Dirichlet series
| L(s) = 1 | + 112·2-s + 165·3-s + 5.37e3·4-s + 909·5-s + 1.84e4·6-s + 3.69e3·7-s + 1.14e5·8-s + 5.63e4·9-s + 1.01e5·10-s + 1.56e4·11-s + 8.87e5·12-s + 6.42e3·13-s + 4.13e5·14-s + 1.49e5·15-s − 9.17e5·16-s + 3.13e5·17-s + 6.31e6·18-s + 1.13e6·19-s + 4.88e6·20-s + 6.09e5·21-s + 1.75e6·22-s + 3.44e6·23-s + 1.89e7·24-s + 6.04e6·25-s + 7.19e5·26-s + 9.34e6·27-s + 1.98e7·28-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 1.17·3-s + 21/2·4-s + 0.650·5-s + 5.82·6-s + 0.581·7-s + 9.89·8-s + 2.86·9-s + 3.21·10-s + 0.322·11-s + 12.3·12-s + 0.0623·13-s + 2.87·14-s + 0.764·15-s − 7/2·16-s + 0.910·17-s + 14.1·18-s + 1.99·19-s + 6.82·20-s + 0.683·21-s + 1.59·22-s + 2.57·23-s + 11.6·24-s + 3.09·25-s + 0.308·26-s + 3.38·27-s + 6.10·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 19^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.20974\times 10^{18}\) |
| Root analytic conductor: | \(4.42395\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 19^{14} ,\ ( \ : [9/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(393.1990624\) |
| \(L(\frac12)\) | \(\approx\) | \(393.1990624\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{4} T + p^{8} T^{2} )^{7} \) |
| 19 | \( 1 - 59696 p T + 2015220901 p^{2} T^{2} - 32602687775808 p^{3} T^{3} + 12134830343874732 p^{5} T^{4} + 3202236035452171264 p^{8} T^{5} - \)\(66\!\cdots\!56\)\( p^{11} T^{6} + \)\(39\!\cdots\!80\)\( p^{15} T^{7} - \)\(66\!\cdots\!56\)\( p^{20} T^{8} + 3202236035452171264 p^{26} T^{9} + 12134830343874732 p^{32} T^{10} - 32602687775808 p^{39} T^{11} + 2015220901 p^{47} T^{12} - 59696 p^{55} T^{13} + p^{63} T^{14} \) | |
| good | 3 | \( 1 - 55 p T - 29135 T^{2} + 1588046 p T^{3} + 451677226 T^{4} - 20893699168 p T^{5} + 544556908943 p^{3} T^{6} - 59316764529179 p^{3} T^{7} - 7368535461248440 p^{4} T^{8} + 90022882066726139 p^{6} T^{9} + 9451308112414253795 p^{6} T^{10} - \)\(49\!\cdots\!44\)\( p^{7} T^{11} + \)\(29\!\cdots\!85\)\( p^{12} T^{12} + \)\(42\!\cdots\!33\)\( p^{9} T^{13} - \)\(10\!\cdots\!46\)\( p^{10} T^{14} + \)\(42\!\cdots\!33\)\( p^{18} T^{15} + \)\(29\!\cdots\!85\)\( p^{30} T^{16} - \)\(49\!\cdots\!44\)\( p^{34} T^{17} + 9451308112414253795 p^{42} T^{18} + 90022882066726139 p^{51} T^{19} - 7368535461248440 p^{58} T^{20} - 59316764529179 p^{66} T^{21} + 544556908943 p^{75} T^{22} - 20893699168 p^{82} T^{23} + 451677226 p^{90} T^{24} + 1588046 p^{100} T^{25} - 29135 p^{108} T^{26} - 55 p^{118} T^{27} + p^{126} T^{28} \) |
