Dirichlet series
| L(s) = 1 | + 58·3-s − 54·5-s + 2.46e3·7-s + 1.68e3·9-s − 1.17e3·13-s − 3.13e3·15-s − 2.51e4·17-s + 6.47e4·19-s + 1.43e5·21-s − 3.99e4·23-s + 6.04e4·25-s + 8.53e4·27-s − 1.33e5·35-s − 6.47e5·37-s − 6.79e4·39-s − 1.15e6·41-s − 1.58e6·43-s − 9.08e4·45-s − 1.41e6·47-s + 3.04e6·49-s − 1.45e6·51-s − 2.11e6·53-s + 3.75e6·57-s + 6.49e6·59-s + 1.93e6·61-s + 4.14e6·63-s + 6.32e4·65-s + ⋯ |
| L(s) = 1 | + 1.24·3-s − 0.193·5-s + 2.71·7-s + 0.769·9-s − 0.147·13-s − 0.239·15-s − 1.24·17-s + 2.16·19-s + 3.37·21-s − 0.684·23-s + 0.774·25-s + 0.834·27-s − 0.524·35-s − 2.10·37-s − 0.183·39-s − 2.61·41-s − 3.04·43-s − 0.148·45-s − 1.99·47-s + 3.69·49-s − 1.53·51-s − 1.95·53-s + 2.68·57-s + 4.11·59-s + 1.09·61-s + 2.08·63-s + 0.0285·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{100} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.46681\times 10^{33}\) |
| Root analytic conductor: | \(7.06976\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{100} \cdot 5^{20} ,\ ( \ : [7/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.007282372963\) |
| \(L(\frac12)\) | \(\approx\) | \(0.007282372963\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 54 T - 11514 p T^{2} - 301874 p T^{3} + 108910573 p^{2} T^{4} - 611018496 p^{4} T^{5} + 12100344264 p^{6} T^{6} - 75665683088 p^{8} T^{7} - 361541973718 p^{11} T^{8} - 3631666625524 p^{13} T^{9} + 4971217749108 p^{16} T^{10} - 3631666625524 p^{20} T^{11} - 361541973718 p^{25} T^{12} - 75665683088 p^{29} T^{13} + 12100344264 p^{34} T^{14} - 611018496 p^{39} T^{15} + 108910573 p^{44} T^{16} - 301874 p^{50} T^{17} - 11514 p^{57} T^{18} + 54 p^{63} T^{19} + p^{70} T^{20} \) | |
| good | 3 | \( 1 - 58 T + 1682 T^{2} - 28466 p T^{3} + 265214 T^{4} + 233083414 T^{5} - 10318518758 T^{6} + 276673227278 p T^{7} - 42509722544531 T^{8} + 564447876115136 T^{9} - 2218089709204768 T^{10} + 875694561274624 p T^{11} + 19700435408516952904 p^{2} T^{12} - \)\(14\!\cdots\!08\)\( p^{4} T^{13} + \)\(54\!\cdots\!52\)\( p^{4} T^{14} - \)\(10\!\cdots\!24\)\( p^{5} T^{15} + \)\(10\!\cdots\!74\)\( p^{6} T^{16} + \)\(27\!\cdots\!56\)\( p^{8} T^{17} - \)\(18\!\cdots\!48\)\( p^{8} T^{18} + \)\(68\!\cdots\!28\)\( p^{9} T^{19} - \)\(19\!\cdots\!08\)\( p^{10} T^{20} + \)\(68\!\cdots\!28\)\( p^{16} T^{21} - \)\(18\!\cdots\!48\)\( p^{22} T^{22} + \)\(27\!\cdots\!56\)\( p^{29} T^{23} + \)\(10\!\cdots\!74\)\( p^{34} T^{24} - \)\(10\!\cdots\!24\)\( p^{40} T^{25} + \)\(54\!\cdots\!52\)\( p^{46} T^{26} - \)\(14\!\cdots\!08\)\( p^{53} T^{27} + 19700435408516952904 p^{58} T^{28} + 875694561274624 p^{64} T^{29} - 2218089709204768 p^{70} T^{30} + 564447876115136 p^{77} T^{31} - 42509722544531 p^{84} T^{32} + 276673227278 p^{92} T^{33} - 10318518758 p^{98} T^{34} + 233083414 p^{105} T^{35} + 265214 p^{112} T^{36} - 28466 p^{120} T^{37} + 1682 p^{126} T^{38} - 58 p^{133} T^{39} + p^{140} T^{40} \) |
| 7 | \( 1 - 2466 T + 3040578 T^{2} - 1811535046 T^{3} + 80750367886 T^{4} + 63825717304670 T^{5} + 1237907600483716730 T^{6} - \)\(18\!\cdots\!14\)\( T^{7} + \)\(10\!\cdots\!63\)\( p T^{8} + \)\(82\!\cdots\!56\)\( T^{9} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!76\)\( T^{11} + \)\(21\!\cdots\!88\)\( T^{12} + \)\(41\!\cdots\!24\)\( T^{13} - \)\(93\!\cdots\!76\)\( T^{14} + \)\(54\!\cdots\!80\)\( T^{15} + \)\(42\!\cdots\!98\)\( T^{16} - \)\(89\!\cdots\!68\)\( T^{17} + \)\(64\!\cdots\!24\)\( T^{18} - \)\(13\!\cdots\!84\)\( T^{19} - \)\(91\!