\cdots\!84\)\( p^{28} T^{27} + \)\(21\!\cdots\!20\)\( p^{32} T^{28} - \)\(32\!\cdots\!64\)\( p^{36} T^{29} - \)\(47\!\cdots\!60\)\( p^{40} T^{30} - \)\(19\!\cdots\!76\)\( p^{44} T^{31} - \)\(43\!\cdots\!11\)\( p^{48} T^{32} - \)\(67\!\cdots\!96\)\( p^{52} T^{33} - \)\(82\!\cdots\!48\)\( p^{56} T^{34} - 1249918289696962656 p^{60} T^{35} + 474170561130434 p^{64} T^{36} + 75339719264 p^{68} T^{37} + 11063808 p^{72} T^{38} + 4704 p^{76} T^{39} + p^{80} T^{40} \) | |
| 71 | \( 1 + 9756 T + 47589768 T^{2} - 41704370580 T^{3} - 2479773902451342 T^{4} - 13728535104929956404 T^{5} - \)\(21\!\cdots\!08\)\( p T^{6} + \)\(24\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!05\)\( T^{8} + \)\(13\!\cdots\!40\)\( T^{9} - \)\(56\!\cdots\!48\)\( T^{10} + \)\(17\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!96\)\( T^{12} + \)\(23\!\cdots\!84\)\( T^{13} - \)\(43\!\cdots\!80\)\( T^{14} - \)\(36\!\cdots\!84\)\( T^{15} - \)\(91\!\cdots\!34\)\( T^{16} + \)\(36\!\cdots\!04\)\( T^{17} + \)\(29\!\cdots\!08\)\( T^{18} + \)\(53\!\cdots\!60\)\( T^{19} - \)\(36\!\cdots\!68\)\( T^{20} + \)\(53\!\cdots\!60\)\( p^{4} T^{21} + \)\(29\!\cdots\!08\)\( p^{8} T^{22} + \)\(36\!\cdots\!04\)\( p^{12} T^{23} - \)\(91\!\cdots\!34\)\( p^{16} T^{24} - \)\(36\!\cdots\!84\)\( p^{20} T^{25} - \)\(43\!\cdots\!80\)\( p^{24} T^{26} + \)\(23\!\cdots\!84\)\( p^{28} T^{27} + \)\(16\!\cdots\!96\)\( p^{32} T^{28} + \)\(17\!\cdots\!04\)\( p^{36} T^{29} - \)\(56\!\cdots\!48\)\( p^{40} T^{30} + \)\(13\!\cdots\!40\)\( p^{44} T^{31} + \)\(13\!\cdots\!05\)\( p^{48} T^{32} + \)\(24\!\cdots\!20\)\( p^{52} T^{33} - \)\(21\!\cdots\!08\)\( p^{57} T^{34} - 13728535104929956404 p^{60} T^{35} - 2479773902451342 p^{64} T^{36} - 41704370580 p^{68} T^{37} + 47589768 p^{72} T^{38} + 9756 p^{76} T^{39} + p^{80} T^{40} \) | |
| 73 | \( 1 - 13692 T + 93735432 T^{2} - 899899866004 T^{3} + 7226667214920326 T^{4} - 28635201202088662068 T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(64\!\cdots\!68\)\( T^{7} - \)\(13\!\cdots\!47\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} - \)\(13\!\cdots\!72\)\( T^{10} + \)\(11\!\cdots\!56\)\( T^{11} - \)\(86\!\cdots\!04\)\( T^{12} + \)\(32\!\cdots\!80\)\( T^{13} - \)\(12\!\cdots\!80\)\( T^{14} + \)\(82\!\cdots\!88\)\( T^{15} - \)\(81\!\cdots\!22\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{17} + \)\(11\!\cdots\!16\)\( T^{18} - \)\(98\!\cdots\!96\)\( T^{19} + \)\(79\!\cdots\!12\)\( T^{20} - \)\(98\!\cdots\!96\)\( p^{4} T^{21} + \)\(11\!\cdots\!16\)\( p^{8} T^{22} - \)\(20\!\cdots\!80\)\( p^{12} T^{23} - \)\(81\!\cdots\!22\)\( p^{16} T^{24} + \)\(82\!\cdots\!88\)\( p^{20} T^{25} - \)\(12\!\cdots\!80\)\( p^{24} T^{26} + \)\(32\!\cdots\!80\)\( p^{28} T^{27} - \)\(86\!\cdots\!04\)\( p^{32} T^{28} + \)\(11\!\cdots\!56\)\( p^{36} T^{29} - \)\(13\!\cdots\!72\)\( p^{40} T^{30} + \)\(28\!\cdots\!84\)\( p^{44} T^{31} - \)\(13\!\cdots\!47\)\( p^{48} T^{32} - \)\(64\!\cdots\!68\)\( p^{52} T^{33} + \)\(11\!\cdots\!32\)\( p^{56} T^{34} - 28635201202088662068 p^{60} T^{35} + 7226667214920326 p^{64} T^{36} - 899899866004 p^{68} T^{37} + 93735432 p^{72} T^{38} - 13692 p^{76} T^{39} + p^{80} T^{40} \) | |