| 5 | \( 1 - 909 T - 5223473 T^{2} + 6842219956 T^{3} + 7631944964557 T^{4} - 17998555332534719 T^{5} - 647245457166208194 T^{6} + \)\(49\!\cdots\!93\)\( p T^{7} + \)\(15\!\cdots\!66\)\( p T^{8} - \)\(17\!\cdots\!13\)\( p^{2} T^{9} - \)\(16\!\cdots\!94\)\( p^{2} T^{10} + \)\(63\!\cdots\!39\)\( p^{3} T^{11} + \)\(48\!\cdots\!62\)\( p^{4} T^{12} - \)\(19\!\cdots\!39\)\( p^{5} T^{13} + \)\(15\!\cdots\!86\)\( p^{6} T^{14} - \)\(19\!\cdots\!39\)\( p^{14} T^{15} + \)\(48\!\cdots\!62\)\( p^{22} T^{16} + \)\(63\!\cdots\!39\)\( p^{30} T^{17} - \)\(16\!\cdots\!94\)\( p^{38} T^{18} - \)\(17\!\cdots\!13\)\( p^{47} T^{19} + \)\(15\!\cdots\!66\)\( p^{55} T^{20} + \)\(49\!\cdots\!93\)\( p^{64} T^{21} - 647245457166208194 p^{72} T^{22} - 17998555332534719 p^{81} T^{23} + 7631944964557 p^{90} T^{24} + 6842219956 p^{99} T^{25} - 5223473 p^{108} T^{26} - 909 p^{117} T^{27} + p^{126} T^{28} \) | |
| 7 | \( ( 1 - 1846 T + 162836395 T^{2} - 395563390220 T^{3} + 1873414615513493 p T^{4} - 816651797390903402 p^{2} T^{5} + \)\(20\!\cdots\!75\)\( p^{3} T^{6} - \)\(89\!\cdots\!72\)\( p^{4} T^{7} + \)\(20\!\cdots\!75\)\( p^{12} T^{8} - 816651797390903402 p^{20} T^{9} + 1873414615513493 p^{28} T^{10} - 395563390220 p^{36} T^{11} + 162836395 p^{45} T^{12} - 1846 p^{54} T^{13} + p^{63} T^{14} )^{2} \) | |
| 11 | \( ( 1 - 7828 T + 10744399421 T^{2} + 73506903973280 T^{3} + 49953305391936986260 T^{4} + \)\(99\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(36\!\cdots\!96\)\( T^{7} + \)\(14\!\cdots\!48\)\( p^{9} T^{8} + \)\(99\!\cdots\!40\)\( p^{18} T^{9} + 49953305391936986260 p^{27} T^{10} + 73506903973280 p^{36} T^{11} + 10744399421 p^{45} T^{12} - 7828 p^{54} T^{13} + p^{63} T^{14} )^{2} \) | |
| 13 | \( 1 - 6423 T - 2441559433 p T^{2} + 11712779229316 p T^{3} + \)\(42\!\cdots\!71\)\( T^{4} + \)\(51\!\cdots\!53\)\( T^{5} - \)\(29\!\cdots\!88\)\( T^{6} - \)\(29\!\cdots\!57\)\( T^{7} + \)\(84\!\cdots\!86\)\( T^{8} + \)\(56\!\cdots\!57\)\( T^{9} + \)\(58\!\cdots\!56\)\( T^{10} - \)\(51\!\cdots\!37\)\( T^{11} - \)\(15\!\cdots\!08\)\( T^{12} + \)\(19\!\cdots\!87\)\( T^{13} + \)\(20\!\cdots\!86\)\( T^{14} + \)\(19\!\cdots\!87\)\( p^{9} T^{15} - \)\(15\!\cdots\!08\)\( p^{18} T^{16} - \)\(51\!\cdots\!37\)\( p^{27} T^{17} + \)\(58\!\cdots\!56\)\( p^{36} T^{18} + \)\(56\!\cdots\!57\)\( p^{45} T^{19} + \)\(84\!\cdots\!86\)\( p^{54} T^{20} - \)\(29\!\cdots\!57\)\( p^{63} T^{21} - \)\(29\!\cdots\!88\)\( p^{72} T^{22} + \)\(51\!\cdots\!53\)\( p^{81} T^{23} + \)\(42\!\cdots\!71\)\( p^{90} T^{24} + 11712779229316 p^{100} T^{25} - 2441559433 p^{109} T^{26} - 6423 p^{117} T^{27} + p^{126} T^{28} \) | |
| 17 | \( 1 - 18451 p T - 468818776657 T^{2} + 96272340434591936 T^{3} + \)\(12\!\cdots\!27\)\( T^{4} - \)\(13\!\cdots\!11\)\( T^{5} - \)\(24\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!15\)\( T^{7} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(25\!\cdots\!43\)\( T^{9} - \)\(51\!\cdots\!04\)\( T^{10} + \)\(19\!\cdots\!23\)\( p T^{11} + \)\(14\!\cdots\!52\)\( p^{3} T^{12} - \)\(42\!\cdots\!33\)\( p^{3} T^{13} - \)\(11\!\cdots\!66\)\( p^{4} T^{14} - \)\(42\!\cdots\!33\)\( p^{12} T^{15} + \)\(14\!