Dirichlet series
| L(s) = 1 | − 22·2-s − 80·4-s + 4.16e3·8-s + 3.83e5·9-s + 2.78e5·13-s + 1.07e6·16-s + 1.92e6·17-s − 8.42e6·18-s − 6.13e6·26-s − 6.60e7·29-s − 1.73e7·32-s − 4.22e7·34-s − 3.06e7·36-s + 1.53e8·37-s + 4.77e8·41-s + 2.99e9·49-s − 2.23e7·52-s + 1.66e9·53-s + 1.45e9·58-s − 4.28e9·61-s − 1.75e8·64-s − 1.53e8·68-s + 1.59e9·72-s − 2.47e9·73-s − 3.37e9·74-s + 6.97e10·81-s − 1.05e10·82-s + ⋯ |
| L(s) = 1 | − 0.687·2-s − 0.0781·4-s + 0.126·8-s + 6.48·9-s + 0.751·13-s + 1.02·16-s + 1.35·17-s − 4.45·18-s − 0.516·26-s − 3.21·29-s − 0.516·32-s − 0.930·34-s − 0.506·36-s + 2.21·37-s + 4.12·41-s + 10.5·49-s − 0.0586·52-s + 3.99·53-s + 2.21·58-s − 5.07·61-s − 0.163·64-s − 0.105·68-s + 0.823·72-s − 1.19·73-s − 1.52·74-s + 19.9·81-s − 2.83·82-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{40}\right)^{s/2} \, \Gamma_{\C}(s+5)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 5^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14910\times 10^{36}\) |
| Root analytic conductor: | \(7.97093\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 5^{40} ,\ ( \ : [5]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(3313.261213\) |
| \(L(\frac12)\) | \(\approx\) | \(3313.261213\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 11 p T + 141 p^{2} T^{2} + 1251 p^{3} T^{3} - 56283 p^{4} T^{4} - 216161 p^{7} T^{5} - 753219 p^{10} T^{6} - 4333981 p^{13} T^{7} + 8669157 p^{16} T^{8} + 20319771 p^{21} T^{9} + 23842081 p^{26} T^{10} + 20319771 p^{31} T^{11} + 8669157 p^{36} T^{12} - 4333981 p^{43} T^{13} - 753219 p^{50} T^{14} - 216161 p^{57} T^{15} - 56283 p^{64} T^{16} + 1251 p^{73} T^{17} + 141 p^{82} T^{18} + 11 p^{91} T^{19} + p^{100} T^{20} \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 - 383056 T^{2} + 77005404038 T^{4} - 132345437205968 p^{4} T^{6} + 4784887966219297727 p^{5} T^{8} - \)\(48\!\cdots\!60\)\( p^{7} T^{10} + \)\(13\!\cdots\!84\)\( p^{8} T^{12} - \)\(36\!\cdots\!56\)\( p^{11} T^{14} + \)\(31\!\cdots\!46\)\( p^{15} T^{16} - \)\(23\!\cdots\!68\)\( p^{17} T^{18} + \)\(53\!\cdots\!08\)\( p^{20} T^{20} - \)\(23\!\cdots\!68\)\( p^{37} T^{22} + \)\(31\!\cdots\!46\)\( p^{55} T^{24} - \)\(36\!\cdots\!56\)\( p^{71} T^{26} + \)\(13\!\cdots\!84\)\( p^{88} T^{28} - \)\(48\!\cdots\!60\)\( p^{107} T^{30} + 4784887966219297727 p^{125} T^{32} - 132345437205968 p^{144} T^{34} + 77005404038 p^{160} T^{36} - 383056 p^{180} T^{38} + p^{200} T^{40} \) |
| 7 | \( 1 - 427376928 p T^{2} + 4418015971776571398 T^{4} - \)\(43\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!81\)\( T^{8} - \)\(37\!\cdots\!80\)\( p^{2} T^{10} + \)\(37\!\cdots\!04\)\( p^{4} T^{12} - \)\(32\!\cdots\!88\)\( p^{6} T^{14} + \)\(24\!\cdots\!62\)\( p^{8} T^{16} - \)\(16\!\cdots\!56\)\( p^{10} T^{18} + \)\(10\!\cdots\!08\)\( p^{12} T^{20} - \)\(16\!\cdots\!56\)\( p^{30} T^{22} + \)\(24\!\cdots\!62\)\( p^{48} T^{24} - \)\(32\!\cdots\!88\)\( p^{66} T^{26} + \)\(37\!\cdots\!04\)\( p^{84} T^{28} - \)\(37\!\cdots\!80\)\( p^{102} T^{30} + \)\(31\!\cdots\!81\)\( p^{120} T^{32} - \)\(43\!\cdots\!28\)\( p^{140} T^{34} + 4418015971776571398 p^{160} T^{36} - 427376928 p^{181} T^{38} + p^{200} T^{40} \) | |
