Dirichlet series
| L(s) = 1 | − 18·2-s + 161·3-s + 972·4-s + 1.53e3·5-s − 2.89e3·6-s − 1.03e3·7-s − 1.02e4·8-s + 4.43e4·9-s − 2.75e4·10-s + 4.22e4·11-s + 1.56e5·12-s − 3.19e5·13-s + 1.86e4·14-s + 2.46e5·15-s + 5.69e5·16-s + 3.24e5·17-s − 7.97e5·18-s − 1.61e4·19-s + 1.49e6·20-s − 1.66e5·21-s − 7.59e5·22-s + 2.63e6·23-s − 1.65e6·24-s + 5.40e6·25-s + 5.75e6·26-s + 1.74e6·27-s − 1.00e6·28-s + ⋯ |
| L(s) = 1 | − 0.795·2-s + 1.14·3-s + 1.89·4-s + 1.09·5-s − 0.912·6-s − 0.163·7-s − 0.886·8-s + 2.25·9-s − 0.872·10-s + 0.869·11-s + 2.17·12-s − 3.10·13-s + 0.129·14-s + 1.25·15-s + 2.17·16-s + 0.942·17-s − 1.79·18-s − 0.0283·19-s + 2.08·20-s − 0.187·21-s − 0.691·22-s + 1.96·23-s − 1.01·24-s + 2.76·25-s + 2.46·26-s + 0.631·27-s − 0.309·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(370983.\) |
| Root analytic conductor: | \(1.89874\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} ,\ ( \ : [9/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(25.29084626\) |
| \(L(\frac12)\) | \(\approx\) | \(25.29084626\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + 148 p T + 73029 p^{3} T^{2} + 14753264 p^{5} T^{3} + 310110650 p^{8} T^{4} + 9799752 p^{12} T^{5} + 310110650 p^{17} T^{6} + 14753264 p^{23} T^{7} + 73029 p^{30} T^{8} + 148 p^{37} T^{9} + p^{45} T^{10} \) |
| good | 2 | \( 1 + 9 p T - 81 p^{3} T^{2} - 2361 p^{3} T^{3} - 5917 p^{4} T^{4} + 69339 p^{6} T^{5} + 141607 p^{10} T^{6} + 198195 p^{10} T^{7} - 3847261 p^{13} T^{8} + 4015647 p^{14} T^{9} + 50232041 p^{16} T^{10} + 4015647 p^{23} T^{11} - 3847261 p^{31} T^{12} + 198195 p^{37} T^{13} + 141607 p^{46} T^{14} + 69339 p^{51} T^{15} - 5917 p^{58} T^{16} - 2361 p^{66} T^{17} - 81 p^{75} T^{18} + 9 p^{82} T^{19} + p^{90} T^{20} \) |
| 3 | \( 1 - 161 T - 18380 T^{2} + 2782717 p T^{3} - 93832370 p^{2} T^{4} - 641117953 p^{3} T^{5} + 292724770226 p^{4} T^{6} - 17873823727963 p^{5} T^{7} + 456733649062009 p^{6} T^{8} + 27621943077444478 p^{7} T^{9} - 2552424382213598924 p^{8} T^{10} + 27621943077444478 p^{16} T^{11} + 456733649062009 p^{24} T^{12} - 17873823727963 p^{32} T^{13} + 292724770226 p^{40} T^{14} - 641117953 p^{48} T^{15} - 93832370 p^{56} T^{16} + 2782717 p^{64} T^{17} - 18380 p^{72} T^{18} - 161 p^{81} T^{19} + p^{90} T^{20} \) | |
| 5 | \( 1 - 1533 T - 3055722 T^{2} + 7403950197 T^{3} - 320742802594 T^{4} - 1324872515586711 p T^{5} + 312688030574125192 p^{2} T^{6} - 41723041724779465239 p^{4} T^{7} + \)\(52\!\cdots\!89\)\( p^{4} T^{8} + \)\(14\!\cdots\!02\)\( p^{5} T^{9} - \)\(89\!\cdots\!56\)\( p^{6} T^{10} + \)\(14\!\cdots\!02\)\( p^{14} T^{11} + \)\(52\!\cdots\!89\)\( p^{22} T^{12} - 41723041724779465239 p^{31} T^{13} + 312688030574125192 p^{38} T^{14} - 1324872515586711 p^{46} T^{15} - 320742802594 p^{54} T^{16} + 7403950197 p^{63} T^{17} - 3055722 p^{72} T^{18} - 1533 p^{81} T^{19} + p^{90} T^{20} \) | |