\cdots\!52\)\( p^{21} T^{16} + \)\(19\!\cdots\!23\)\( p^{28} T^{17} - \)\(51\!\cdots\!04\)\( p^{36} T^{18} - \)\(25\!\cdots\!43\)\( p^{45} T^{19} + \)\(36\!\cdots\!46\)\( p^{54} T^{20} + \)\(17\!\cdots\!15\)\( p^{63} T^{21} - \)\(24\!\cdots\!56\)\( p^{72} T^{22} - \)\(13\!\cdots\!11\)\( p^{81} T^{23} + \)\(12\!\cdots\!27\)\( p^{90} T^{24} + 96272340434591936 p^{99} T^{25} - 468818776657 p^{108} T^{26} - 18451 p^{118} T^{27} + p^{126} T^{28} \) | |
| 23 | \( 1 - 3449345 T + 1487353643681 T^{2} + 11807928096761327694 T^{3} - \)\(24\!\cdots\!09\)\( T^{4} + \)\(16\!\cdots\!87\)\( T^{5} + \)\(21\!\cdots\!82\)\( T^{6} - \)\(84\!\cdots\!91\)\( T^{7} + \)\(14\!\cdots\!92\)\( T^{8} - \)\(45\!\cdots\!03\)\( T^{9} - \)\(26\!\cdots\!26\)\( T^{10} + \)\(46\!\cdots\!79\)\( T^{11} - \)\(15\!\cdots\!40\)\( T^{12} - \)\(46\!\cdots\!33\)\( T^{13} + \)\(92\!\cdots\!54\)\( T^{14} - \)\(46\!\cdots\!33\)\( p^{9} T^{15} - \)\(15\!\cdots\!40\)\( p^{18} T^{16} + \)\(46\!\cdots\!79\)\( p^{27} T^{17} - \)\(26\!\cdots\!26\)\( p^{36} T^{18} - \)\(45\!\cdots\!03\)\( p^{45} T^{19} + \)\(14\!\cdots\!92\)\( p^{54} T^{20} - \)\(84\!\cdots\!91\)\( p^{63} T^{21} + \)\(21\!\cdots\!82\)\( p^{72} T^{22} + \)\(16\!\cdots\!87\)\( p^{81} T^{23} - \)\(24\!\cdots\!09\)\( p^{90} T^{24} + 11807928096761327694 p^{99} T^{25} + 1487353643681 p^{108} T^{26} - 3449345 p^{117} T^{27} + p^{126} T^{28} \) | |
| 29 | \( 1 + 7002615 T - 21892482805901 T^{2} - 3445114076256756656 p T^{3} + \)\(15\!\cdots\!93\)\( T^{4} + \)\(53\!\cdots\!81\)\( T^{5} - \)\(18\!\cdots\!26\)\( T^{6} + \)\(13\!\cdots\!01\)\( T^{7} + \)\(57\!\cdots\!46\)\( T^{8} + \)\(72\!\cdots\!31\)\( T^{9} + \)\(26\!\cdots\!38\)\( T^{10} + \)\(24\!\cdots\!71\)\( T^{11} + \)\(25\!\cdots\!78\)\( T^{12} + \)\(19\!\cdots\!61\)\( T^{13} + \)\(24\!\cdots\!34\)\( T^{14} + \)\(19\!\cdots\!61\)\( p^{9} T^{15} + \)\(25\!\cdots\!78\)\( p^{18} T^{16} + \)\(24\!\cdots\!71\)\( p^{27} T^{17} + \)\(26\!\cdots\!38\)\( p^{36} T^{18} + \)\(72\!\cdots\!31\)\( p^{45} T^{19} + \)\(57\!\cdots\!46\)\( p^{54} T^{20} + \)\(13\!\cdots\!01\)\( p^{63} T^{21} - \)\(18\!\cdots\!26\)\( p^{72} T^{22} + \)\(53\!\cdots\!81\)\( p^{81} T^{23} + \)\(15\!\cdots\!93\)\( p^{90} T^{24} - 3445114076256756656 p^{100} T^{25} - 21892482805901 p^{108} T^{26} + 7002615 p^{117} T^{27} + p^{126} T^{28} \) | |
| 31 | \( ( 1 + 301074 p T + 148134985391023 T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!35\)\( T^{4} + \)\(57\!\cdots\!94\)\( T^{5} + \)\(40\!\cdots\!61\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(40\!\cdots\!61\)\( p^{9} T^{8} + \)\(57\!\cdots\!94\)\( p^{18} T^{9} + \)\(10\!\cdots\!35\)\( p^{27} T^{10} + \)\(10\!\cdots\!40\)\( p^{36} T^{11} + 148134985391023 p^{45} T^{12} + 301074 p^{55} T^{13} + p^{63} T^{14} )^{2} \) | |