| 79 | \( ( 1 + 32 p T + 192348362 T^{2} + 512208031520 T^{3} + 20696924271533613 T^{4} + 52712332908391428928 T^{5} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(37\!\cdots\!68\)\( T^{7} + \)\(86\!\cdots\!38\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} + \)\(37\!\cdots\!60\)\( T^{10} + \)\(19\!\cdots\!12\)\( p^{4} T^{11} + \)\(86\!\cdots\!38\)\( p^{8} T^{12} + \)\(37\!\cdots\!68\)\( p^{12} T^{13} + \)\(15\!\cdots\!04\)\( p^{16} T^{14} + 52712332908391428928 p^{20} T^{15} + 20696924271533613 p^{24} T^{16} + 512208031520 p^{28} T^{17} + 192348362 p^{32} T^{18} + 32 p^{37} T^{19} + p^{40} T^{20} )^{2} \) | |
| 83 | \( 1 + 24708 T + 305242632 T^{2} + 3314813064852 T^{3} + 34464644359174386 T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!04\)\( T^{7} + \)\(18\!\cdots\!05\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} + \)\(11\!\cdots\!20\)\( T^{10} + \)\(95\!\cdots\!32\)\( T^{11} + \)\(74\!\cdots\!76\)\( T^{12} + \)\(58\!\cdots\!92\)\( T^{13} + \)\(44\!\cdots\!56\)\( T^{14} + \)\(33\!\cdots\!48\)\( T^{15} + \)\(24\!\cdots\!74\)\( T^{16} + \)\(17\!\cdots\!44\)\( T^{17} + \)\(12\!\cdots\!20\)\( T^{18} + \)\(85\!\cdots\!56\)\( T^{19} + \)\(59\!\cdots\!16\)\( T^{20} + \)\(85\!\cdots\!56\)\( p^{4} T^{21} + \)\(12\!\cdots\!20\)\( p^{8} T^{22} + \)\(17\!\cdots\!44\)\( p^{12} T^{23} + \)\(24\!\cdots\!74\)\( p^{16} T^{24} + \)\(33\!\cdots\!48\)\( p^{20} T^{25} + \)\(44\!\cdots\!56\)\( p^{24} T^{26} + \)\(58\!\cdots\!92\)\( p^{28} T^{27} + \)\(74\!\cdots\!76\)\( p^{32} T^{28} + \)\(95\!\cdots\!32\)\( p^{36} T^{29} + \)\(11\!\cdots\!20\)\( p^{40} T^{30} + \)\(14\!\cdots\!84\)\( p^{44} T^{31} + \)\(18\!\cdots\!05\)\( p^{48} T^{32} + \)\(22\!\cdots\!04\)\( p^{52} T^{33} + \)\(26\!\cdots\!00\)\( p^{56} T^{34} + \)\(31\!\cdots\!00\)\( p^{60} T^{35} + 34464644359174386 p^{64} T^{36} + 3314813064852 p^{68} T^{37} + 305242632 p^{72} T^{38} + 24708 p^{76} T^{39} + p^{80} T^{40} \) | |
| 89 | \( 1 + 36000 T + 648000000 T^{2} + 8732630210136 T^{3} + 91893272860844550 T^{4} + \)\(69\!\cdots\!60\)\( T^{5} + \)\(36\!\cdots\!48\)\( T^{6} + \)\(40\!\cdots\!56\)\( T^{7} - \)\(19\!\cdots\!63\)\( T^{8} - \)\(24\!\cdots\!16\)\( T^{9} - \)\(16\!\cdots\!84\)\( T^{10} - \)\(56\!\cdots\!08\)\( T^{11} + \)\(64\!\cdots\!40\)\( T^{12} + \)\(13\!\cdots\!28\)\( T^{13} + \)\(14\!\cdots\!44\)\( T^{14} + \)\(12\!\cdots\!76\)\( T^{15} + \)\(77\!\cdots\!06\)\( T^{16} + \)\(20\!\cdots\!28\)\( T^{17} - \)\(17\!\cdots\!16\)\( T^{18} - \)\(36\!\cdots\!60\)\( T^{19} - \)\(37\!\cdots\!88\)\( T^{20} - \)\(36\!\cdots\!60\)\( p^{4} T^{21} - \)\(17\!\cdots\!16\)\( p^{8} T^{22} + \)\(20\!\cdots\!28\)\( p^{12} T^{23} + \)\(77\!\cdots\!06\)\( p^{16} T^{24} + \)\(12\!\cdots\!76\)\( p^{20} T^{25} + \)\(14\!\cdots\!44\)\( p^{24} T^{26} + \)\(13\!\cdots\!28\)\( p^{28} T^{27} + \)\(64\!\cdots\!40\)\( p^{32} T^{28} - \)\(56\!\cdots\!08\)\( p^{36} T^{29} - \)\(16\!\cdots\!84\)\( p^{40} T^{30} - \)\(24\!\cdots\!16\)\( p^{44} T^{31} - \)\(19\!\cdots\!63\)\( p^{48} T^{32} + \)\(40\!\cdots\!56\)\( p^{52} T^{33} + \)\(36\!\cdots\!48\)\( p^{56} T^{34} + \)\(69\!\cdots\!60\)\( p^{60} T^{35} + 91893272860844550 p^{64} T^{36} + 8732630210136 p^{68} T^{37} + 648000000 p^{72} T^{38} + 36000 p^{76} T^{39} + p^{80} T^{40} \) | |