| 11 | \( 1 - 42213 T - 6326783460 T^{2} + 327121353142323 T^{3} + 17750681164843604918 T^{4} - \)\(93\!\cdots\!51\)\( T^{5} - \)\(48\!\cdots\!18\)\( T^{6} + \)\(92\!\cdots\!83\)\( T^{7} + \)\(18\!\cdots\!73\)\( T^{8} - \)\(42\!\cdots\!58\)\( T^{9} - \)\(56\!\cdots\!76\)\( T^{10} - \)\(42\!\cdots\!58\)\( p^{9} T^{11} + \)\(18\!\cdots\!73\)\( p^{18} T^{12} + \)\(92\!\cdots\!83\)\( p^{27} T^{13} - \)\(48\!\cdots\!18\)\( p^{36} T^{14} - \)\(93\!\cdots\!51\)\( p^{45} T^{15} + 17750681164843604918 p^{54} T^{16} + 327121353142323 p^{63} T^{17} - 6326783460 p^{72} T^{18} - 42213 p^{81} T^{19} + p^{90} T^{20} \) | |
| 13 | \( ( 1 + 159838 T + 52194785337 T^{2} + 6011850787968872 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(92\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!66\)\( p^{9} T^{6} + 6011850787968872 p^{18} T^{7} + 52194785337 p^{27} T^{8} + 159838 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 17 | \( 1 - 324681 T - 328834212334 T^{2} + 62765598850452981 T^{3} + \)\(38\!\cdots\!26\)\( p T^{4} - \)\(30\!\cdots\!55\)\( T^{5} - \)\(12\!\cdots\!16\)\( T^{6} + \)\(59\!\cdots\!33\)\( T^{7} + \)\(18\!\cdots\!29\)\( T^{8} - \)\(67\!\cdots\!18\)\( T^{9} - \)\(22\!\cdots\!44\)\( T^{10} - \)\(67\!\cdots\!18\)\( p^{9} T^{11} + \)\(18\!\cdots\!29\)\( p^{18} T^{12} + \)\(59\!\cdots\!33\)\( p^{27} T^{13} - \)\(12\!\cdots\!16\)\( p^{36} T^{14} - \)\(30\!\cdots\!55\)\( p^{45} T^{15} + \)\(38\!\cdots\!26\)\( p^{55} T^{16} + 62765598850452981 p^{63} T^{17} - 328834212334 p^{72} T^{18} - 324681 p^{81} T^{19} + p^{90} T^{20} \) | |
| 19 | \( 1 + 16121 T - 434040959924 T^{2} - 420368046459486167 T^{3} + \)\(18\!\cdots\!38\)\( T^{4} + \)\(22\!\cdots\!55\)\( T^{5} + \)\(44\!\cdots\!02\)\( T^{6} - \)\(88\!\cdots\!79\)\( T^{7} - \)\(41\!\cdots\!79\)\( T^{8} + \)\(95\!\cdots\!02\)\( T^{9} + \)\(20\!\cdots\!04\)\( T^{10} + \)\(95\!\cdots\!02\)\( p^{9} T^{11} - \)\(41\!\cdots\!79\)\( p^{18} T^{12} - \)\(88\!\cdots\!79\)\( p^{27} T^{13} + \)\(44\!\cdots\!02\)\( p^{36} T^{14} + \)\(22\!\cdots\!55\)\( p^{45} T^{15} + \)\(18\!\cdots\!38\)\( p^{54} T^{16} - 420368046459486167 p^{63} T^{17} - 434040959924 p^{72} T^{18} + 16121 p^{81} T^{19} + p^{90} T^{20} \) | |