| 37 | \( ( 1 + 3433040 T + 360370797123145 T^{2} - \)\(96\!\cdots\!48\)\( T^{3} + \)\(63\!\cdots\!59\)\( T^{4} - \)\(35\!\cdots\!32\)\( T^{5} + \)\(11\!\cdots\!75\)\( T^{6} - \)\(43\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!75\)\( p^{9} T^{8} - \)\(35\!\cdots\!32\)\( p^{18} T^{9} + \)\(63\!\cdots\!59\)\( p^{27} T^{10} - \)\(96\!\cdots\!48\)\( p^{36} T^{11} + 360370797123145 p^{45} T^{12} + 3433040 p^{54} T^{13} + p^{63} T^{14} )^{2} \) | |
| 41 | \( 1 - 3564107 T - 1773733816349839 T^{2} + \)\(90\!\cdots\!54\)\( T^{3} + \)\(16\!\cdots\!04\)\( T^{4} - \)\(94\!\cdots\!96\)\( T^{5} - \)\(11\!\cdots\!87\)\( T^{6} + \)\(55\!\cdots\!75\)\( T^{7} + \)\(60\!\cdots\!64\)\( T^{8} - \)\(21\!\cdots\!71\)\( T^{9} - \)\(27\!\cdots\!41\)\( T^{10} + \)\(53\!\cdots\!92\)\( T^{11} + \)\(11\!\cdots\!95\)\( T^{12} - \)\(62\!\cdots\!43\)\( T^{13} - \)\(39\!\cdots\!22\)\( T^{14} - \)\(62\!\cdots\!43\)\( p^{9} T^{15} + \)\(11\!\cdots\!95\)\( p^{18} T^{16} + \)\(53\!\cdots\!92\)\( p^{27} T^{17} - \)\(27\!\cdots\!41\)\( p^{36} T^{18} - \)\(21\!\cdots\!71\)\( p^{45} T^{19} + \)\(60\!\cdots\!64\)\( p^{54} T^{20} + \)\(55\!\cdots\!75\)\( p^{63} T^{21} - \)\(11\!\cdots\!87\)\( p^{72} T^{22} - \)\(94\!\cdots\!96\)\( p^{81} T^{23} + \)\(16\!\cdots\!04\)\( p^{90} T^{24} + \)\(90\!\cdots\!54\)\( p^{99} T^{25} - 1773733816349839 p^{108} T^{26} - 3564107 p^{117} T^{27} + p^{126} T^{28} \) | |
| 43 | \( 1 - 19837521 T - 1509371539523275 T^{2} + \)\(57\!\cdots\!50\)\( T^{3} + \)\(60\!\cdots\!57\)\( T^{4} - \)\(55\!\cdots\!51\)\( T^{5} + \)\(49\!\cdots\!16\)\( T^{6} + \)\(22\!\cdots\!33\)\( T^{7} - \)\(56\!\cdots\!60\)\( T^{8} + \)\(33\!\cdots\!57\)\( T^{9} + \)\(12\!\cdots\!68\)\( T^{10} - \)\(69\!\cdots\!55\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} + \)\(20\!\cdots\!03\)\( T^{13} - \)\(12\!\cdots\!90\)\( T^{14} + \)\(20\!\cdots\!03\)\( p^{9} T^{15} + \)\(13\!\cdots\!30\)\( p^{18} T^{16} - \)\(69\!\cdots\!55\)\( p^{27} T^{17} + \)\(12\!\cdots\!68\)\( p^{36} T^{18} + \)\(33\!\cdots\!57\)\( p^{45} T^{19} - \)\(56\!\cdots\!60\)\( p^{54} T^{20} + \)\(22\!\cdots\!33\)\( p^{63} T^{21} + \)\(49\!\cdots\!16\)\( p^{72} T^{22} - \)\(55\!\cdots\!51\)\( p^{81} T^{23} + \)\(60\!\cdots\!57\)\( p^{90} T^{24} + \)\(57\!\cdots\!50\)\( p^{99} T^{25} - 1509371539523275 p^{108} T^{26} - 19837521 p^{117} T^{27} + p^{126} T^{28} \) | |
| 47 | \( 1 - 60353825 T - 2620632466749919 T^{2} + \)\(14\!\cdots\!26\)\( T^{3} + \)\(66\!\cdots\!83\)\( T^{4} - \)\(17\!\cdots\!33\)\( T^{5} - \)\(14\!\cdots\!10\)\( T^{6} + \)\(24\!\cdots\!69\)\( T^{7} + \)\(20\!\cdots\!12\)\( T^{8} - \)\(23\!\cdots\!51\)\( T^{9} - \)\(28\!\cdots\!46\)\( T^{10} + \)\(25\!\cdots\!63\)\( T^{11} + \)\(40\!\cdots\!76\)\( T^{12} - \)\(18\!\cdots\!49\)\( T^{13} - \)\(46\!\cdots\!66\)\( T^{14} - \)\(18\!\cdots\!49\)\( p^{9} T^{15} + \)\(40\!\cdots\!76\)\( p^{18} T^{16} + \)\(25\!