| 97 | \( 1 - 50012 T + 1250600072 T^{2} - 22640719416692 T^{3} + 374452218284704006 T^{4} - \)\(60\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!24\)\( T^{6} - \)\(12\!\cdots\!36\)\( T^{7} + \)\(16\!\cdots\!17\)\( T^{8} - \)\(21\!\cdots\!28\)\( T^{9} + \)\(26\!\cdots\!32\)\( T^{10} - \)\(31\!\cdots\!96\)\( T^{11} + \)\(36\!\cdots\!44\)\( T^{12} - \)\(41\!\cdots\!04\)\( T^{13} + \)\(45\!\cdots\!16\)\( T^{14} - \)\(48\!\cdots\!88\)\( T^{15} + \)\(50\!\cdots\!90\)\( T^{16} - \)\(50\!\cdots\!96\)\( T^{17} + \)\(50\!\cdots\!28\)\( T^{18} - \)\(49\!\cdots\!76\)\( T^{19} + \)\(47\!\cdots\!24\)\( T^{20} - \)\(49\!\cdots\!76\)\( p^{4} T^{21} + \)\(50\!\cdots\!28\)\( p^{8} T^{22} - \)\(50\!\cdots\!96\)\( p^{12} T^{23} + \)\(50\!\cdots\!90\)\( p^{16} T^{24} - \)\(48\!\cdots\!88\)\( p^{20} T^{25} + \)\(45\!\cdots\!16\)\( p^{24} T^{26} - \)\(41\!\cdots\!04\)\( p^{28} T^{27} + \)\(36\!\cdots\!44\)\( p^{32} T^{28} - \)\(31\!\cdots\!96\)\( p^{36} T^{29} + \)\(26\!\cdots\!32\)\( p^{40} T^{30} - \)\(21\!\cdots\!28\)\( p^{44} T^{31} + \)\(16\!\cdots\!17\)\( p^{48} T^{32} - \)\(12\!\cdots\!36\)\( p^{52} T^{33} + \)\(91\!\cdots\!24\)\( p^{56} T^{34} - \)\(60\!\cdots\!52\)\( p^{60} T^{35} + 374452218284704006 p^{64} T^{36} - 22640719416692 p^{68} T^{37} + 1250600072 p^{72} T^{38} - 50012 p^{76} T^{39} + p^{80} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.64462855924011763066993579641, −2.60258134612032465408700106784, −2.56487414250605582795134832887, −2.52803935147116536265641460351, −2.47986450412030259638970478020, −2.44945120962235741826858481936, −2.11140452653785743868614569657, −1.85353005149474058836004056141, −1.84899427688821475464608887563, −1.82937540613067678467155963512, −1.81887718089371900108628078389, −1.69570384664525355752186937873, −1.64557471662293249777578958899, −1.24338494762895988061052666745, −1.19570359007844755347932927938, −0.912317347052984556862876228748, −0.823089618032236632816033912097, −0.78726964986329520113861952905, −0.77468836102483979804974298519, −0.70744340366295474961971927299, −0.68255469668027603357802883204, −0.51778617057840728700112136829, −0.38081814352997705483669248014, −0.18599498228617313607110483223, −0.14840772630651013020755392450, 0.14840772630651013020755392450, 0.18599498228617313607110483223, 0.38081814352997705483669248014, 0.51778617057840728700112136829, 0.68255469668027603357802883204, 0.70744340366295474961971927299, 0.77468836102483979804974298519, 0.78726964986329520113861952905, 0.823089618032236632816033912097, 0.912317347052984556862876228748, 1.19570359007844755347932927938, 1.24338494762895988061052666745, 1.64557471662293249777578958899, 1.69570384664525355752186937873, 1.81887718089371900108628078389, 1.82937540613067678467155963512, 1.84899427688821475464608887563, 1.85353005149474058836004056141, 2.11140452653785743868614569657, 2.44945120962235741826858481936, 2.47986450412030259638970478020, 2.52803935147116536265641460351, 2.56487414250605582795134832887, 2.60258134612032465408700106784, 2.64462855924011763066993579641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.