Dirichlet series
| L(s) = 1 | − 4·2-s + 2·3-s + 8·4-s − 8·6-s − 2·7-s − 8·8-s − 18·9-s + 32·11-s + 16·12-s − 8·13-s + 8·14-s + 4·16-s − 62·17-s + 72·18-s + 30·19-s − 4·21-s − 128·22-s − 18·23-s − 16·24-s + 35·25-s + 32·26-s − 76·27-s − 16·28-s + 100·29-s + 132·31-s − 16·32-s + 64·33-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2/3·3-s + 2·4-s − 4/3·6-s − 2/7·7-s − 8-s − 2·9-s + 2.90·11-s + 4/3·12-s − 0.615·13-s + 4/7·14-s + 1/4·16-s − 3.64·17-s + 4·18-s + 1.57·19-s − 0.190·21-s − 5.81·22-s − 0.782·23-s − 2/3·24-s + 7/5·25-s + 1.23·26-s − 2.81·27-s − 4/7·28-s + 3.44·29-s + 4.25·31-s − 1/2·32-s + 1.93·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(140.890\) |
| Root analytic conductor: | \(1.16721\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.060790793\) |
| \(L(\frac12)\) | \(\approx\) | \(1.060790793\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 7 p T^{2} + 2 p^{2} T^{3} + 51 p^{2} T^{4} - 14 p^{3} T^{5} - 33 p^{4} T^{6} + 4 p^{5} T^{7} + 176 p^{5} T^{8} + 4 p^{7} T^{9} - 33 p^{8} T^{10} - 14 p^{9} T^{11} + 51 p^{10} T^{12} + 2 p^{12} T^{13} - 7 p^{13} T^{14} + p^{16} T^{16} \) | |
| good | 3 | \( 1 - 2 T + 22 T^{2} - 4 T^{3} + 238 T^{4} + 128 T^{5} + 3661 T^{6} + 820 T^{7} + 37409 T^{8} + 77452 T^{9} + 238852 T^{10} + 1206344 T^{11} + 2544805 T^{12} + 12363716 T^{13} + 18204073 T^{14} + 155891242 T^{15} + 76610278 T^{16} + 155891242 p^{2} T^{17} + 18204073 p^{4} T^{18} + 12363716 p^{6} T^{19} + 2544805 p^{8} T^{20} + 1206344 p^{10} T^{21} + 238852 p^{12} T^{22} + 77452 p^{14} T^{23} + 37409 p^{16} T^{24} + 820 p^{18} T^{25} + 3661 p^{20} T^{26} + 128 p^{22} T^{27} + 238 p^{24} T^{28} - 4 p^{26} T^{29} + 22 p^{28} T^{30} - 2 p^{30} T^{31} + p^{32} T^{32} \) |
| 7 | \( 1 + 2 T + 2 T^{2} + 194 T^{3} + 1903 T^{4} - 4378 T^{5} + 6256 T^{6} + 322620 T^{7} + 1931219 T^{8} - 13909652 T^{9} + 28584812 T^{10} + 95863098 p T^{11} - 16856202915 T^{12} - 68987219646 T^{13} + 72379738978 T^{14} - 1624210209802 T^{15} - 60143089352512 T^{16} - 1624210209802 p^{2} T^{17} + 72379738978 p^{4} T^{18} - 68987219646 p^{6} T^{19} - 16856202915 p^{8} T^{20} + 95863098 p^{11} T^{21} + 28584812 p^{12} T^{22} - 13909652 p^{14} T^{23} + 1931219 p^{16} T^{24} + 322620 p^{18} T^{25} + 6256 p^{20} T^{26} - 4378 p^{22} T^{27} + 1903 p^{24} T^{28} + 194 p^{26} T^{29} + 2 p^{28} T^{30} + 2 p^{30} T^{31} + p^{32} T^{32} \) | |