| 11 | \( 1 - 216890736420 T^{2} + \)\(24\!\cdots\!50\)\( T^{4} - \)\(15\!\cdots\!00\)\( p^{2} T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(49\!\cdots\!04\)\( T^{10} + \)\(19\!\cdots\!60\)\( T^{12} - \)\(70\!\cdots\!00\)\( T^{14} + \)\(22\!\cdots\!50\)\( T^{16} - \)\(66\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!56\)\( T^{20} - \)\(66\!\cdots\!60\)\( p^{20} T^{22} + \)\(22\!\cdots\!50\)\( p^{40} T^{24} - \)\(70\!\cdots\!00\)\( p^{60} T^{26} + \)\(19\!\cdots\!60\)\( p^{80} T^{28} - \)\(49\!\cdots\!04\)\( p^{100} T^{30} + \)\(10\!\cdots\!45\)\( p^{120} T^{32} - \)\(15\!\cdots\!00\)\( p^{142} T^{34} + \)\(24\!\cdots\!50\)\( p^{160} T^{36} - 216890736420 p^{180} T^{38} + p^{200} T^{40} \) | |
| 13 | \( ( 1 - 139432 T + 539604593094 T^{2} - 125888889228743848 T^{3} + \)\(14\!\cdots\!97\)\( T^{4} - \)\(42\!\cdots\!52\)\( T^{5} + \)\(26\!\cdots\!24\)\( T^{6} - \)\(92\!\cdots\!88\)\( T^{7} + \)\(38\!\cdots\!02\)\( T^{8} - \)\(15\!\cdots\!52\)\( T^{9} + \)\(52\!\cdots\!64\)\( T^{10} - \)\(15\!\cdots\!52\)\( p^{10} T^{11} + \)\(38\!\cdots\!02\)\( p^{20} T^{12} - \)\(92\!\cdots\!88\)\( p^{30} T^{13} + \)\(26\!\cdots\!24\)\( p^{40} T^{14} - \)\(42\!\cdots\!52\)\( p^{50} T^{15} + \)\(14\!\cdots\!97\)\( p^{60} T^{16} - 125888889228743848 p^{70} T^{17} + 539604593094 p^{80} T^{18} - 139432 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 17 | \( ( 1 - 960828 T + 9354882982494 T^{2} - 22061040593629788 p^{2} T^{3} + \)\(42\!\cdots\!97\)\( T^{4} - \)\(25\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!24\)\( T^{6} - \)\(89\!\cdots\!32\)\( T^{7} + \)\(35\!\cdots\!02\)\( T^{8} - \)\(24\!\cdots\!88\)\( T^{9} + \)\(77\!\cdots\!64\)\( T^{10} - \)\(24\!\cdots\!88\)\( p^{10} T^{11} + \)\(35\!\cdots\!02\)\( p^{20} T^{12} - \)\(89\!\cdots\!32\)\( p^{30} T^{13} + \)\(13\!\cdots\!24\)\( p^{40} T^{14} - \)\(25\!\cdots\!48\)\( p^{50} T^{15} + \)\(42\!\cdots\!97\)\( p^{60} T^{16} - 22061040593629788 p^{72} T^{17} + 9354882982494 p^{80} T^{18} - 960828 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 19 | \( 1 - 61514763121780 T^{2} + \)\(18\!\cdots\!30\)\( T^{4} - \)\(36\!\cdots\!80\)\( T^{6} + \)\(52\!\cdots\!05\)\( T^{8} - \)\(60\!\cdots\!04\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{12} - \)\(49\!\cdots\!40\)\( T^{14} + \)\(37\!\cdots\!90\)\( T^{16} - \)\(26\!\cdots\!40\)\( T^{18} + \)\(16\!\cdots\!56\)\( T^{20} - \)\(26\!\cdots\!40\)\( p^{20} T^{22} + \)\(37\!\cdots\!90\)\( p^{40} T^{24} - \)\(49\!\cdots\!40\)\( p^{60} T^{26} + \)\(58\!\cdots\!40\)\( p^{80} T^{28} - \)\(60\!\cdots\!04\)\( p^{100} T^{30} + \)\(52\!\cdots\!05\)\( p^{120} T^{32} - \)\(36\!\cdots\!80\)\( p^{140} T^{34} + \)\(18\!\cdots\!30\)\( p^{160} T^{36} - 61514763121780 p^{180} T^{38} + p^{200} T^{40} \) | |
