Dirichlet series
| L(s) = 1 | − 460·3-s + 1.66e4·4-s − 8.00e4·7-s + 2.29e4·9-s − 7.64e6·12-s − 2.62e6·13-s + 1.13e8·16-s + 8.18e7·19-s + 3.68e7·21-s − 1.44e8·27-s − 1.33e9·28-s + 6.83e8·31-s + 3.80e8·36-s + 7.86e7·37-s + 1.20e9·39-s + 6.51e9·43-s − 5.23e10·48-s − 8.65e10·49-s − 4.35e10·52-s − 3.76e10·57-s − 9.72e10·61-s − 1.83e9·63-s + 4.22e11·64-s + 1.95e11·67-s + 7.55e11·73-s + 1.36e12·76-s + 4.51e11·79-s + ⋯ |
| L(s) = 1 | − 0.631·3-s + 4.05·4-s − 0.680·7-s + 0.0431·9-s − 2.55·12-s − 0.543·13-s + 6.78·16-s + 1.74·19-s + 0.429·21-s − 0.374·27-s − 2.76·28-s + 0.770·31-s + 0.175·36-s + 0.0306·37-s + 0.342·39-s + 1.03·43-s − 4.27·48-s − 6.25·49-s − 2.20·52-s − 1.09·57-s − 1.88·61-s − 0.0293·63-s + 6.14·64-s + 2.16·67-s + 4.99·73-s + 7.06·76-s + 1.85·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+6)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.37723\times 10^{29}\) |
| Root analytic conductor: | \(8.27946\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{32} ,\ ( \ : [6]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(55.75310582\) |
| \(L(\frac12)\) | \(\approx\) | \(55.75310582\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 460 T + 62890 p T^{2} + 24576640 p^{2} T^{3} - 598431613 p^{5} T^{4} - 47395372760 p^{8} T^{5} + 11267704030 p^{12} T^{6} + 99076145740 p^{16} T^{7} - 1027768936396 p^{22} T^{8} + 99076145740 p^{28} T^{9} + 11267704030 p^{36} T^{10} - 47395372760 p^{44} T^{11} - 598431613 p^{53} T^{12} + 24576640 p^{62} T^{13} + 62890 p^{73} T^{14} + 460 p^{84} T^{15} + p^{96} T^{16} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - 16615 T^{2} + 81150921 p T^{4} - 76791339925 p^{4} T^{6} + 62659223940941 p^{7} T^{8} - 44561340163175415 p^{10} T^{10} + 14190321148959457361 p^{14} T^{12} - \)\(51\!\cdots\!25\)\( p^{21} T^{14} + \)\(54\!\cdots\!35\)\( p^{33} T^{16} - \)\(51\!\cdots\!25\)\( p^{45} T^{18} + 14190321148959457361 p^{62} T^{20} - 44561340163175415 p^{82} T^{22} + 62659223940941 p^{103} T^{24} - 76791339925 p^{124} T^{26} + 81150921 p^{145} T^{28} - 16615 p^{168} T^{30} + p^{192} T^{32} \) |
| 7 | \( ( 1 + 5720 p T + 45672097710 T^{2} + 1822394265146720 T^{3} + \)\(18\!\cdots\!63\)\( p T^{4} + \)\(38\!\cdots\!20\)\( p T^{5} + \)\(49\!\cdots\!10\)\( p^{2} T^{6} + \)\(89\!\cdots\!00\)\( p^{3} T^{7} + \)\(15\!\cdots\!16\)\( p^{4} T^{8} + \)\(89\!\cdots\!00\)\( p^{15} T^{9} + \)\(49\!\cdots\!10\)\( p^{26} T^{10} + \)\(38\!\cdots\!20\)\( p^{37} T^{11} + \)\(18\!\cdots\!63\)\( p^{49} T^{12} + 1822394265146720 p^{60} T^{13} + 45672097710 p^{72} T^{14} + 5720 p^{85} T^{15} + p^{96} T^{16} )^{2} \) | |
| 11 | \( 1 - 2374855313516 p T^{2} + \)\(32\!\cdots\!30\)\( p T^{4} - \)\(24\!\cdots\!00\)\( p^{3} T^{6} + \)\(14\!\cdots\!55\)\( p^{5} T^{8} - \)\(68\!\cdots\!48\)\( p^{7} T^{10} + \)\(26\!\cdots\!78\)\( p^{9} T^{12} - \)\(87\!\cdots\!60\)\( p^{11} T^{14} + \)\(24\!\cdots\!00\)\( p^{13} T^{16} - \)\(87\!\cdots\!60\)\( p^{35} T^{18} + \)\(26\!\cdots\!78\)\( p^{57} T^{20} - \)\(68\!\cdots\!48\)\( p^{79} T^{22} + \)\(14\!\cdots\!55\)\( p^{101} T^{24} - \)\(24\!\cdots\!00\)\( p^{123} T^{26} + \)\(32\!\cdots\!30\)\( p^{145} T^{28} - 2374855313516 p^{169} T^{30} + p^{192} T^{32} \) | |