\cdots\!63\)\( p^{27} T^{17} - \)\(28\!\cdots\!46\)\( p^{36} T^{18} - \)\(23\!\cdots\!51\)\( p^{45} T^{19} + \)\(20\!\cdots\!12\)\( p^{54} T^{20} + \)\(24\!\cdots\!69\)\( p^{63} T^{21} - \)\(14\!\cdots\!10\)\( p^{72} T^{22} - \)\(17\!\cdots\!33\)\( p^{81} T^{23} + \)\(66\!\cdots\!83\)\( p^{90} T^{24} + \)\(14\!\cdots\!26\)\( p^{99} T^{25} - 2620632466749919 p^{108} T^{26} - 60353825 p^{117} T^{27} + p^{126} T^{28} \) | |
| 53 | \( 1 - 54744235 T - 8539409640633709 T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(38\!\cdots\!91\)\( T^{4} - \)\(10\!\cdots\!91\)\( T^{5} - \)\(98\!\cdots\!36\)\( T^{6} - \)\(72\!\cdots\!37\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} + \)\(12\!\cdots\!09\)\( T^{9} + \)\(22\!\cdots\!08\)\( T^{10} - \)\(36\!\cdots\!77\)\( T^{11} - \)\(19\!\cdots\!20\)\( T^{12} + \)\(96\!\cdots\!43\)\( p T^{13} + \)\(70\!\cdots\!90\)\( T^{14} + \)\(96\!\cdots\!43\)\( p^{10} T^{15} - \)\(19\!\cdots\!20\)\( p^{18} T^{16} - \)\(36\!\cdots\!77\)\( p^{27} T^{17} + \)\(22\!\cdots\!08\)\( p^{36} T^{18} + \)\(12\!\cdots\!09\)\( p^{45} T^{19} + \)\(11\!\cdots\!22\)\( p^{54} T^{20} - \)\(72\!\cdots\!37\)\( p^{63} T^{21} - \)\(98\!\cdots\!36\)\( p^{72} T^{22} - \)\(10\!\cdots\!91\)\( p^{81} T^{23} + \)\(38\!\cdots\!91\)\( p^{90} T^{24} + \)\(38\!\cdots\!48\)\( p^{99} T^{25} - 8539409640633709 p^{108} T^{26} - 54744235 p^{117} T^{27} + p^{126} T^{28} \) | |
| 59 | \( 1 - 164456585 T + 645894830577137 T^{2} - \)\(90\!\cdots\!30\)\( T^{3} + \)\(16\!\cdots\!66\)\( T^{4} - \)\(67\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!93\)\( T^{6} - \)\(18\!\cdots\!05\)\( T^{7} - \)\(83\!\cdots\!48\)\( T^{8} + \)\(11\!\cdots\!75\)\( T^{9} + \)\(61\!\cdots\!19\)\( T^{10} + \)\(73\!\cdots\!60\)\( T^{11} - \)\(30\!\cdots\!83\)\( T^{12} + \)\(71\!\cdots\!75\)\( T^{13} - \)\(14\!\cdots\!02\)\( T^{14} + \)\(71\!\cdots\!75\)\( p^{9} T^{15} - \)\(30\!\cdots\!83\)\( p^{18} T^{16} + \)\(73\!\cdots\!60\)\( p^{27} T^{17} + \)\(61\!\cdots\!19\)\( p^{36} T^{18} + \)\(11\!\cdots\!75\)\( p^{45} T^{19} - \)\(83\!\cdots\!48\)\( p^{54} T^{20} - \)\(18\!\cdots\!05\)\( p^{63} T^{21} + \)\(10\!\cdots\!93\)\( p^{72} T^{22} - \)\(67\!\cdots\!40\)\( p^{81} T^{23} + \)\(16\!\cdots\!66\)\( p^{90} T^{24} - \)\(90\!\cdots\!30\)\( p^{99} T^{25} + 645894830577137 p^{108} T^{26} - 164456585 p^{117} T^{27} + p^{126} T^{28} \) | |
| 61 | \( 1 - 49328881 T - 42356131219314273 T^{2} + \)\(13\!\cdots\!32\)\( T^{3} + \)\(90\!\cdots\!57\)\( T^{4} - \)\(16\!\cdots\!99\)\( T^{5} - \)\(13\!\cdots\!30\)\( T^{6} + \)\(17\!\cdots\!17\)\( T^{7} + \)\(14\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!61\)\( T^{9} - \)\(93\!\cdots\!10\)\( T^{10} + \)\(55\!\cdots\!99\)\( T^{11} + \)\(63\!\cdots\!02\)\( T^{12} + \)\(45\!\cdots\!53\)\( T^{13} + \)\(46\!\cdots\!98\)\( T^{14} + \)\(45\!