| 23 | \( 1 - 409708172800576 T^{2} + \)\(88\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!68\)\( T^{6} + \)\(14\!\cdots\!81\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(10\!\cdots\!04\)\( T^{12} - \)\(68\!\cdots\!72\)\( T^{14} + \)\(39\!\cdots\!62\)\( T^{16} - \)\(19\!\cdots\!64\)\( T^{18} + \)\(86\!\cdots\!08\)\( T^{20} - \)\(19\!\cdots\!64\)\( p^{20} T^{22} + \)\(39\!\cdots\!62\)\( p^{40} T^{24} - \)\(68\!\cdots\!72\)\( p^{60} T^{26} + \)\(10\!\cdots\!04\)\( p^{80} T^{28} - \)\(13\!\cdots\!20\)\( p^{100} T^{30} + \)\(14\!\cdots\!81\)\( p^{120} T^{32} - \)\(13\!\cdots\!68\)\( p^{140} T^{34} + \)\(88\!\cdots\!98\)\( p^{160} T^{36} - 409708172800576 p^{180} T^{38} + p^{200} T^{40} \) | |
| 29 | \( ( 1 + 33007444 T + 3138879712022718 T^{2} + \)\(74\!\cdots\!92\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} + \)\(79\!\cdots\!80\)\( T^{5} + \)\(36\!\cdots\!64\)\( T^{6} + \)\(57\!\cdots\!68\)\( T^{7} + \)\(22\!\cdots\!42\)\( T^{8} + \)\(31\!\cdots\!16\)\( T^{9} + \)\(10\!\cdots\!08\)\( T^{10} + \)\(31\!\cdots\!16\)\( p^{10} T^{11} + \)\(22\!\cdots\!42\)\( p^{20} T^{12} + \)\(57\!\cdots\!68\)\( p^{30} T^{13} + \)\(36\!\cdots\!64\)\( p^{40} T^{14} + \)\(79\!\cdots\!80\)\( p^{50} T^{15} + \)\(42\!\cdots\!21\)\( p^{60} T^{16} + \)\(74\!\cdots\!92\)\( p^{70} T^{17} + 3138879712022718 p^{80} T^{18} + 33007444 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 31 | \( 1 - 4715933080035780 T^{2} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(28\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!05\)\( T^{8} - \)\(73\!\cdots\!04\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(11\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!90\)\( T^{16} - \)\(10\!\cdots\!40\)\( T^{18} + \)\(93\!\cdots\!56\)\( T^{20} - \)\(10\!\cdots\!40\)\( p^{20} T^{22} + \)\(11\!\cdots\!90\)\( p^{40} T^{24} - \)\(11\!\cdots\!40\)\( p^{60} T^{26} + \)\(95\!\cdots\!40\)\( p^{80} T^{28} - \)\(73\!\cdots\!04\)\( p^{100} T^{30} + \)\(49\!\cdots\!05\)\( p^{120} T^{32} - \)\(28\!\cdots\!80\)\( p^{140} T^{34} + \)\(13\!\cdots\!30\)\( p^{160} T^{36} - 4715933080035780 p^{180} T^{38} + p^{200} T^{40} \) | |