\cdots\!72\)\( T^{20} - \)\(13\!\cdots\!84\)\( p^{7} T^{21} + \)\(64\!\cdots\!24\)\( p^{14} T^{22} - \)\(89\!\cdots\!68\)\( p^{21} T^{23} + \)\(42\!\cdots\!98\)\( p^{28} T^{24} + \)\(54\!\cdots\!80\)\( p^{35} T^{25} - \)\(93\!\cdots\!76\)\( p^{42} T^{26} + \)\(41\!\cdots\!24\)\( p^{49} T^{27} + \)\(21\!\cdots\!88\)\( p^{56} T^{28} + \)\(14\!\cdots\!76\)\( p^{63} T^{29} - \)\(10\!\cdots\!80\)\( p^{70} T^{30} + \)\(82\!\cdots\!56\)\( p^{77} T^{31} + \)\(10\!\cdots\!63\)\( p^{85} T^{32} - \)\(18\!\cdots\!14\)\( p^{91} T^{33} + 1237907600483716730 p^{98} T^{34} + 63825717304670 p^{105} T^{35} + 80750367886 p^{112} T^{36} - 1811535046 p^{119} T^{37} + 3040578 p^{126} T^{38} - 2466 p^{133} T^{39} + p^{140} T^{40} \) | |
| 11 | \( 1 - 155375456 T^{2} + 12138542646745750 T^{4} - \)\(63\!\cdots\!52\)\( T^{6} + \)\(25\!\cdots\!97\)\( T^{8} - \)\(74\!\cdots\!20\)\( p T^{10} + \)\(22\!\cdots\!08\)\( T^{12} - \)\(54\!\cdots\!08\)\( T^{14} + \)\(10\!\cdots\!50\)\( p T^{16} - \)\(24\!\cdots\!44\)\( T^{18} + \)\(48\!\cdots\!28\)\( T^{20} - \)\(24\!\cdots\!44\)\( p^{14} T^{22} + \)\(10\!\cdots\!50\)\( p^{29} T^{24} - \)\(54\!\cdots\!08\)\( p^{42} T^{26} + \)\(22\!\cdots\!08\)\( p^{56} T^{28} - \)\(74\!\cdots\!20\)\( p^{71} T^{30} + \)\(25\!\cdots\!97\)\( p^{84} T^{32} - \)\(63\!\cdots\!52\)\( p^{98} T^{34} + 12138542646745750 p^{112} T^{36} - 155375456 p^{126} T^{38} + p^{140} T^{40} \) | |
| 13 | \( 1 + 1172 T + 686792 T^{2} - 313989518964 T^{3} - 3212129452443754 T^{4} - 19811183424016885604 T^{5} + \)\(21\!\cdots\!56\)\( p T^{6} - \)\(14\!\cdots\!08\)\( T^{7} - \)\(24\!\cdots\!35\)\( T^{8} - \)\(10\!\cdots\!56\)\( T^{9} + \)\(21\!\cdots\!20\)\( T^{10} - \)\(26\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!56\)\( T^{12} + \)\(69\!\cdots\!64\)\( T^{13} + \)\(43\!\cdots\!28\)\( T^{14} + \)\(39\!\cdots\!24\)\( T^{15} - \)\(13\!\cdots\!30\)\( T^{16} - \)\(19\!\cdots\!76\)\( p T^{17} - \)\(10\!\cdots\!24\)\( T^{18} - \)\(64\!\cdots\!28\)\( T^{19} + \)\(12\!\cdots\!32\)\( T^{20} - \)\(64\!\cdots\!28\)\( p^{7} T^{21} - \)\(10\!\cdots\!24\)\( p^{14} T^{22} - \)\(19\!\cdots\!76\)\( p^{22} T^{23} - \)\(13\!\cdots\!30\)\( p^{28} T^{24} + \)\(39\!\cdots\!24\)\( p^{35} T^{25} + \)\(43\!\cdots\!28\)\( p^{42} T^{26} + \)\(69\!\cdots\!64\)\( p^{49} T^{27} + \)\(74\!\cdots\!56\)\( p^{56} T^{28} - \)\(26\!\cdots\!04\)\( p^{63} T^{29} + \)\(21\!\cdots\!20\)\( p^{70} T^{30} - \)\(10\!\cdots\!56\)\( p^{77} T^{31} - \)\(24\!\cdots\!35\)\( p^{84} T^{32} - \)\(14\!\cdots\!08\)\( p^{91} T^{33} + \)\(21\!\cdots\!56\)\( p^{99} T^{34} - 19811183424016885604 p^{105} T^{35} - 3212129452443754 p^{112} T^{36} - 313989518964 p^{119} T^{37} + 686792 p^{126} T^{38} + 1172 p^{133} T^{39} + p^{140} T^{40} \) | |
| 17 | \( 1 + 25136 T + 315909248 T^{2} - 11900876851792 T^{3} - 9718471582121546 T^{4} + \)\(66\!\cdots\!20\)\( T^{5} + \)\(24\!\cdots\!60\)\( T^{6} - \)\(17\!\cdots\!20\)\( T^{7} - \)\(58\!\cdots\!95\)\( T^{8} + \)\(57\!\cdots\!20\)\( T^{9} + \)\(78\!\cdots\!80\)\( T^{10} + \)\(78\!\cdots\!80\)\( T^{11} - \)\(10\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!68\)\( T^{13} + \)\(68\!\cdots\!56\)\( T^{14} + \)\(26\!\cdots\!56\)\( T^{15} + \)\(18\!\cdots\!18\)\( T^{16} - \)\(35\!\cdots\!60\)\( T^{17} - \)\(58\!\cdots\!80\)\( T^{18} + \)\(27\!\cdots\!80\)\( T^{19} + \)\(84\!\cdots\!40\)\( T^{20} + \)\(27\!\cdots\!80\)\( p^{7} T^{21} - \)\(58\!\cdots\!80\)\( p^{14} T^{22} - \)\(35\!\cdots\!60\)\( p^{21} T^{23} + \)\(18\!