| 13 | \( ( 1 + 1310540 T + 77217299741670 T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!21\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!30\)\( T^{6} + \)\(82\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!96\)\( T^{8} + \)\(82\!\cdots\!80\)\( p^{12} T^{9} + \)\(80\!\cdots\!30\)\( p^{24} T^{10} + \)\(43\!\cdots\!00\)\( p^{36} T^{11} + \)\(29\!\cdots\!21\)\( p^{48} T^{12} + \)\(13\!\cdots\!80\)\( p^{60} T^{13} + 77217299741670 p^{72} T^{14} + 1310540 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 17 | \( 1 - 6402873579909460 T^{2} + \)\(20\!\cdots\!62\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(61\!\cdots\!93\)\( T^{8} - \)\(71\!\cdots\!40\)\( T^{10} + \)\(39\!\cdots\!02\)\( p T^{12} - \)\(51\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!20\)\( T^{16} - \)\(51\!\cdots\!00\)\( p^{24} T^{18} + \)\(39\!\cdots\!02\)\( p^{49} T^{20} - \)\(71\!\cdots\!40\)\( p^{72} T^{22} + \)\(61\!\cdots\!93\)\( p^{96} T^{24} - \)\(40\!\cdots\!00\)\( p^{120} T^{26} + \)\(20\!\cdots\!62\)\( p^{144} T^{28} - 6402873579909460 p^{168} T^{30} + p^{192} T^{32} \) | |
| 19 | \( ( 1 - 40946020 T + 9937867863785772 T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!78\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} - \)\(49\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!95\)\( T^{8} - \)\(49\!\cdots\!00\)\( p^{12} T^{9} + \)\(14\!\cdots\!64\)\( p^{24} T^{10} - \)\(15\!\cdots\!80\)\( p^{36} T^{11} + \)\(45\!\cdots\!78\)\( p^{48} T^{12} - \)\(33\!\cdots\!00\)\( p^{60} T^{13} + 9937867863785772 p^{72} T^{14} - 40946020 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 23 | \( 1 - 120326821409886640 T^{2} + \)\(86\!\cdots\!92\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!48\)\( T^{8} - \)\(67\!\cdots\!60\)\( T^{10} + \)\(38\!\cdots\!56\)\( p^{2} T^{12} - \)\(19\!\cdots\!00\)\( p^{4} T^{14} + \)\(85\!\cdots\!30\)\( p^{6} T^{16} - \)\(19\!\cdots\!00\)\( p^{28} T^{18} + \)\(38\!\cdots\!56\)\( p^{50} T^{20} - \)\(67\!\cdots\!60\)\( p^{72} T^{22} + \)\(19\!\cdots\!48\)\( p^{96} T^{24} - \)\(45\!\cdots\!00\)\( p^{120} T^{26} + \)\(86\!\cdots\!92\)\( p^{144} T^{28} - 120326821409886640 p^{168} T^{30} + p^{192} T^{32} \) | |
| 29 | \( 1 - 1948875241292844016 T^{2} + \)\(23\!\cdots\!60\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!60\)\( T^{8} - \)\(81\!\cdots\!28\)\( T^{10} + \)\(40\!\cdots\!48\)\( T^{12} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!90\)\( T^{16} - \)\(17\!\cdots\!20\)\( p^{24} T^{18} + \)\(40\!\cdots\!48\)\( p^{48} T^{20} - \)\(81\!\cdots\!28\)\( p^{72} T^{22} + \)\(14\!\cdots\!60\)\( p^{96} T^{24} - \)\(20\!\cdots\!20\)\( p^{120} T^{26} + \)\(23\!\cdots\!60\)\( p^{144} T^{28} - 1948875241292844016 p^{168} T^{30} + p^{192} T^{32} \) | |
| 31 | \( ( 1 - 341803456 T + 3164151354058306710 T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(55\!\cdots\!85\)\( T^{4} - \)\(18\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!18\)\( T^{6} - \)\(21\!\cdots\!20\)\( T^{7} + \)\(59\!\cdots\!40\)\( T^{8} - \)\(21\!\cdots\!20\)\( p^{12} T^{9} + \)\(66\!\cdots\!18\)\( p^{24} T^{10} - \)\(18\!\cdots\!28\)\( p^{36} T^{11} + \)\(55\!\cdots\!85\)\( p^{48} T^{12} - \)\(10\!