\cdots\!53\)\( p^{9} T^{15} + \)\(63\!\cdots\!02\)\( p^{18} T^{16} + \)\(55\!\cdots\!99\)\( p^{27} T^{17} - \)\(93\!\cdots\!10\)\( p^{36} T^{18} - \)\(19\!\cdots\!61\)\( p^{45} T^{19} + \)\(14\!\cdots\!34\)\( p^{54} T^{20} + \)\(17\!\cdots\!17\)\( p^{63} T^{21} - \)\(13\!\cdots\!30\)\( p^{72} T^{22} - \)\(16\!\cdots\!99\)\( p^{81} T^{23} + \)\(90\!\cdots\!57\)\( p^{90} T^{24} + \)\(13\!\cdots\!32\)\( p^{99} T^{25} - 42356131219314273 p^{108} T^{26} - 49328881 p^{117} T^{27} + p^{126} T^{28} \) | |
| 67 | \( 1 + 171522309 T - 140691473513169943 T^{2} - \)\(14\!\cdots\!30\)\( T^{3} + \)\(12\!\cdots\!98\)\( T^{4} + \)\(66\!\cdots\!04\)\( T^{5} - \)\(79\!\cdots\!11\)\( T^{6} - \)\(13\!\cdots\!59\)\( T^{7} + \)\(37\!\cdots\!04\)\( T^{8} - \)\(11\!\cdots\!71\)\( T^{9} - \)\(13\!\cdots\!53\)\( T^{10} + \)\(12\!\cdots\!96\)\( T^{11} + \)\(43\!\cdots\!57\)\( T^{12} - \)\(21\!\cdots\!47\)\( T^{13} - \)\(12\!\cdots\!18\)\( T^{14} - \)\(21\!\cdots\!47\)\( p^{9} T^{15} + \)\(43\!\cdots\!57\)\( p^{18} T^{16} + \)\(12\!\cdots\!96\)\( p^{27} T^{17} - \)\(13\!\cdots\!53\)\( p^{36} T^{18} - \)\(11\!\cdots\!71\)\( p^{45} T^{19} + \)\(37\!\cdots\!04\)\( p^{54} T^{20} - \)\(13\!\cdots\!59\)\( p^{63} T^{21} - \)\(79\!\cdots\!11\)\( p^{72} T^{22} + \)\(66\!\cdots\!04\)\( p^{81} T^{23} + \)\(12\!\cdots\!98\)\( p^{90} T^{24} - \)\(14\!\cdots\!30\)\( p^{99} T^{25} - 140691473513169943 p^{108} T^{26} + 171522309 p^{117} T^{27} + p^{126} T^{28} \) | |
| 71 | \( 1 + 74596055 T - 278659361842615555 T^{2} - \)\(10\!\cdots\!78\)\( T^{3} + \)\(43\!\cdots\!13\)\( T^{4} + \)\(68\!\cdots\!33\)\( T^{5} - \)\(48\!\cdots\!88\)\( T^{6} - \)\(15\!\cdots\!35\)\( T^{7} + \)\(40\!\cdots\!84\)\( T^{8} - \)\(95\!\cdots\!39\)\( T^{9} - \)\(28\!\cdots\!92\)\( T^{10} + \)\(86\!\cdots\!97\)\( T^{11} + \)\(16\!\cdots\!06\)\( T^{12} - \)\(20\!\cdots\!13\)\( T^{13} - \)\(80\!\cdots\!78\)\( T^{14} - \)\(20\!\cdots\!13\)\( p^{9} T^{15} + \)\(16\!\cdots\!06\)\( p^{18} T^{16} + \)\(86\!\cdots\!97\)\( p^{27} T^{17} - \)\(28\!\cdots\!92\)\( p^{36} T^{18} - \)\(95\!\cdots\!39\)\( p^{45} T^{19} + \)\(40\!\cdots\!84\)\( p^{54} T^{20} - \)\(15\!\cdots\!35\)\( p^{63} T^{21} - \)\(48\!\cdots\!88\)\( p^{72} T^{22} + \)\(68\!\cdots\!33\)\( p^{81} T^{23} + \)\(43\!\cdots\!13\)\( p^{90} T^{24} - \)\(10\!\cdots\!78\)\( p^{99} T^{25} - 278659361842615555 p^{108} T^{26} + 74596055 p^{117} T^{27} + p^{126} T^{28} \) | |