| 23 | \( 1 - 2638863 T - 480527621872 T^{2} - 311022444212974755 T^{3} + \)\(13\!\cdots\!02\)\( T^{4} - \)\(34\!\cdots\!73\)\( T^{5} - \)\(90\!\cdots\!82\)\( T^{6} - \)\(28\!\cdots\!95\)\( T^{7} + \)\(11\!\cdots\!53\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{9} + \)\(39\!\cdots\!12\)\( T^{10} + \)\(12\!\cdots\!50\)\( p^{9} T^{11} + \)\(11\!\cdots\!53\)\( p^{18} T^{12} - \)\(28\!\cdots\!95\)\( p^{27} T^{13} - \)\(90\!\cdots\!82\)\( p^{36} T^{14} - \)\(34\!\cdots\!73\)\( p^{45} T^{15} + \)\(13\!\cdots\!02\)\( p^{54} T^{16} - 311022444212974755 p^{63} T^{17} - 480527621872 p^{72} T^{18} - 2638863 p^{81} T^{19} + p^{90} T^{20} \) | |
| 29 | \( ( 1 - 7646250 T + 88492818721737 T^{2} - \)\(45\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!74\)\( T^{4} - \)\(98\!\cdots\!56\)\( T^{5} + \)\(27\!\cdots\!74\)\( p^{9} T^{6} - \)\(45\!\cdots\!12\)\( p^{18} T^{7} + 88492818721737 p^{27} T^{8} - 7646250 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 31 | \( 1 - 19179237 T + 120960263219896 T^{2} - \)\(28\!\cdots\!01\)\( T^{3} + \)\(22\!\cdots\!62\)\( T^{4} - \)\(27\!\cdots\!71\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} - \)\(34\!\cdots\!25\)\( T^{7} + \)\(39\!\cdots\!25\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{9} + \)\(10\!\cdots\!20\)\( T^{10} - \)\(24\!\cdots\!10\)\( p^{9} T^{11} + \)\(39\!\cdots\!25\)\( p^{18} T^{12} - \)\(34\!\cdots\!25\)\( p^{27} T^{13} + \)\(10\!\cdots\!50\)\( p^{36} T^{14} - \)\(27\!\cdots\!71\)\( p^{45} T^{15} + \)\(22\!\cdots\!62\)\( p^{54} T^{16} - \)\(28\!\cdots\!01\)\( p^{63} T^{17} + 120960263219896 p^{72} T^{18} - 19179237 p^{81} T^{19} + p^{90} T^{20} \) | |
| 37 | \( 1 - 39566985 T + 448975929019726 T^{2} - \)\(17\!\cdots\!23\)\( T^{3} + \)\(74\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!03\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!29\)\( T^{7} + \)\(18\!\cdots\!53\)\( T^{8} - \)\(52\!\cdots\!46\)\( T^{9} - \)\(22\!\cdots\!36\)\( T^{10} - \)\(52\!\cdots\!46\)\( p^{9} T^{11} + \)\(18\!\cdots\!53\)\( p^{18} T^{12} + \)\(18\!\cdots\!29\)\( p^{27} T^{13} + \)\(10\!\cdots\!96\)\( p^{36} T^{14} - \)\(13\!\cdots\!03\)\( p^{45} T^{15} + \)\(74\!\cdots\!22\)\( p^{54} T^{16} - \)\(17\!\cdots\!23\)\( p^{63} T^{17} + 448975929019726 p^{72} T^{18} - 39566985 p^{81} T^{19} + p^{90} T^{20} \) | |
| 41 | \( ( 1 + 26718426 T + 918529945217061 T^{2} + \)\(23\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!46\)\( T^{4} + \)\(93\!\cdots\!60\)\( T^{5} + \)\(56\!\cdots\!46\)\( p^{9} T^{6} + \)\(23\!\cdots\!36\)\( p^{18} T^{7} + 918529945217061 p^{27} T^{8} + 26718426 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 50917996 T + 2693125333085463 T^{2} - \)\(86\!\cdots\!48\)\( T^{3} + \)\(25\!\cdots\!62\)\( T^{4} - \)\(60\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!62\)\( p^{9} T^{6} - \)\(86\!\cdots\!48\)\( p^{18} T^{7} + 2693125333085463 p^{27} T^{8} - 50917996 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 47 | \( 1 - 32509659 T - 3410631284393464 T^{2} + \)\(10\!