\cdots\!20\)\( p^{60} T^{13} + 3164151354058306710 p^{72} T^{14} - 341803456 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 39320680 T + 20976020527294770720 T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!96\)\( T^{4} - \)\(42\!\cdots\!40\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} - \)\(53\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!46\)\( T^{8} - \)\(53\!\cdots\!80\)\( p^{12} T^{9} + \)\(21\!\cdots\!80\)\( p^{24} T^{10} - \)\(42\!\cdots\!40\)\( p^{36} T^{11} + \)\(24\!\cdots\!96\)\( p^{48} T^{12} - \)\(20\!\cdots\!00\)\( p^{60} T^{13} + 20976020527294770720 p^{72} T^{14} - 39320680 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 41 | \( 1 - \)\(14\!\cdots\!36\)\( T^{2} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(34\!\cdots\!78\)\( T^{12} - \)\(95\!\cdots\!60\)\( T^{14} + \)\(23\!\cdots\!00\)\( T^{16} - \)\(95\!\cdots\!60\)\( p^{24} T^{18} + \)\(34\!\cdots\!78\)\( p^{48} T^{20} - \)\(10\!\cdots\!08\)\( p^{72} T^{22} + \)\(29\!\cdots\!05\)\( p^{96} T^{24} - \)\(65\!\cdots\!00\)\( p^{120} T^{26} + \)\(11\!\cdots\!30\)\( p^{144} T^{28} - \)\(14\!\cdots\!36\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 43 | \( ( 1 - 3257879440 T + \)\(17\!\cdots\!90\)\( T^{2} - \)\(16\!\cdots\!40\)\( p T^{3} + \)\(16\!\cdots\!01\)\( T^{4} - \)\(66\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} - \)\(38\!\cdots\!00\)\( T^{7} + \)\(49\!\cdots\!96\)\( T^{8} - \)\(38\!\cdots\!00\)\( p^{12} T^{9} + \)\(10\!\cdots\!10\)\( p^{24} T^{10} - \)\(66\!\cdots\!40\)\( p^{36} T^{11} + \)\(16\!\cdots\!01\)\( p^{48} T^{12} - \)\(16\!\cdots\!40\)\( p^{61} T^{13} + \)\(17\!\cdots\!90\)\( p^{72} T^{14} - 3257879440 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 47 | \( 1 - \)\(28\!\cdots\!60\)\( T^{2} + \)\(96\!\cdots\!32\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!28\)\( T^{8} - \)\(71\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!04\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(18\!\cdots\!70\)\( T^{16} - \)\(14\!\cdots\!00\)\( p^{24} T^{18} + \)\(10\!\cdots\!04\)\( p^{48} T^{20} - \)\(71\!\cdots\!40\)\( p^{72} T^{22} + \)\(42\!\cdots\!28\)\( p^{96} T^{24} - \)\(20\!\cdots\!00\)\( p^{120} T^{26} + \)\(96\!\cdots\!32\)\( p^{144} T^{28} - \)\(28\!\cdots\!60\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 53 | \( 1 - \)\(32\!\cdots\!60\)\( T^{2} + \)\(55\!\cdots\!32\)\( T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(65\!\cdots\!28\)\( T^{8} - \)\(51\!\cdots\!40\)\( T^{10} + \)\(35\!\cdots\!04\)\( T^{12} - \)\(20\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!70\)\( T^{16} - \)\(20\!\cdots\!00\)\( p^{24} T^{18} + \)\(35\!\cdots\!04\)\( p^{48} T^{20} - \)\(51\!\cdots\!40\)\( p^{72} T^{22} + \)\(65\!\cdots\!28\)\( p^{96} T^{24} - \)\(67\!\cdots\!00\)\( p^{120} T^{26} + \)\(55\!\cdots\!32\)\( p^{144} T^{28} - \)\(32\!\cdots\!60\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 59 | \( 1 - \)\(15\!\cdots\!36\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{8} - \)\(41\!\cdots\!08\)\( T^{10} + \)\(75\!\cdots\!28\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{14} + \)\(18\!\cdots\!50\)\( T^{16} - \)\(11\!\cdots\!60\)\( p^{24} T^{18} + \)\(75\!\cdots\!28\)\( p^{48} T^{20} - \)\(41\!\cdots\!08\)\( p^{72} T^{22} + \)\(17\!