| 73 | \( 1 - 58695287 T - 215198725080579879 T^{2} - \)\(10\!\cdots\!94\)\( T^{3} + \)\(26\!\cdots\!56\)\( T^{4} + \)\(32\!\cdots\!48\)\( T^{5} - \)\(17\!\cdots\!27\)\( T^{6} - \)\(40\!\cdots\!77\)\( T^{7} + \)\(59\!\cdots\!52\)\( T^{8} + \)\(30\!\cdots\!25\)\( T^{9} + \)\(24\!\cdots\!11\)\( T^{10} - \)\(15\!\cdots\!56\)\( T^{11} - \)\(49\!\cdots\!25\)\( T^{12} + \)\(36\!\cdots\!81\)\( T^{13} + \)\(38\!\cdots\!22\)\( T^{14} + \)\(36\!\cdots\!81\)\( p^{9} T^{15} - \)\(49\!\cdots\!25\)\( p^{18} T^{16} - \)\(15\!\cdots\!56\)\( p^{27} T^{17} + \)\(24\!\cdots\!11\)\( p^{36} T^{18} + \)\(30\!\cdots\!25\)\( p^{45} T^{19} + \)\(59\!\cdots\!52\)\( p^{54} T^{20} - \)\(40\!\cdots\!77\)\( p^{63} T^{21} - \)\(17\!\cdots\!27\)\( p^{72} T^{22} + \)\(32\!\cdots\!48\)\( p^{81} T^{23} + \)\(26\!\cdots\!56\)\( p^{90} T^{24} - \)\(10\!\cdots\!94\)\( p^{99} T^{25} - 215198725080579879 p^{108} T^{26} - 58695287 p^{117} T^{27} + p^{126} T^{28} \) | |
| 79 | \( 1 - 121854617 T - 293022562823571531 T^{2} + \)\(92\!\cdots\!38\)\( T^{3} + \)\(37\!\cdots\!05\)\( T^{4} + \)\(25\!\cdots\!03\)\( p T^{5} - \)\(56\!\cdots\!24\)\( T^{6} - \)\(49\!\cdots\!75\)\( T^{7} - \)\(42\!\cdots\!84\)\( T^{8} - \)\(56\!\cdots\!55\)\( T^{9} + \)\(56\!\cdots\!04\)\( T^{10} + \)\(21\!\cdots\!41\)\( T^{11} - \)\(18\!\cdots\!62\)\( T^{12} - \)\(15\!\cdots\!53\)\( T^{13} - \)\(83\!\cdots\!50\)\( T^{14} - \)\(15\!\cdots\!53\)\( p^{9} T^{15} - \)\(18\!\cdots\!62\)\( p^{18} T^{16} + \)\(21\!\cdots\!41\)\( p^{27} T^{17} + \)\(56\!\cdots\!04\)\( p^{36} T^{18} - \)\(56\!\cdots\!55\)\( p^{45} T^{19} - \)\(42\!\cdots\!84\)\( p^{54} T^{20} - \)\(49\!\cdots\!75\)\( p^{63} T^{21} - \)\(56\!\cdots\!24\)\( p^{72} T^{22} + \)\(25\!\cdots\!03\)\( p^{82} T^{23} + \)\(37\!\cdots\!05\)\( p^{90} T^{24} + \)\(92\!\cdots\!38\)\( p^{99} T^{25} - 293022562823571531 p^{108} T^{26} - 121854617 p^{117} T^{27} + p^{126} T^{28} \) | |
| 83 | \( ( 1 + 366303628 T + 534324485457307121 T^{2} + \)\(12\!\cdots\!36\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} + \)\(49\!\cdots\!04\)\( T^{5} + \)\(50\!\cdots\!16\)\( T^{6} + \)\(91\!\cdots\!24\)\( T^{7} + \)\(50\!\cdots\!16\)\( p^{9} T^{8} + \)\(49\!\cdots\!04\)\( p^{18} T^{9} + \)\(20\!\cdots\!56\)\( p^{27} T^{10} + \)\(12\!\cdots\!36\)\( p^{36} T^{11} + 534324485457307121 p^{45} T^{12} + 366303628 p^{54} T^{13} + p^{63} T^{14} )^{2} \) | |
| 89 | \( 1 + 1652463181 T + 722928101622300623 T^{2} - \)\(87\!\cdots\!28\)\( T^{3} - \)\(13\!\cdots\!97\)\( T^{4} - \)\(62\!\cdots\!79\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!99\)\( T^{7} + \)\(26\!\cdots\!02\)\( T^{8} + \)\(79\!\cdots\!77\)\( T^{9} - \)\(90\!\cdots\!36\)\( T^{10} - \)\(64\!\cdots\!49\)\( T^{11} - \)\(11\!\cdots\!64\)\( T^{12} + \)\(12\!\cdots\!23\)\( T^{13} + \)\(12\!\cdots\!46\)\( T^{14} + \)\(12\!\cdots\!23\)\( p^{9} T^{15} - \)\(11\!\cdots\!64\)\( p^{18} T^{16} - \)\(64\!\cdots\!49\)\( p^{27} T^{17} - \)\(90\!\cdots\!36\)\( p^{36} T^{18} + \)\(79\!\cdots\!77\)\( p^{45} T^{19} + \)\(26\!\cdots\!02\)\( p^{54} T^{20} + \)\(45\!\cdots\!99\)\( p^{63} T^{21} + \)\(20\!\cdots\!84\)\( p^{72} T^{22} - \)\(62\!\cdots\!79\)\( p^{81} T^{23} - \)\(13\!\cdots\!97\)\( p^{90} T^{24} - \)\(87\!\cdots\!28\)\( p^{99} T^{25} + 722928101622300623 p^{108} T^{26} + 1652463181 p^{117} T^{27} + p^{126} T^{28} \) | |