| 37 | \( ( 1 - 76810328 T + 25057252174723014 T^{2} - \)\(18\!\cdots\!52\)\( T^{3} + \)\(32\!\cdots\!97\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(27\!\cdots\!44\)\( T^{6} - \)\(48\!\cdots\!56\)\( p T^{7} + \)\(18\!\cdots\!02\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(97\!\cdots\!84\)\( T^{10} - \)\(10\!\cdots\!28\)\( p^{10} T^{11} + \)\(18\!\cdots\!02\)\( p^{20} T^{12} - \)\(48\!\cdots\!56\)\( p^{31} T^{13} + \)\(27\!\cdots\!44\)\( p^{40} T^{14} - \)\(22\!\cdots\!68\)\( p^{50} T^{15} + \)\(32\!\cdots\!97\)\( p^{60} T^{16} - \)\(18\!\cdots\!52\)\( p^{70} T^{17} + 25057252174723014 p^{80} T^{18} - 76810328 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 238703080 T + 2971474532107910 p T^{2} - \)\(22\!\cdots\!60\)\( T^{3} + \)\(64\!\cdots\!05\)\( T^{4} - \)\(97\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} - \)\(26\!\cdots\!80\)\( T^{7} + \)\(44\!\cdots\!30\)\( T^{8} - \)\(48\!\cdots\!40\)\( T^{9} + \)\(69\!\cdots\!56\)\( T^{10} - \)\(48\!\cdots\!40\)\( p^{10} T^{11} + \)\(44\!\cdots\!30\)\( p^{20} T^{12} - \)\(26\!\cdots\!80\)\( p^{30} T^{13} + \)\(20\!\cdots\!40\)\( p^{40} T^{14} - \)\(97\!\cdots\!04\)\( p^{50} T^{15} + \)\(64\!\cdots\!05\)\( p^{60} T^{16} - \)\(22\!\cdots\!60\)\( p^{70} T^{17} + 2971474532107910 p^{81} T^{18} - 238703080 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 43 | \( 1 - 188176174082270896 T^{2} + \)\(18\!\cdots\!98\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(59\!\cdots\!81\)\( T^{8} - \)\(24\!\cdots\!20\)\( T^{10} + \)\(87\!\cdots\!04\)\( T^{12} - \)\(27\!\cdots\!12\)\( T^{14} + \)\(76\!\cdots\!62\)\( T^{16} - \)\(19\!\cdots\!44\)\( T^{18} + \)\(43\!\cdots\!08\)\( T^{20} - \)\(19\!\cdots\!44\)\( p^{20} T^{22} + \)\(76\!\cdots\!62\)\( p^{40} T^{24} - \)\(27\!\cdots\!12\)\( p^{60} T^{26} + \)\(87\!\cdots\!04\)\( p^{80} T^{28} - \)\(24\!\cdots\!20\)\( p^{100} T^{30} + \)\(59\!\cdots\!81\)\( p^{120} T^{32} - \)\(11\!\cdots\!28\)\( p^{140} T^{34} + \)\(18\!\cdots\!98\)\( p^{160} T^{36} - 188176174082270896 p^{180} T^{38} + p^{200} T^{40} \) | |
| 47 | \( 1 - 477244207749268096 T^{2} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!01\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{10} + \)\(77\!\cdots\!84\)\( T^{12} - \)\(36\!\cdots\!12\)\( T^{14} + \)\(12\!\cdots\!02\)\( T^{16} - \)\(33\!\cdots\!44\)\( T^{18} + \)\(10\!\cdots\!08\)\( T^{20} - \)\(33\!\cdots\!44\)\( p^{20} T^{22} + \)\(12\!\cdots\!02\)\( p^{40} T^{24} - \)\(36\!\cdots\!12\)\( p^{60} T^{26} + \)\(77\!\cdots\!84\)\( p^{80} T^{28} - \)\(12\!\cdots\!20\)\( p^{100} T^{30} + \)\(16\!\cdots\!01\)\( p^{120} T^{32} - \)\(16\!\cdots\!28\)\( p^{140} T^{34} + \)\(10\!\cdots\!58\)\( p^{160} T^{36} - 477244207749268096 p^{180} T^{38} + p^{200} T^{40} \) | |
| 53 | \( ( 1 - 834745912 T + 1240521839141847174 T^{2} - \)\(79\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!97\)\( T^{4} - \)\(33\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!04\)\( T^{6} - \)\(79\!\cdots\!28\)\( T^{7} + \)\(38\!\cdots\!02\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{9} + \)\(65\!\cdots\!44\)\( T^{10} - \)\(14\!\cdots\!52\)\( p^{10} T^{11} + \)\(38\!\cdots\!02\)\( p^{20} T^{12} - \)\(79\!