\cdots\!18\)\( p^{28} T^{24} + \)\(26\!\cdots\!56\)\( p^{35} T^{25} + \)\(68\!\cdots\!56\)\( p^{42} T^{26} - \)\(30\!\cdots\!68\)\( p^{49} T^{27} - \)\(10\!\cdots\!08\)\( p^{56} T^{28} + \)\(78\!\cdots\!80\)\( p^{63} T^{29} + \)\(78\!\cdots\!80\)\( p^{70} T^{30} + \)\(57\!\cdots\!20\)\( p^{77} T^{31} - \)\(58\!\cdots\!95\)\( p^{84} T^{32} - \)\(17\!\cdots\!20\)\( p^{91} T^{33} + \)\(24\!\cdots\!60\)\( p^{98} T^{34} + \)\(66\!\cdots\!20\)\( p^{105} T^{35} - 9718471582121546 p^{112} T^{36} - 11900876851792 p^{119} T^{37} + 315909248 p^{126} T^{38} + 25136 p^{133} T^{39} + p^{140} T^{40} \) | |
| 19 | \( ( 1 - 32392 T + 6808159262 T^{2} - 218132006965784 T^{3} + 21986522110533225701 T^{4} - \)\(67\!\cdots\!00\)\( T^{5} + \)\(44\!\cdots\!36\)\( T^{6} - \)\(12\!\cdots\!36\)\( T^{7} + \)\(63\!\cdots\!22\)\( T^{8} - \)\(16\!\cdots\!28\)\( T^{9} + \)\(65\!\cdots\!52\)\( T^{10} - \)\(16\!\cdots\!28\)\( p^{7} T^{11} + \)\(63\!\cdots\!22\)\( p^{14} T^{12} - \)\(12\!\cdots\!36\)\( p^{21} T^{13} + \)\(44\!\cdots\!36\)\( p^{28} T^{14} - \)\(67\!\cdots\!00\)\( p^{35} T^{15} + 21986522110533225701 p^{42} T^{16} - 218132006965784 p^{49} T^{17} + 6808159262 p^{56} T^{18} - 32392 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 23 | \( 1 + 39922 T + 796883042 T^{2} - 282586659050298 T^{3} - 24146722462484821266 T^{4} + \)\(21\!\cdots\!78\)\( T^{5} + \)\(67\!\cdots\!90\)\( T^{6} + \)\(49\!\cdots\!42\)\( T^{7} - \)\(97\!\cdots\!43\)\( T^{8} - \)\(16\!\cdots\!48\)\( T^{9} - \)\(30\!\cdots\!84\)\( T^{10} + \)\(55\!\cdots\!76\)\( T^{11} + \)\(48\!\cdots\!68\)\( T^{12} - \)\(72\!\cdots\!36\)\( T^{13} - \)\(14\!\cdots\!32\)\( T^{14} - \)\(10\!\cdots\!44\)\( T^{15} + \)\(74\!\cdots\!46\)\( T^{16} + \)\(34\!\cdots\!68\)\( T^{17} + \)\(10\!\cdots\!68\)\( T^{18} - \)\(17\!\cdots\!08\)\( T^{19} - \)\(64\!\cdots\!32\)\( T^{20} - \)\(17\!\cdots\!08\)\( p^{7} T^{21} + \)\(10\!\cdots\!68\)\( p^{14} T^{22} + \)\(34\!\cdots\!68\)\( p^{21} T^{23} + \)\(74\!\cdots\!46\)\( p^{28} T^{24} - \)\(10\!\cdots\!44\)\( p^{35} T^{25} - \)\(14\!\cdots\!32\)\( p^{42} T^{26} - \)\(72\!\cdots\!36\)\( p^{49} T^{27} + \)\(48\!\cdots\!68\)\( p^{56} T^{28} + \)\(55\!\cdots\!76\)\( p^{63} T^{29} - \)\(30\!\cdots\!84\)\( p^{70} T^{30} - \)\(16\!\cdots\!48\)\( p^{77} T^{31} - \)\(97\!\cdots\!43\)\( p^{84} T^{32} + \)\(49\!\cdots\!42\)\( p^{91} T^{33} + \)\(67\!\cdots\!90\)\( p^{98} T^{34} + \)\(21\!\cdots\!78\)\( p^{105} T^{35} - 24146722462484821266 p^{112} T^{36} - 282586659050298 p^{119} T^{37} + 796883042 p^{126} T^{38} + 39922 p^{133} T^{39} + p^{140} T^{40} \) | |
| 29 | \( 1 - 214056087860 T^{2} + \)\(21\!\cdots\!42\)\( T^{4} - \)\(14\!\cdots\!48\)\( T^{6} + \)\(68\!\cdots\!13\)\( T^{8} - \)\(25\!\cdots\!20\)\( T^{10} + \)\(78\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!52\)\( T^{14} + \)\(47\!\cdots\!78\)\( T^{16} - \)\(95\!\cdots\!40\)\( T^{18} + \)\(17\!\cdots\!88\)\( T^{20} - \)\(95\!\cdots\!40\)\( p^{14} T^{22} + \)\(47\!\cdots\!78\)\( p^{28} T^{24} - \)\(20\!\cdots\!52\)\( p^{42} T^{26} + \)\(78\!\cdots\!52\)\( p^{56} T^{28} - \)\(25\!\cdots\!20\)\( p^{70} T^{30} + \)\(68\!\cdots\!13\)\( p^{84} T^{32} - \)\(14\!\cdots\!48\)\( p^{98} T^{34} + \)\(21\!\cdots\!42\)\( p^{112} T^{36} - 214056087860 p^{126} T^{38} + p^{140} T^{40} \) | |