\cdots\!80\)\( p^{96} T^{24} - \)\(54\!\cdots\!00\)\( p^{120} T^{26} + \)\(11\!\cdots\!80\)\( p^{144} T^{28} - \)\(15\!\cdots\!36\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 61 | \( ( 1 + 48612327044 T + \)\(10\!\cdots\!90\)\( T^{2} + \)\(30\!\cdots\!60\)\( p T^{3} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(30\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!98\)\( T^{6} - \)\(21\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!80\)\( T^{8} - \)\(21\!\cdots\!80\)\( p^{12} T^{9} + \)\(12\!\cdots\!98\)\( p^{24} T^{10} - \)\(30\!\cdots\!48\)\( p^{36} T^{11} + \)\(41\!\cdots\!65\)\( p^{48} T^{12} + \)\(30\!\cdots\!60\)\( p^{61} T^{13} + \)\(10\!\cdots\!90\)\( p^{72} T^{14} + 48612327044 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 97945801900 T + \)\(57\!\cdots\!80\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{5} + \)\(22\!\cdots\!20\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!51\)\( T^{8} - \)\(12\!\cdots\!60\)\( p^{12} T^{9} + \)\(22\!\cdots\!20\)\( p^{24} T^{10} - \)\(10\!\cdots\!20\)\( p^{36} T^{11} + \)\(14\!\cdots\!26\)\( p^{48} T^{12} - \)\(46\!\cdots\!20\)\( p^{60} T^{13} + \)\(57\!\cdots\!80\)\( p^{72} T^{14} - 97945801900 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 71 | \( 1 - \)\(15\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(65\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!60\)\( T^{8} - \)\(78\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!48\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(75\!\cdots\!90\)\( T^{16} - \)\(42\!\cdots\!20\)\( p^{24} T^{18} + \)\(19\!\cdots\!48\)\( p^{48} T^{20} - \)\(78\!\cdots\!28\)\( p^{72} T^{22} + \)\(25\!\cdots\!60\)\( p^{96} T^{24} - \)\(65\!\cdots\!20\)\( p^{120} T^{26} + \)\(12\!\cdots\!60\)\( p^{144} T^{28} - \)\(15\!\cdots\!16\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 73 | \( ( 1 - 377718679900 T + \)\(20\!\cdots\!30\)\( p T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!41\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!10\)\( T^{6} - \)\(40\!\cdots\!80\)\( T^{7} + \)\(74\!\cdots\!76\)\( T^{8} - \)\(40\!\cdots\!80\)\( p^{12} T^{9} + \)\(26\!\cdots\!10\)\( p^{24} T^{10} - \)\(12\!\cdots\!60\)\( p^{36} T^{11} + \)\(76\!\cdots\!41\)\( p^{48} T^{12} - \)\(31\!\cdots\!60\)\( p^{60} T^{13} + \)\(20\!\cdots\!30\)\( p^{73} T^{14} - 377718679900 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 225645006160 T + \)\(13\!\cdots\!52\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!88\)\( T^{4} - \)\(73\!\cdots\!40\)\( T^{5} + \)\(54\!\cdots\!24\)\( T^{6} - \)\(96\!\cdots\!00\)\( T^{7} + \)\(49\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!00\)\( p^{12} T^{9} + \)\(54\!\cdots\!24\)\( p^{24} T^{10} - \)\(73\!\cdots\!40\)\( p^{36} T^{11} + \)\(68\!\cdots\!88\)\( p^{48} T^{12} - \)\(18\!\cdots\!00\)\( p^{60} T^{13} + \)\(13\!\cdots\!52\)\( p^{72} T^{14} - 225645006160 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