\cdots\!28\)\( p^{30} T^{13} + \)\(19\!\cdots\!04\)\( p^{40} T^{14} - \)\(33\!\cdots\!92\)\( p^{50} T^{15} + \)\(65\!\cdots\!97\)\( p^{60} T^{16} - \)\(79\!\cdots\!28\)\( p^{70} T^{17} + 1240521839141847174 p^{80} T^{18} - 834745912 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 59 | \( 1 - 5624284467879494900 T^{2} + \)\(15\!\cdots\!90\)\( T^{4} - \)\(28\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!25\)\( T^{8} - \)\(38\!\cdots\!04\)\( T^{10} + \)\(32\!\cdots\!00\)\( T^{12} - \)\(23\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!70\)\( T^{16} - \)\(86\!\cdots\!00\)\( T^{18} + \)\(45\!\cdots\!56\)\( T^{20} - \)\(86\!\cdots\!00\)\( p^{20} T^{22} + \)\(15\!\cdots\!70\)\( p^{40} T^{24} - \)\(23\!\cdots\!20\)\( p^{60} T^{26} + \)\(32\!\cdots\!00\)\( p^{80} T^{28} - \)\(38\!\cdots\!04\)\( p^{100} T^{30} + \)\(37\!\cdots\!25\)\( p^{120} T^{32} - \)\(28\!\cdots\!40\)\( p^{140} T^{34} + \)\(15\!\cdots\!90\)\( p^{160} T^{36} - 5624284467879494900 p^{180} T^{38} + p^{200} T^{40} \) | |
| 61 | \( ( 1 + 2141583040 T + 6667611743725968150 T^{2} + \)\(92\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!85\)\( T^{4} + \)\(16\!\cdots\!96\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!50\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(15\!\cdots\!56\)\( T^{10} + \)\(14\!\cdots\!20\)\( p^{10} T^{11} + \)\(20\!\cdots\!50\)\( p^{20} T^{12} + \)\(17\!\cdots\!00\)\( p^{30} T^{13} + \)\(22\!\cdots\!80\)\( p^{40} T^{14} + \)\(16\!\cdots\!96\)\( p^{50} T^{15} + \)\(16\!\cdots\!85\)\( p^{60} T^{16} + \)\(92\!\cdots\!00\)\( p^{70} T^{17} + 6667611743725968150 p^{80} T^{18} + 2141583040 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 67 | \( 1 - 15499583630895038256 T^{2} + \)\(12\!\cdots\!38\)\( T^{4} - \)\(63\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!61\)\( T^{8} - \)\(89\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!24\)\( T^{12} - \)\(67\!\cdots\!32\)\( T^{14} + \)\(15\!\cdots\!22\)\( T^{16} - \)\(32\!\cdots\!84\)\( T^{18} + \)\(62\!\cdots\!08\)\( T^{20} - \)\(32\!\cdots\!84\)\( p^{20} T^{22} + \)\(15\!\cdots\!22\)\( p^{40} T^{24} - \)\(67\!\cdots\!32\)\( p^{60} T^{26} + \)\(26\!\cdots\!24\)\( p^{80} T^{28} - \)\(89\!\cdots\!20\)\( p^{100} T^{30} + \)\(26\!\cdots\!61\)\( p^{120} T^{32} - \)\(63\!\cdots\!08\)\( p^{140} T^{34} + \)\(12\!\cdots\!38\)\( p^{160} T^{36} - 15499583630895038256 p^{180} T^{38} + p^{200} T^{40} \) | |
| 71 | \( 1 - 44284185599594329860 T^{2} + \)\(96\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!85\)\( T^{8} - \)\(12\!\cdots\!04\)\( T^{10} + \)\(83\!\cdots\!80\)\( T^{12} - \)\(46\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!10\)\( T^{16} - \)\(91\!\cdots\!80\)\( T^{18} + \)\(32\!\cdots\!56\)\( T^{20} - \)\(91\!\cdots\!80\)\( p^{20} T^{22} + \)\(22\!\cdots\!10\)\( p^{40} T^{24} - \)\(46\!\cdots\!60\)\( p^{60} T^{26} + \)\(83\!\cdots\!80\)\( p^{80} T^{28} - \)\(12\!\cdots\!04\)\( p^{100} T^{30} + \)\(14\!\cdots\!85\)\( p^{120} T^{32} - \)\(13\!\cdots\!20\)\( p^{140} T^{34} + \)\(96\!\cdots\!70\)\( p^{160} T^{36} - 44284185599594329860 p^{180} T^{38} + p^{200} T^{40} \) | |