| 31 | \( 1 - 354387960480 T^{2} + \)\(62\!\cdots\!42\)\( T^{4} - \)\(72\!\cdots\!32\)\( T^{6} + \)\(62\!\cdots\!53\)\( T^{8} - \)\(42\!\cdots\!60\)\( T^{10} + \)\(23\!\cdots\!72\)\( T^{12} - \)\(11\!\cdots\!48\)\( T^{14} + \)\(43\!\cdots\!78\)\( T^{16} - \)\(14\!\cdots\!00\)\( T^{18} + \)\(44\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!00\)\( p^{14} T^{22} + \)\(43\!\cdots\!78\)\( p^{28} T^{24} - \)\(11\!\cdots\!48\)\( p^{42} T^{26} + \)\(23\!\cdots\!72\)\( p^{56} T^{28} - \)\(42\!\cdots\!60\)\( p^{70} T^{30} + \)\(62\!\cdots\!53\)\( p^{84} T^{32} - \)\(72\!\cdots\!32\)\( p^{98} T^{34} + \)\(62\!\cdots\!42\)\( p^{112} T^{36} - 354387960480 p^{126} T^{38} + p^{140} T^{40} \) | |
| 37 | \( 1 + 647408 T + 209568559232 T^{2} + 89242611105741808 T^{3} + \)\(44\!\cdots\!90\)\( T^{4} + \)\(38\!\cdots\!72\)\( p T^{5} + \)\(39\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(52\!\cdots\!57\)\( T^{8} + \)\(10\!\cdots\!32\)\( T^{9} + \)\(25\!\cdots\!80\)\( T^{10} + \)\(94\!\cdots\!88\)\( T^{11} + \)\(18\!\cdots\!24\)\( T^{12} - \)\(63\!\cdots\!12\)\( T^{13} - \)\(15\!\cdots\!80\)\( p T^{14} - \)\(29\!\cdots\!80\)\( T^{15} - \)\(25\!\cdots\!54\)\( T^{16} - \)\(10\!\cdots\!64\)\( T^{17} - \)\(28\!\cdots\!16\)\( T^{18} - \)\(10\!\cdots\!12\)\( T^{19} - \)\(38\!\cdots\!36\)\( T^{20} - \)\(10\!\cdots\!12\)\( p^{7} T^{21} - \)\(28\!\cdots\!16\)\( p^{14} T^{22} - \)\(10\!\cdots\!64\)\( p^{21} T^{23} - \)\(25\!\cdots\!54\)\( p^{28} T^{24} - \)\(29\!\cdots\!80\)\( p^{35} T^{25} - \)\(15\!\cdots\!80\)\( p^{43} T^{26} - \)\(63\!\cdots\!12\)\( p^{49} T^{27} + \)\(18\!\cdots\!24\)\( p^{56} T^{28} + \)\(94\!\cdots\!88\)\( p^{63} T^{29} + \)\(25\!\cdots\!80\)\( p^{70} T^{30} + \)\(10\!\cdots\!32\)\( p^{77} T^{31} + \)\(52\!\cdots\!57\)\( p^{84} T^{32} + \)\(15\!\cdots\!20\)\( p^{91} T^{33} + \)\(39\!\cdots\!64\)\( p^{98} T^{34} + \)\(38\!\cdots\!72\)\( p^{106} T^{35} + \)\(44\!\cdots\!90\)\( p^{112} T^{36} + 89242611105741808 p^{119} T^{37} + 209568559232 p^{126} T^{38} + 647408 p^{133} T^{39} + p^{140} T^{40} \) | |
| 41 | \( ( 1 + 575942 T + 977657095278 T^{2} + 357461430642746942 T^{3} + \)\(41\!\cdots\!09\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} + \)\(27\!\cdots\!92\)\( T^{7} + \)\(29\!\cdots\!38\)\( T^{8} + \)\(55\!\cdots\!72\)\( T^{9} + \)\(59\!\cdots\!12\)\( T^{10} + \)\(55\!\cdots\!72\)\( p^{7} T^{11} + \)\(29\!\cdots\!38\)\( p^{14} T^{12} + \)\(27\!\cdots\!92\)\( p^{21} T^{13} + \)\(12\!\cdots\!64\)\( p^{28} T^{14} + \)\(11\!\cdots\!80\)\( p^{35} T^{15} + \)\(41\!\cdots\!09\)\( p^{42} T^{16} + 357461430642746942 p^{49} T^{17} + 977657095278 p^{56} T^{18} + 575942 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 43 | \( 1 + 1589406 T + 1263105716418 T^{2} + 990112346346798690 T^{3} + \)\(85\!\cdots\!30\)\( T^{4} + \)\(59\!\cdots\!74\)\( T^{5} + \)\(35\!\cdots\!54\)\( T^{6} + \)\(23\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!73\)\( T^{8} + \)\(68\!\cdots\!76\)\( T^{9} + \)\(32\!\cdots\!60\)\( T^{10} + \)\(16\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!48\)\( T^{12} + \)\(78\!\cdots\!16\)\( T^{13} - \)\(19\!\cdots\!36\)\( T^{14} - \)\(48\!\cdots\!44\)\( T^{15} - \)\(66\!\cdots\!74\)\( T^{16} - \)\(47\!\cdots\!64\)\( T^{17} - \)\(25\!\cdots\!32\)\( T^{18} - \)\(15\!\cdots\!80\)\( T^{19} - \)\(92\!\cdots\!56\)\( T^{20} - \)\(15\!\cdots\!80\)\( p^{7} T^{21} - \)\(25\!\cdots\!32\)\( p^{14} T^{22} - \)\(47\!\cdots\!64\)\( p^{21} T^{23} - \)\(66\!\cdots\!74\)\( p^{28} T^{24} - \)\(48\!\cdots\!44\)\( p^{35} T^{25} - \)\(19\!\cdots\!36\)\( p^{42} T^{26} + \)\(78\!\cdots\!16\)\( p^{49} T^{27} + \)\(59\!\cdots\!48\)\( p^{56} T^{28} + \)\(16\!\cdots\!40\)\( p^{63} T^{29} + \)\(32\!\cdots\!60\)\( p^{70} T^{30} + \)\(68\!\cdots\!76\)\( p^{77} T^{31} + \)\(14\!\cdots\!73\)\( p^{84} T^{32} + \)\(23\!\cdots\!50\)\( p^{91} T^{33} + \)\(35\!\cdots\!54\)\( p^{98} T^{34} + \)\(59\!\cdots\!74\)\( p^{105} T^{35} + \)\(85\!\cdots\!30\)\( p^{112} T^{36} + 990112346346798690 p^{119} T^{37} + 1263105716418 p^{126} T^{38} + 1589406 p^{133} T^{39} + p^{140} T^{40} \) | |