| 11 | \( 1 - 32 T + 311 T^{2} - 280 T^{3} + 25040 T^{4} - 1279836 T^{5} + 16492702 T^{6} - 35246116 T^{7} + 12103010 T^{8} - 27077399240 T^{9} + 444117609725 T^{10} - 1661544256620 T^{11} - 2997040907925 T^{12} - 428490487832240 T^{13} + 8705210851769310 T^{14} - 44648771996261860 T^{15} - 39232285361881900 T^{16} - 44648771996261860 p^{2} T^{17} + 8705210851769310 p^{4} T^{18} - 428490487832240 p^{6} T^{19} - 2997040907925 p^{8} T^{20} - 1661544256620 p^{10} T^{21} + 444117609725 p^{12} T^{22} - 27077399240 p^{14} T^{23} + 12103010 p^{16} T^{24} - 35246116 p^{18} T^{25} + 16492702 p^{20} T^{26} - 1279836 p^{22} T^{27} + 25040 p^{24} T^{28} - 280 p^{26} T^{29} + 311 p^{28} T^{30} - 32 p^{30} T^{31} + p^{32} T^{32} \) | |
| 13 | \( 1 + 8 T - 408 T^{2} + 1246 T^{3} + 154073 T^{4} - 53392 T^{5} - 14943744 T^{6} + 136437880 T^{7} + 2161124259 T^{8} + 39087616392 T^{9} + 656128569372 T^{10} - 5151678207876 T^{11} - 39649654549605 T^{12} + 2738591517267766 T^{13} + 26266948888870908 T^{14} + 7177333027493672 T^{15} - 36923773419092912 T^{16} + 7177333027493672 p^{2} T^{17} + 26266948888870908 p^{4} T^{18} + 2738591517267766 p^{6} T^{19} - 39649654549605 p^{8} T^{20} - 5151678207876 p^{10} T^{21} + 656128569372 p^{12} T^{22} + 39087616392 p^{14} T^{23} + 2161124259 p^{16} T^{24} + 136437880 p^{18} T^{25} - 14943744 p^{20} T^{26} - 53392 p^{22} T^{27} + 154073 p^{24} T^{28} + 1246 p^{26} T^{29} - 408 p^{28} T^{30} + 8 p^{30} T^{31} + p^{32} T^{32} \) | |
| 17 | \( 1 + 62 T + 1597 T^{2} + 36384 T^{3} + 62124 p T^{4} + 23573022 T^{5} + 414465036 T^{6} + 8360646090 T^{7} + 168082839244 T^{8} + 2887160089828 T^{9} + 51685218984187 T^{10} + 1006071764420186 T^{11} + 18939725531373815 T^{12} + 341876916956340004 T^{13} + 6132835487649258408 T^{14} + \)\(11\!\cdots\!68\)\( T^{15} + \)\(19\!\cdots\!08\)\( T^{16} + \)\(11\!\cdots\!68\)\( p^{2} T^{17} + 6132835487649258408 p^{4} T^{18} + 341876916956340004 p^{6} T^{19} + 18939725531373815 p^{8} T^{20} + 1006071764420186 p^{10} T^{21} + 51685218984187 p^{12} T^{22} + 2887160089828 p^{14} T^{23} + 168082839244 p^{16} T^{24} + 8360646090 p^{18} T^{25} + 414465036 p^{20} T^{26} + 23573022 p^{22} T^{27} + 62124 p^{25} T^{28} + 36384 p^{26} T^{29} + 1597 p^{28} T^{30} + 62 p^{30} T^{31} + p^{32} T^{32} \) | |
| 19 | \( 1 - 30 T + 2034 T^{2} - 56610 T^{3} + 1934975 T^{4} - 48188310 T^{5} + 1133912760 T^{6} - 25586281660 T^{7} + 475825509875 T^{8} - 10504860191160 T^{9} + 183994479506092 T^{10} - 221295647236630 p T^{11} + 84985205091126893 T^{12} - 1838105219654083030 T^{13} + 41116661316090464650 T^{14} - \)\(78\!\cdots\!30\)\( T^{15} + \)\(16\!\cdots\!80\)\( T^{16} - \)\(78\!\cdots\!30\)\( p^{2} T^{17} + 41116661316090464650 p^{4} T^{18} - 1838105219654083030 p^{6} T^{19} + 84985205091126893 p^{8} T^{20} - 221295647236630 p^{11} T^{21} + 183994479506092 p^{12} T^{22} - 10504860191160 p^{14} T^{23} + 475825509875 p^{16} T^{24} - 25586281660 p^{18} T^{25} + 1133912760 p^{20} T^{26} - 48188310 p^{22} T^{27} + 1934975 p^{24} T^{28} - 56610 p^{26} T^{29} + 2034 p^{28} T^{30} - 30 p^{30} T^{31} + p^{32} T^{32} \) | |