| 73 | \( ( 1 + 1237143828 T + 18224089167775758174 T^{2} + \)\(29\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!97\)\( T^{4} + \)\(33\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!04\)\( T^{6} + \)\(24\!\cdots\!72\)\( T^{7} + \)\(68\!\cdots\!02\)\( T^{8} + \)\(12\!\cdots\!28\)\( T^{9} + \)\(33\!\cdots\!44\)\( T^{10} + \)\(12\!\cdots\!28\)\( p^{10} T^{11} + \)\(68\!\cdots\!02\)\( p^{20} T^{12} + \)\(24\!\cdots\!72\)\( p^{30} T^{13} + \)\(11\!\cdots\!04\)\( p^{40} T^{14} + \)\(33\!\cdots\!68\)\( p^{50} T^{15} + \)\(16\!\cdots\!97\)\( p^{60} T^{16} + \)\(29\!\cdots\!52\)\( p^{70} T^{17} + 18224089167775758174 p^{80} T^{18} + 1237143828 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 79 | \( 1 - 94707825699194455380 T^{2} + \)\(47\!\cdots\!30\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!05\)\( T^{8} - \)\(96\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{12} - \)\(26\!\cdots\!40\)\( T^{14} + \)\(35\!\cdots\!90\)\( T^{16} - \)\(40\!\cdots\!40\)\( T^{18} + \)\(41\!\cdots\!56\)\( T^{20} - \)\(40\!\cdots\!40\)\( p^{20} T^{22} + \)\(35\!\cdots\!90\)\( p^{40} T^{24} - \)\(26\!\cdots\!40\)\( p^{60} T^{26} + \)\(17\!\cdots\!40\)\( p^{80} T^{28} - \)\(96\!\cdots\!04\)\( p^{100} T^{30} + \)\(44\!\cdots\!05\)\( p^{120} T^{32} - \)\(16\!\cdots\!80\)\( p^{140} T^{34} + \)\(47\!\cdots\!30\)\( p^{160} T^{36} - 94707825699194455380 p^{180} T^{38} + p^{200} T^{40} \) | |
| 83 | \( 1 - \)\(12\!\cdots\!76\)\( T^{2} + \)\(87\!\cdots\!98\)\( T^{4} - \)\(42\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{10} + \)\(13\!\cdots\!04\)\( T^{12} - \)\(31\!\cdots\!72\)\( T^{14} + \)\(65\!\cdots\!62\)\( T^{16} - \)\(11\!\cdots\!64\)\( T^{18} + \)\(19\!\cdots\!08\)\( T^{20} - \)\(11\!\cdots\!64\)\( p^{20} T^{22} + \)\(65\!\cdots\!62\)\( p^{40} T^{24} - \)\(31\!\cdots\!72\)\( p^{60} T^{26} + \)\(13\!\cdots\!04\)\( p^{80} T^{28} - \)\(50\!\cdots\!20\)\( p^{100} T^{30} + \)\(16\!\cdots\!81\)\( p^{120} T^{32} - \)\(42\!\cdots\!68\)\( p^{140} T^{34} + \)\(87\!\cdots\!98\)\( p^{160} T^{36} - \)\(12\!\cdots\!76\)\( p^{180} T^{38} + p^{200} T^{40} \) | |