| 47 | \( 1 + 1417862 T + 1005166325522 T^{2} + 1240239832229276466 T^{3} + \)\(77\!\cdots\!50\)\( T^{4} - \)\(27\!\cdots\!10\)\( T^{5} - \)\(17\!\cdots\!42\)\( T^{6} - \)\(14\!\cdots\!58\)\( T^{7} - \)\(16\!\cdots\!27\)\( T^{8} + \)\(79\!\cdots\!72\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!76\)\( T^{11} + \)\(14\!\cdots\!72\)\( T^{12} + \)\(44\!\cdots\!04\)\( T^{13} + \)\(18\!\cdots\!08\)\( T^{14} - \)\(22\!\cdots\!20\)\( T^{15} - \)\(20\!\cdots\!30\)\( T^{16} + \)\(42\!\cdots\!32\)\( T^{17} + \)\(68\!\cdots\!52\)\( T^{18} + \)\(84\!\cdots\!96\)\( T^{19} + \)\(10\!\cdots\!88\)\( T^{20} + \)\(84\!\cdots\!96\)\( p^{7} T^{21} + \)\(68\!\cdots\!52\)\( p^{14} T^{22} + \)\(42\!\cdots\!32\)\( p^{21} T^{23} - \)\(20\!\cdots\!30\)\( p^{28} T^{24} - \)\(22\!\cdots\!20\)\( p^{35} T^{25} + \)\(18\!\cdots\!08\)\( p^{42} T^{26} + \)\(44\!\cdots\!04\)\( p^{49} T^{27} + \)\(14\!\cdots\!72\)\( p^{56} T^{28} + \)\(14\!\cdots\!76\)\( p^{63} T^{29} + \)\(11\!\cdots\!80\)\( p^{70} T^{30} + \)\(79\!\cdots\!72\)\( p^{77} T^{31} - \)\(16\!\cdots\!27\)\( p^{84} T^{32} - \)\(14\!\cdots\!58\)\( p^{91} T^{33} - \)\(17\!\cdots\!42\)\( p^{98} T^{34} - \)\(27\!\cdots\!10\)\( p^{105} T^{35} + \)\(77\!\cdots\!50\)\( p^{112} T^{36} + 1240239832229276466 p^{119} T^{37} + 1005166325522 p^{126} T^{38} + 1417862 p^{133} T^{39} + p^{140} T^{40} \) | |
| 53 | \( 1 + 39912 p T + 796483872 p^{2} T^{2} + 6206825688530927800 T^{3} + \)\(83\!\cdots\!70\)\( T^{4} + \)\(44\!\cdots\!84\)\( T^{5} + \)\(99\!\cdots\!64\)\( T^{6} + \)\(92\!\cdots\!20\)\( T^{7} - \)\(53\!\cdots\!87\)\( T^{8} + \)\(31\!\cdots\!16\)\( T^{9} + \)\(32\!\cdots\!00\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(20\!\cdots\!88\)\( T^{12} + \)\(95\!\cdots\!96\)\( T^{13} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(12\!\cdots\!16\)\( T^{15} + \)\(19\!\cdots\!86\)\( T^{16} - \)\(26\!\cdots\!44\)\( T^{17} + \)\(12\!\cdots\!68\)\( T^{18} + \)\(12\!\cdots\!60\)\( T^{19} - \)\(43\!\cdots\!76\)\( T^{20} + \)\(12\!\cdots\!60\)\( p^{7} T^{21} + \)\(12\!\cdots\!68\)\( p^{14} T^{22} - \)\(26\!\cdots\!44\)\( p^{21} T^{23} + \)\(19\!\cdots\!86\)\( p^{28} T^{24} + \)\(12\!\cdots\!16\)\( p^{35} T^{25} - \)\(20\!\cdots\!56\)\( p^{42} T^{26} + \)\(95\!\cdots\!96\)\( p^{49} T^{27} + \)\(20\!\cdots\!88\)\( p^{56} T^{28} - \)\(19\!\cdots\!80\)\( p^{63} T^{29} + \)\(32\!\cdots\!00\)\( p^{70} T^{30} + \)\(31\!\cdots\!16\)\( p^{77} T^{31} - \)\(53\!\cdots\!87\)\( p^{84} T^{32} + \)\(92\!\cdots\!20\)\( p^{91} T^{33} + \)\(99\!\cdots\!64\)\( p^{98} T^{34} + \)\(44\!\cdots\!84\)\( p^{105} T^{35} + \)\(83\!\cdots\!70\)\( p^{112} T^{36} + 6206825688530927800 p^{119} T^{37} + 796483872 p^{128} T^{38} + 39912 p^{134} T^{39} + p^{140} T^{40} \) | |