\cdots\!49\)\( T^{3} + \)\(68\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!25\)\( T^{5} - \)\(11\!\cdots\!42\)\( T^{6} + \)\(13\!\cdots\!77\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} - \)\(49\!\cdots\!58\)\( T^{9} - \)\(21\!\cdots\!56\)\( T^{10} - \)\(49\!\cdots\!58\)\( p^{9} T^{11} + \)\(16\!\cdots\!61\)\( p^{18} T^{12} + \)\(13\!\cdots\!77\)\( p^{27} T^{13} - \)\(11\!\cdots\!42\)\( p^{36} T^{14} - \)\(15\!\cdots\!25\)\( p^{45} T^{15} + \)\(68\!\cdots\!10\)\( p^{54} T^{16} + \)\(10\!\cdots\!49\)\( p^{63} T^{17} - 3410631284393464 p^{72} T^{18} - 32509659 p^{81} T^{19} + p^{90} T^{20} \) | |
| 53 | \( 1 + 25714707 T - 5546541912716410 T^{2} - \)\(63\!\cdots\!55\)\( T^{3} - \)\(57\!\cdots\!02\)\( T^{4} + \)\(20\!\cdots\!13\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} + \)\(36\!\cdots\!85\)\( T^{7} - \)\(87\!\cdots\!67\)\( T^{8} - \)\(15\!\cdots\!94\)\( T^{9} - \)\(13\!\cdots\!68\)\( T^{10} - \)\(15\!\cdots\!94\)\( p^{9} T^{11} - \)\(87\!\cdots\!67\)\( p^{18} T^{12} + \)\(36\!\cdots\!85\)\( p^{27} T^{13} + \)\(13\!\cdots\!12\)\( p^{36} T^{14} + \)\(20\!\cdots\!13\)\( p^{45} T^{15} - \)\(57\!\cdots\!02\)\( p^{54} T^{16} - \)\(63\!\cdots\!55\)\( p^{63} T^{17} - 5546541912716410 p^{72} T^{18} + 25714707 p^{81} T^{19} + p^{90} T^{20} \) | |
| 59 | \( 1 - 46776513 T - 27103348513629988 T^{2} + \)\(28\!\cdots\!83\)\( T^{3} + \)\(37\!\cdots\!62\)\( T^{4} - \)\(54\!\cdots\!63\)\( T^{5} - \)\(15\!\cdots\!10\)\( T^{6} + \)\(60\!\cdots\!99\)\( T^{7} - \)\(16\!\cdots\!03\)\( T^{8} - \)\(22\!\cdots\!46\)\( T^{9} + \)\(32\!\cdots\!56\)\( T^{10} - \)\(22\!\cdots\!46\)\( p^{9} T^{11} - \)\(16\!\cdots\!03\)\( p^{18} T^{12} + \)\(60\!\cdots\!99\)\( p^{27} T^{13} - \)\(15\!\cdots\!10\)\( p^{36} T^{14} - \)\(54\!\cdots\!63\)\( p^{45} T^{15} + \)\(37\!\cdots\!62\)\( p^{54} T^{16} + \)\(28\!\cdots\!83\)\( p^{63} T^{17} - 27103348513629988 p^{72} T^{18} - 46776513 p^{81} T^{19} + p^{90} T^{20} \) | |
| 61 | \( 1 + 113075039 T - 40080640716766298 T^{2} - \)\(29\!\cdots\!31\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} + \)\(48\!\cdots\!45\)\( T^{5} - \)\(22\!\cdots\!44\)\( T^{6} - \)\(47\!\cdots\!63\)\( T^{7} + \)\(35\!\cdots\!01\)\( T^{8} + \)\(19\!\cdots\!06\)\( T^{9} - \)\(45\!\cdots\!48\)\( T^{10} + \)\(19\!\cdots\!06\)\( p^{9} T^{11} + \)\(35\!\cdots\!01\)\( p^{18} T^{12} - \)\(47\!\cdots\!63\)\( p^{27} T^{13} - \)\(22\!\cdots\!44\)\( p^{36} T^{14} + \)\(48\!\cdots\!45\)\( p^{45} T^{15} + \)\(11\!\cdots\!98\)\( p^{54} T^{16} - \)\(29\!\cdots\!31\)\( p^{63} T^{17} - 40080640716766298 p^{72} T^{18} + 113075039 p^{81} T^{19} + p^{90} T^{20} \) | |