| 89 | \( ( 1 - 1505925796 T + \)\(18\!\cdots\!98\)\( T^{2} - \)\(27\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{5} + \)\(94\!\cdots\!04\)\( T^{6} - \)\(86\!\cdots\!12\)\( T^{7} + \)\(40\!\cdots\!62\)\( T^{8} - \)\(27\!\cdots\!44\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} - \)\(27\!\cdots\!44\)\( p^{10} T^{11} + \)\(40\!\cdots\!62\)\( p^{20} T^{12} - \)\(86\!\cdots\!12\)\( p^{30} T^{13} + \)\(94\!\cdots\!04\)\( p^{40} T^{14} - \)\(20\!\cdots\!20\)\( p^{50} T^{15} + \)\(16\!\cdots\!81\)\( p^{60} T^{16} - \)\(27\!\cdots\!28\)\( p^{70} T^{17} + \)\(18\!\cdots\!98\)\( p^{80} T^{18} - 1505925796 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 19992251028 T + \)\(57\!\cdots\!14\)\( T^{2} - \)\(80\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!97\)\( T^{4} - \)\(14\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!44\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!02\)\( T^{8} - \)\(13\!\cdots\!48\)\( T^{9} + \)\(13\!\cdots\!84\)\( T^{10} - \)\(13\!\cdots\!48\)\( p^{10} T^{11} + \)\(17\!\cdots\!02\)\( p^{20} T^{12} - \)\(16\!\cdots\!92\)\( p^{30} T^{13} + \)\(18\!\cdots\!44\)\( p^{40} T^{14} - \)\(14\!\cdots\!28\)\( p^{50} T^{15} + \)\(13\!\cdots\!97\)\( p^{60} T^{16} - \)\(80\!\cdots\!12\)\( p^{70} T^{17} + \)\(57\!\cdots\!14\)\( p^{80} T^{18} - 19992251028 p^{90} T^{19} + p^{100} T^{20} )^{2} \) | |
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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
−1.76861502029858739907536005493, −1.73948593693483507608482112845, −1.56517371976408911050701669109, −1.55366488377459135627982723681, −1.54918593462244731881890130079, −1.50319473480162360189079791443, −1.46234674094803414833118321117, −1.16426540998004408532895740511, −1.16164971802128024349097027021, −1.09878517324683902703170473718, −1.01850425960262197806090115982, −0.960412325443603645608824963726, −0.912602541531428637596534055876, −0.870527814581453313113219034404, −0.856629555057288619368377076417, −0.828746718191272511306223077951, −0.799488008092810647402495931289, −0.76226838581154158795444291790, −0.48433095996616068503541650723, −0.48376943447907721443327641487, −0.38507277279556639758493736185, −0.35957592946081191798880830139, −0.30240381307754578558069599178, −0.29727799331088416476887939817, −0.17926352295572370729433923731, 0.17926352295572370729433923731, 0.29727799331088416476887939817, 0.30240381307754578558069599178, 0.35957592946081191798880830139, 0.38507277279556639758493736185, 0.48376943447907721443327641487, 0.48433095996616068503541650723, 0.76226838581154158795444291790, 0.799488008092810647402495931289, 0.828746718191272511306223077951, 0.856629555057288619368377076417, 0.870527814581453313113219034404, 0.912602541531428637596534055876, 0.960412325443603645608824963726, 1.01850425960262197806090115982, 1.09878517324683902703170473718, 1.16164971802128024349097027021, 1.16426540998004408532895740511, 1.46234674094803414833118321117, 1.50319473480162360189079791443, 1.54918593462244731881890130079, 1.55366488377459135627982723681, 1.56517371976408911050701669109, 1.73948593693483507608482112845, 1.76861502029858739907536005493
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.