| 59 | \( ( 1 - 3249288 T + 10146405408526 T^{2} - 23600639727955731160 T^{3} + \)\(55\!\cdots\!21\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} - \)\(52\!\cdots\!44\)\( p T^{7} + \)\(52\!\cdots\!94\)\( T^{8} - \)\(77\!\cdots\!04\)\( T^{9} + \)\(12\!\cdots\!28\)\( T^{10} - \)\(77\!\cdots\!04\)\( p^{7} T^{11} + \)\(52\!\cdots\!94\)\( p^{14} T^{12} - \)\(52\!\cdots\!44\)\( p^{22} T^{13} + \)\(19\!\cdots\!92\)\( p^{28} T^{14} - \)\(10\!\cdots\!16\)\( p^{35} T^{15} + \)\(55\!\cdots\!21\)\( p^{42} T^{16} - 23600639727955731160 p^{49} T^{17} + 10146405408526 p^{56} T^{18} - 3249288 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 968282 T + 5482392324014 T^{2} - 7025858435107045338 T^{3} + \)\(76\!\cdots\!53\)\( p T^{4} - \)\(46\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!36\)\( T^{7} + \)\(78\!\cdots\!30\)\( T^{8} - \)\(74\!\cdots\!08\)\( T^{9} + \)\(19\!\cdots\!64\)\( T^{10} - \)\(74\!\cdots\!08\)\( p^{7} T^{11} + \)\(78\!\cdots\!30\)\( p^{14} T^{12} - \)\(19\!\cdots\!36\)\( p^{21} T^{13} + \)\(15\!\cdots\!32\)\( p^{28} T^{14} - \)\(46\!\cdots\!08\)\( p^{35} T^{15} + \)\(76\!\cdots\!53\)\( p^{43} T^{16} - 7025858435107045338 p^{49} T^{17} + 5482392324014 p^{56} T^{18} - 968282 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 67 | \( 1 - 1634150 T + 1335223111250 T^{2} + 50049565458787288022 T^{3} - \)\(10\!\cdots\!14\)\( T^{4} - \)\(61\!\cdots\!18\)\( T^{5} + \)\(14\!\cdots\!42\)\( T^{6} - \)\(32\!\cdots\!38\)\( T^{7} - \)\(55\!\cdots\!75\)\( T^{8} + \)\(38\!\cdots\!32\)\( T^{9} - \)\(53\!\cdots\!36\)\( T^{10} - \)\(20\!\cdots\!84\)\( T^{11} + \)\(83\!\cdots\!16\)\( T^{12} - \)\(36\!\cdots\!84\)\( T^{13} - \)\(51\!\cdots\!56\)\( T^{14} + \)\(14\!\cdots\!72\)\( T^{15} + \)\(65\!\cdots\!90\)\( T^{16} - \)\(10\!\cdots\!72\)\( T^{17} + \)\(18\!\cdots\!44\)\( T^{18} + \)\(29\!\cdots\!12\)\( T^{19} - \)\(17\!\cdots\!76\)\( T^{20} + \)\(29\!\cdots\!12\)\( p^{7} T^{21} + \)\(18\!\cdots\!44\)\( p^{14} T^{22} - \)\(10\!\cdots\!72\)\( p^{21} T^{23} + \)\(65\!\cdots\!90\)\( p^{28} T^{24} + \)\(14\!\cdots\!72\)\( p^{35} T^{25} - \)\(51\!\cdots\!56\)\( p^{42} T^{26} - \)\(36\!\cdots\!84\)\( p^{49} T^{27} + \)\(83\!\cdots\!16\)\( p^{56} T^{28} - \)\(20\!\cdots\!84\)\( p^{63} T^{29} - \)\(53\!\cdots\!36\)\( p^{70} T^{30} + \)\(38\!\cdots\!32\)\( p^{77} T^{31} - \)\(55\!\cdots\!75\)\( p^{84} T^{32} - \)\(32\!\cdots\!38\)\( p^{91} T^{33} + \)\(14\!\cdots\!42\)\( p^{98} T^{34} - \)\(61\!\cdots\!18\)\( p^{105} T^{35} - \)\(10\!\cdots\!14\)\( p^{112} T^{36} + 50049565458787288022 p^{119} T^{37} + 1335223111250 p^{126} T^{38} - 1634150 p^{133} T^{39} + p^{140} T^{40} \) | |
| 71 | \( 1 - 113490809626464 T^{2} + \)\(63\!\cdots\!30\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(62\!\cdots\!73\)\( T^{8} - \)\(13\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!60\)\( T^{12} - \)\(35\!\cdots\!52\)\( T^{14} + \)\(46\!\cdots\!46\)\( T^{16} - \)\(51\!\cdots\!48\)\( T^{18} + \)\(50\!\cdots\!72\)\( T^{20} - \)\(51\!\cdots\!48\)\( p^{14} T^{22} + \)\(46\!\cdots\!46\)\( p^{28} T^{24} - \)\(35\!\cdots\!52\)\( p^{42} T^{26} + \)\(23\!\cdots\!60\)\( p^{56} T^{28} - \)\(13\!\cdots\!88\)\( p^{70} T^{30} + \)\(62\!\cdots\!73\)\( p^{84} T^{32} - \)\(23\!\cdots\!92\)\( p^{98} T^{34} + \)\(63\!\cdots\!30\)\( p^{112} T^{36} - 113490809626464 p^{126} T^{38} + p^{140} T^{40} \) | |