| 67 | \( 1 + 126707879 T - 43788049157476460 T^{2} - \)\(33\!\cdots\!93\)\( T^{3} + \)\(41\!\cdots\!94\)\( T^{4} - \)\(50\!\cdots\!91\)\( T^{5} - \)\(15\!\cdots\!78\)\( T^{6} - \)\(40\!\cdots\!93\)\( T^{7} + \)\(55\!\cdots\!05\)\( T^{8} + \)\(14\!\cdots\!02\)\( T^{9} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(14\!\cdots\!02\)\( p^{9} T^{11} + \)\(55\!\cdots\!05\)\( p^{18} T^{12} - \)\(40\!\cdots\!93\)\( p^{27} T^{13} - \)\(15\!\cdots\!78\)\( p^{36} T^{14} - \)\(50\!\cdots\!91\)\( p^{45} T^{15} + \)\(41\!\cdots\!94\)\( p^{54} T^{16} - \)\(33\!\cdots\!93\)\( p^{63} T^{17} - 43788049157476460 p^{72} T^{18} + 126707879 p^{81} T^{19} + p^{90} T^{20} \) | |
| 71 | \( ( 1 + 594368016 T + 309193099286863459 T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(31\!\cdots\!50\)\( T^{4} + \)\(69\!\cdots\!04\)\( T^{5} + \)\(31\!\cdots\!50\)\( p^{9} T^{6} + \)\(10\!\cdots\!72\)\( p^{18} T^{7} + 309193099286863459 p^{27} T^{8} + 594368016 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 73 | \( 1 + 859257651 T + 199070264658270994 T^{2} + \)\(11\!\cdots\!21\)\( T^{3} + \)\(18\!\cdots\!02\)\( T^{4} + \)\(85\!\cdots\!25\)\( T^{5} + \)\(51\!\cdots\!68\)\( T^{6} + \)\(92\!\cdots\!93\)\( T^{7} + \)\(17\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!82\)\( T^{9} + \)\(16\!\cdots\!12\)\( T^{10} + \)\(29\!\cdots\!82\)\( p^{9} T^{11} + \)\(17\!\cdots\!81\)\( p^{18} T^{12} + \)\(92\!\cdots\!93\)\( p^{27} T^{13} + \)\(51\!\cdots\!68\)\( p^{36} T^{14} + \)\(85\!\cdots\!25\)\( p^{45} T^{15} + \)\(18\!\cdots\!02\)\( p^{54} T^{16} + \)\(11\!\cdots\!21\)\( p^{63} T^{17} + 199070264658270994 p^{72} T^{18} + 859257651 p^{81} T^{19} + p^{90} T^{20} \) | |
| 79 | \( 1 + 527065417 T - 105527969267521112 T^{2} + \)\(26\!\cdots\!45\)\( T^{3} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(24\!\cdots\!85\)\( T^{5} + \)\(26\!\cdots\!82\)\( T^{6} + \)\(34\!\cdots\!09\)\( T^{7} - \)\(57\!\cdots\!11\)\( T^{8} + \)\(65\!\cdots\!50\)\( T^{9} + \)\(17\!\cdots\!24\)\( T^{10} + \)\(65\!\cdots\!50\)\( p^{9} T^{11} - \)\(57\!\cdots\!11\)\( p^{18} T^{12} + \)\(34\!\cdots\!09\)\( p^{27} T^{13} + \)\(26\!\cdots\!82\)\( p^{36} T^{14} - \)\(24\!\cdots\!85\)\( p^{45} T^{15} + \)\(64\!\cdots\!58\)\( p^{54} T^{16} + \)\(26\!\cdots\!45\)\( p^{63} T^{17} - 105527969267521112 p^{72} T^{18} + 527065417 p^{81} T^{19} + p^{90} T^{20} \) | |
| 83 | \( ( 1 + 72431604 T + 478832888086868191 T^{2} + \)\(61\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!02\)\( p^{9} T^{6} + \)\(61\!\cdots\!88\)\( p^{18} T^{7} + 478832888086868191 p^{27} T^{8} + 72431604 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