| 97 | \( 1 - 248805607 T - 4330873524811728051 T^{2} + \)\(91\!\cdots\!74\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!76\)\( T^{5} - \)\(17\!\cdots\!23\)\( T^{6} + \)\(24\!\cdots\!47\)\( T^{7} + \)\(22\!\cdots\!76\)\( T^{8} - \)\(23\!\cdots\!15\)\( T^{9} - \)\(24\!\cdots\!81\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(23\!\cdots\!67\)\( T^{12} - \)\(46\!\cdots\!27\)\( p T^{13} - \)\(18\!\cdots\!46\)\( T^{14} - \)\(46\!\cdots\!27\)\( p^{10} T^{15} + \)\(23\!\cdots\!67\)\( p^{18} T^{16} + \)\(15\!\cdots\!08\)\( p^{27} T^{17} - \)\(24\!\cdots\!81\)\( p^{36} T^{18} - \)\(23\!\cdots\!15\)\( p^{45} T^{19} + \)\(22\!\cdots\!76\)\( p^{54} T^{20} + \)\(24\!\cdots\!47\)\( p^{63} T^{21} - \)\(17\!\cdots\!23\)\( p^{72} T^{22} - \)\(18\!\cdots\!76\)\( p^{81} T^{23} + \)\(10\!\cdots\!88\)\( p^{90} T^{24} + \)\(91\!\cdots\!74\)\( p^{99} T^{25} - 4330873524811728051 p^{108} T^{26} - 248805607 p^{117} T^{27} + p^{126} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60178337047165907387846722625, −3.54145633464881430197419973603, −3.38316575584333023210925511638, −3.12728617614770863168083020453, −3.01329490038640654737124803567, −2.96840505022850825840962237212, −2.90950268552546036938702666720, −2.90764634952274908534103279740, −2.78920085881188143698677835832, −2.44014324148002376447540416119, −2.24775316939789714917808115813, −2.16330331409301756283520941983, −1.78833856981667030523428377321, −1.78415191173452650446935879944, −1.78218195325426928155738320250, −1.46115901389038491751261584149, −1.44027023573525054588409789926, −1.31795381633769844133222010767, −1.09096860594422330507313248152, −0.981853144602188827030834619454, −0.829612006890533559189950114411, −0.61641145819741989410479122217, −0.48723474737076303915771116546, −0.23877665349747503045790958693, −0.16749062894081562426239628646, 0.16749062894081562426239628646, 0.23877665349747503045790958693, 0.48723474737076303915771116546, 0.61641145819741989410479122217, 0.829612006890533559189950114411, 0.981853144602188827030834619454, 1.09096860594422330507313248152, 1.31795381633769844133222010767, 1.44027023573525054588409789926, 1.46115901389038491751261584149, 1.78218195325426928155738320250, 1.78415191173452650446935879944, 1.78833856981667030523428377321, 2.16330331409301756283520941983, 2.24775316939789714917808115813, 2.44014324148002376447540416119, 2.78920085881188143698677835832, 2.90764634952274908534103279740, 2.90950268552546036938702666720, 2.96840505022850825840962237212, 3.01329490038640654737124803567, 3.12728617614770863168083020453, 3.38316575584333023210925511638, 3.54145633464881430197419973603, 3.60178337047165907387846722625
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.