| 73 | \( 1 + 2406004 T + 2894427624008 T^{2} + 70630819591091484700 T^{3} + \)\(13\!\cdots\!30\)\( T^{4} - \)\(51\!\cdots\!96\)\( T^{5} + \)\(85\!\cdots\!76\)\( T^{6} - \)\(30\!\cdots\!16\)\( T^{7} - \)\(39\!\cdots\!87\)\( T^{8} + \)\(26\!\cdots\!24\)\( T^{9} - \)\(26\!\cdots\!80\)\( T^{10} - \)\(85\!\cdots\!40\)\( T^{11} + \)\(76\!\cdots\!24\)\( T^{12} + \)\(16\!\cdots\!68\)\( T^{13} - \)\(13\!\cdots\!84\)\( T^{14} + \)\(28\!\cdots\!96\)\( T^{15} + \)\(68\!\cdots\!34\)\( T^{16} - \)\(20\!\cdots\!32\)\( T^{17} + \)\(36\!\cdots\!84\)\( p^{2} T^{18} + \)\(12\!\cdots\!60\)\( p T^{19} - \)\(11\!\cdots\!96\)\( T^{20} + \)\(12\!\cdots\!60\)\( p^{8} T^{21} + \)\(36\!\cdots\!84\)\( p^{16} T^{22} - \)\(20\!\cdots\!32\)\( p^{21} T^{23} + \)\(68\!\cdots\!34\)\( p^{28} T^{24} + \)\(28\!\cdots\!96\)\( p^{35} T^{25} - \)\(13\!\cdots\!84\)\( p^{42} T^{26} + \)\(16\!\cdots\!68\)\( p^{49} T^{27} + \)\(76\!\cdots\!24\)\( p^{56} T^{28} - \)\(85\!\cdots\!40\)\( p^{63} T^{29} - \)\(26\!\cdots\!80\)\( p^{70} T^{30} + \)\(26\!\cdots\!24\)\( p^{77} T^{31} - \)\(39\!\cdots\!87\)\( p^{84} T^{32} - \)\(30\!\cdots\!16\)\( p^{91} T^{33} + \)\(85\!\cdots\!76\)\( p^{98} T^{34} - \)\(51\!\cdots\!96\)\( p^{105} T^{35} + \)\(13\!\cdots\!30\)\( p^{112} T^{36} + 70630819591091484700 p^{119} T^{37} + 2894427624008 p^{126} T^{38} + 2406004 p^{133} T^{39} + p^{140} T^{40} \) | |
| 79 | \( ( 1 + 5874552 T + 125098797360534 T^{2} + \)\(51\!\cdots\!72\)\( T^{3} + \)\(71\!\cdots\!09\)\( T^{4} + \)\(23\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(79\!\cdots\!80\)\( T^{7} + \)\(77\!\cdots\!10\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!80\)\( p^{7} T^{11} + \)\(77\!\cdots\!10\)\( p^{14} T^{12} + \)\(79\!\cdots\!80\)\( p^{21} T^{13} + \)\(27\!\cdots\!00\)\( p^{28} T^{14} + \)\(23\!\cdots\!88\)\( p^{35} T^{15} + \)\(71\!\cdots\!09\)\( p^{42} T^{16} + \)\(51\!\cdots\!72\)\( p^{49} T^{17} + 125098797360534 p^{56} T^{18} + 5874552 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 83 | \( 1 + 13229814 T + 87513989237298 T^{2} + \)\(44\!\cdots\!46\)\( T^{3} + \)\(28\!\cdots\!54\)\( T^{4} + \)\(12\!\cdots\!54\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} - \)\(17\!\cdots\!18\)\( T^{7} - \)\(18\!\cdots\!87\)\( T^{8} - \)\(19\!\cdots\!72\)\( T^{9} - \)\(13\!\cdots\!40\)\( T^{10} - \)\(63\!\cdots\!84\)\( T^{11} - \)\(34\!\cdots\!68\)\( T^{12} - \)\(19\!\cdots\!44\)\( T^{13} - \)\(64\!\cdots\!68\)\( T^{14} - \)\(49\!\cdots\!52\)\( T^{15} + \)\(30\!\cdots\!86\)\( T^{16} + \)\(77\!\cdots\!68\)\( T^{17} + \)\(88\!\cdots\!48\)\( T^{18} + \)\(55\!\cdots\!48\)\( T^{19} + \)\(26\!\cdots\!28\)\( T^{20} + \)\(55\!\cdots\!48\)\( p^{7} T^{21} + \)\(88\!\cdots\!48\)\( p^{14} T^{22} + \)\(77\!\cdots\!68\)\( p^{21} T^{23} + \)\(30\!\cdots\!86\)\( p^{28} T^{24} - \)\(49\!\cdots\!52\)\( p^{35} T^{25} - \)\(64\!\cdots\!68\)\( p^{42} T^{26} - \)\(19\!\cdots\!44\)\( p^{49} T^{27} - \)\(34\!\cdots\!68\)\( p^{56} T^{28} - \)\(63\!\cdots\!84\)\( p^{63} T^{29} - \)\(13\!\cdots\!40\)\( p^{70} T^{30} - \)\(19\!\cdots\!72\)\( p^{77} T^{31} - \)\(18\!\cdots\!87\)\( p^{84} T^{32} - \)\(17\!\cdots\!18\)\( p^{91} T^{33} + \)\(12\!\cdots\!22\)\( p^{98} T^{34} + \)\(12\!\cdots\!54\)\( p^{105} T^{35} + \)\(28\!\cdots\!54\)\( p^{112} T^{36} + \)\(44\!\cdots\!46\)\( p^{119} T^{37} + 87513989237298 p^{126} T^{38} + 13229814 p^{133} T^{39} + p^{140} T^{40} \) | |
| 89 | \( 1 - 529620656263476 T^{2} + \)\(14\!\cdots\!78\)\( T^{4} - \)\(25\!\cdots\!88\)\( T^{6} + \)\(34\!\cdots\!01\)\( T^{8} - \)\(36\!\cdots\!40\)\( T^{10} + \)\(32\!\cdots\!04\)\( T^{12} - \)\(24\!\cdots\!52\)\( T^{14} + \)\(15\!\cdots\!82\)\( T^{16} - \)\(84\!\cdots\!44\)\( T^{18} + \)\(40\!\cdots\!68\)\( T^{20} - \)\(84\!\cdots\!44\)\( p^{14} T^{22} + \)\(15\!\cdots\!82\)\( p^{28} T^{24} - \)\(24\!\cdots\!52\)\( p^{42} T^{26} + \)\(32\!\cdots\!04\)\( p^{56} T^{28} - \)\(36\!\cdots\!40\)\( p^{70} T^{30} + \)\(34\!\cdots\!01\)\( p^{84} T^{32} - \)\(25\!\cdots\!88\)\( p^{98} T^{34} + \)\(14\!\cdots\!78\)\( p^{112} T^{36} - 529620656263476 p^{126} T^{38} + p^{140} T^{40} \) | |