| 89 | \( 1 - 1661554797 T + 593701432891643778 T^{2} + \)\(11\!\cdots\!53\)\( T^{3} + \)\(34\!\cdots\!50\)\( T^{4} - \)\(23\!\cdots\!83\)\( T^{5} - \)\(69\!\cdots\!72\)\( T^{6} + \)\(66\!\cdots\!81\)\( T^{7} - \)\(20\!\cdots\!67\)\( T^{8} + \)\(28\!\cdots\!94\)\( T^{9} - \)\(25\!\cdots\!60\)\( T^{10} + \)\(28\!\cdots\!94\)\( p^{9} T^{11} - \)\(20\!\cdots\!67\)\( p^{18} T^{12} + \)\(66\!\cdots\!81\)\( p^{27} T^{13} - \)\(69\!\cdots\!72\)\( p^{36} T^{14} - \)\(23\!\cdots\!83\)\( p^{45} T^{15} + \)\(34\!\cdots\!50\)\( p^{54} T^{16} + \)\(11\!\cdots\!53\)\( p^{63} T^{17} + 593701432891643778 p^{72} T^{18} - 1661554797 p^{81} T^{19} + p^{90} T^{20} \) | |
| 97 | \( ( 1 - 434885094 T + 2848708413464846333 T^{2} - \)\(88\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!86\)\( T^{4} - \)\(83\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!86\)\( p^{9} T^{6} - \)\(88\!\cdots\!24\)\( p^{18} T^{7} + 2848708413464846333 p^{27} T^{8} - 434885094 p^{36} T^{9} + p^{45} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87453610745396513767716940570, −7.66272054885647040106170760717, −7.50562393348574146132687568386, −7.44447565576539831575856360004, −7.30600903137144338908787238906, −7.05709208054214529132260120731, −6.66734352418330640950634354107, −6.46103341581772464492173359665, −6.13954108144853489870871027378, −5.99336204531668498070537649723, −5.91543591936753400083781865257, −4.86994364224670782644975698177, −4.77405787646261198101472917143, −4.68124674324458104047962861095, −4.60343439957740483311392502826, −4.06763266954747862157287758756, −3.17932847120696140692856041184, −2.84092081527992741803542361822, −2.73196361127006012210716566321, −2.65236173140569496900342592272, −2.31640850783122798197191403210, −1.44004437017363897511066140888, −1.18089554118922301207582774195, −1.10702421639492505080252350268, −0.73083778939571590925591274738, 0.73083778939571590925591274738, 1.10702421639492505080252350268, 1.18089554118922301207582774195, 1.44004437017363897511066140888, 2.31640850783122798197191403210, 2.65236173140569496900342592272, 2.73196361127006012210716566321, 2.84092081527992741803542361822, 3.17932847120696140692856041184, 4.06763266954747862157287758756, 4.60343439957740483311392502826, 4.68124674324458104047962861095, 4.77405787646261198101472917143, 4.86994364224670782644975698177, 5.91543591936753400083781865257, 5.99336204531668498070537649723, 6.13954108144853489870871027378, 6.46103341581772464492173359665, 6.66734352418330640950634354107, 7.05709208054214529132260120731, 7.30600903137144338908787238906, 7.44447565576539831575856360004, 7.50562393348574146132687568386, 7.66272054885647040106170760717, 7.87453610745396513767716940570
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.