| 83 | \( 1 - \)\(55\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!62\)\( T^{4} - \)\(31\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!93\)\( T^{8} - \)\(72\!\cdots\!40\)\( T^{10} + \)\(97\!\cdots\!34\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!20\)\( T^{16} - \)\(12\!\cdots\!00\)\( p^{24} T^{18} + \)\(97\!\cdots\!34\)\( p^{48} T^{20} - \)\(72\!\cdots\!40\)\( p^{72} T^{22} + \)\(50\!\cdots\!93\)\( p^{96} T^{24} - \)\(31\!\cdots\!00\)\( p^{120} T^{26} + \)\(15\!\cdots\!62\)\( p^{144} T^{28} - \)\(55\!\cdots\!60\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 89 | \( 1 - \)\(59\!\cdots\!76\)\( T^{2} + \)\(22\!\cdots\!30\)\( T^{4} - \)\(83\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!05\)\( T^{8} - \)\(89\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!98\)\( T^{12} - \)\(61\!\cdots\!60\)\( T^{14} + \)\(16\!\cdots\!00\)\( T^{16} - \)\(61\!\cdots\!60\)\( p^{24} T^{18} + \)\(22\!\cdots\!98\)\( p^{48} T^{20} - \)\(89\!\cdots\!08\)\( p^{72} T^{22} + \)\(30\!\cdots\!05\)\( p^{96} T^{24} - \)\(83\!\cdots\!00\)\( p^{120} T^{26} + \)\(22\!\cdots\!30\)\( p^{144} T^{28} - \)\(59\!\cdots\!76\)\( p^{168} T^{30} + p^{192} T^{32} \) | |
| 97 | \( ( 1 - 154873171060 T + \)\(27\!\cdots\!30\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!61\)\( T^{4} - \)\(69\!\cdots\!40\)\( T^{5} + \)\(39\!\cdots\!70\)\( T^{6} + \)\(65\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!96\)\( T^{8} + \)\(65\!\cdots\!60\)\( p^{12} T^{9} + \)\(39\!\cdots\!70\)\( p^{24} T^{10} - \)\(69\!\cdots\!40\)\( p^{36} T^{11} + \)\(40\!\cdots\!61\)\( p^{48} T^{12} - \)\(20\!\cdots\!60\)\( p^{60} T^{13} + \)\(27\!\cdots\!30\)\( p^{72} T^{14} - 154873171060 p^{84} T^{15} + p^{96} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.30377474449760891609152877351, −2.21766241293356430751636813774, −2.18560230838203229083557928808, −2.10983651691426824928456461573, −2.08921280984113911062954206552, −1.87292759676288546347521505033, −1.60899433897985619523698784936, −1.60009008766413871635902401064, −1.52407292070147691297975133884, −1.40182429461826085898264728740, −1.35893641781195396077363121923, −1.28812203359639116761881706325, −1.19765430331493753083570041817, −1.18350564829619176097948233999, −1.14720280846164967775925284741, −0.965157887203912002769379786481, −0.71171718498201210543215462123, −0.70884906948695208566800392494, −0.61849151487097835697571216565, −0.48071342854803385136240790703, −0.38819059886500648654468782016, −0.35497443401978241099400806540, −0.32776562821210824679053636581, −0.18127493813420824206220168623, −0.10318860555425566984197054964, 0.10318860555425566984197054964, 0.18127493813420824206220168623, 0.32776562821210824679053636581, 0.35497443401978241099400806540, 0.38819059886500648654468782016, 0.48071342854803385136240790703, 0.61849151487097835697571216565, 0.70884906948695208566800392494, 0.71171718498201210543215462123, 0.965157887203912002769379786481, 1.14720280846164967775925284741, 1.18350564829619176097948233999, 1.19765430331493753083570041817, 1.28812203359639116761881706325, 1.35893641781195396077363121923, 1.40182429461826085898264728740, 1.52407292070147691297975133884, 1.60009008766413871635902401064, 1.60899433897985619523698784936, 1.87292759676288546347521505033, 2.08921280984113911062954206552, 2.10983651691426824928456461573, 2.18560230838203229083557928808, 2.21766241293356430751636813774, 2.30377474449760891609152877351
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.