| 97 | \( 1 - 24630564 T + 303332341479048 T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!58\)\( T^{4} - \)\(58\!\cdots\!88\)\( T^{5} + \)\(46\!\cdots\!48\)\( T^{6} - \)\(38\!\cdots\!68\)\( T^{7} + \)\(98\!\cdots\!41\)\( T^{8} + \)\(25\!\cdots\!48\)\( T^{9} - \)\(41\!\cdots\!12\)\( T^{10} + \)\(61\!\cdots\!04\)\( T^{11} - \)\(82\!\cdots\!76\)\( T^{12} + \)\(69\!\cdots\!44\)\( T^{13} - \)\(48\!\cdots\!72\)\( T^{14} + \)\(35\!\cdots\!96\)\( T^{15} - \)\(44\!\cdots\!58\)\( T^{16} - \)\(25\!\cdots\!20\)\( T^{17} + \)\(34\!\cdots\!08\)\( T^{18} - \)\(44\!\cdots\!92\)\( T^{19} + \)\(50\!\cdots\!68\)\( T^{20} - \)\(44\!\cdots\!92\)\( p^{7} T^{21} + \)\(34\!\cdots\!08\)\( p^{14} T^{22} - \)\(25\!\cdots\!20\)\( p^{21} T^{23} - \)\(44\!\cdots\!58\)\( p^{28} T^{24} + \)\(35\!\cdots\!96\)\( p^{35} T^{25} - \)\(48\!\cdots\!72\)\( p^{42} T^{26} + \)\(69\!\cdots\!44\)\( p^{49} T^{27} - \)\(82\!\cdots\!76\)\( p^{56} T^{28} + \)\(61\!\cdots\!04\)\( p^{63} T^{29} - \)\(41\!\cdots\!12\)\( p^{70} T^{30} + \)\(25\!\cdots\!48\)\( p^{77} T^{31} + \)\(98\!\cdots\!41\)\( p^{84} T^{32} - \)\(38\!\cdots\!68\)\( p^{91} T^{33} + \)\(46\!\cdots\!48\)\( p^{98} T^{34} - \)\(58\!\cdots\!88\)\( p^{105} T^{35} + \)\(59\!\cdots\!58\)\( p^{112} T^{36} - \)\(40\!\cdots\!80\)\( p^{119} T^{37} + 303332341479048 p^{126} T^{38} - 24630564 p^{133} T^{39} + p^{140} T^{40} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.98050229094656361239537187559, −1.90009751478653929533970491088, −1.82577363410621954871927974580, −1.81475051969689521059017458975, −1.71097726028357265924740786756, −1.54357111526638499626581584578, −1.41000430781501888315337398163, −1.32947257237228857215747913886, −1.31159222425906887588252531204, −1.19485764843328985773141166642, −1.15730069744251346430774529468, −1.15421896524718615633269003907, −1.12016374153563000582784467781, −1.08802085157323419224723096387, −0.815068851050532690154293469567, −0.807880988943844627036114482591, −0.70135948509436961878313098569, −0.69410928606380518396534239310, −0.62933449640372121385645318697, −0.43750299864976500499081982890, −0.30749627515957512998707419222, −0.18576943322749667945267519893, −0.13456541019486738442194250746, −0.04074223081672386081822997403, −0.009841785149539491369753285798, 0.009841785149539491369753285798, 0.04074223081672386081822997403, 0.13456541019486738442194250746, 0.18576943322749667945267519893, 0.30749627515957512998707419222, 0.43750299864976500499081982890, 0.62933449640372121385645318697, 0.69410928606380518396534239310, 0.70135948509436961878313098569, 0.807880988943844627036114482591, 0.815068851050532690154293469567, 1.08802085157323419224723096387, 1.12016374153563000582784467781, 1.15421896524718615633269003907, 1.15730069744251346430774529468, 1.19485764843328985773141166642, 1.31159222425906887588252531204, 1.32947257237228857215747913886, 1.41000430781501888315337398163, 1.54357111526638499626581584578, 1.71097726028357265924740786756, 1.81475051969689521059017458975, 1.82577363410621954871927974580, 1.90009751478653929533970491088, 1.98050229094656361239537187559
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.