| 23 | \( 1 + 18 T + 932 T^{2} + 9866 T^{3} + 710628 T^{4} + 16202328 T^{5} + 23577787 p T^{6} + 12140414750 T^{7} + 326834473949 T^{8} + 9662822614212 T^{9} + 229872097333952 T^{10} + 6320239490383634 T^{11} + 137455640756351135 T^{12} + 3811758320527751276 T^{13} + 86208124512570196863 T^{14} + \)\(20\!\cdots\!72\)\( T^{15} + \)\(44\!\cdots\!78\)\( T^{16} + \)\(20\!\cdots\!72\)\( p^{2} T^{17} + 86208124512570196863 p^{4} T^{18} + 3811758320527751276 p^{6} T^{19} + 137455640756351135 p^{8} T^{20} + 6320239490383634 p^{10} T^{21} + 229872097333952 p^{12} T^{22} + 9662822614212 p^{14} T^{23} + 326834473949 p^{16} T^{24} + 12140414750 p^{18} T^{25} + 23577787 p^{21} T^{26} + 16202328 p^{22} T^{27} + 710628 p^{24} T^{28} + 9866 p^{26} T^{29} + 932 p^{28} T^{30} + 18 p^{30} T^{31} + p^{32} T^{32} \) | |
| 29 | \( 1 - 100 T + 9264 T^{2} - 536510 T^{3} + 28912705 T^{4} - 1178229540 T^{5} + 45544181980 T^{6} - 1372993134540 T^{7} + 39931784295555 T^{8} - 834296794343660 T^{9} + 17333962050516952 T^{10} - 134373037392029400 T^{11} + 1145801917255275763 T^{12} + \)\(11\!\cdots\!70\)\( T^{13} - \)\(42\!\cdots\!60\)\( T^{14} + \)\(55\!\cdots\!80\)\( T^{15} + \)\(12\!\cdots\!40\)\( T^{16} + \)\(55\!\cdots\!80\)\( p^{2} T^{17} - \)\(42\!\cdots\!60\)\( p^{4} T^{18} + \)\(11\!\cdots\!70\)\( p^{6} T^{19} + 1145801917255275763 p^{8} T^{20} - 134373037392029400 p^{10} T^{21} + 17333962050516952 p^{12} T^{22} - 834296794343660 p^{14} T^{23} + 39931784295555 p^{16} T^{24} - 1372993134540 p^{18} T^{25} + 45544181980 p^{20} T^{26} - 1178229540 p^{22} T^{27} + 28912705 p^{24} T^{28} - 536510 p^{26} T^{29} + 9264 p^{28} T^{30} - 100 p^{30} T^{31} + p^{32} T^{32} \) | |
| 31 | \( 1 - 132 T + 8456 T^{2} - 411940 T^{3} + 20688750 T^{4} - 1004410316 T^{5} + 40153103477 T^{6} - 1393921802456 T^{7} + 48452285556185 T^{8} - 1611236897501190 T^{9} + 44575775518757750 T^{10} - 1039905597691436520 T^{11} + 22863368448906938825 T^{12} - \)\(39\!\cdots\!40\)\( T^{13} + \)\(43\!\cdots\!85\)\( T^{14} + \)\(21\!\cdots\!90\)\( T^{15} - \)\(85\!\cdots\!50\)\( T^{16} + \)\(21\!\cdots\!90\)\( p^{2} T^{17} + \)\(43\!\cdots\!85\)\( p^{4} T^{18} - \)\(39\!\cdots\!40\)\( p^{6} T^{19} + 22863368448906938825 p^{8} T^{20} - 1039905597691436520 p^{10} T^{21} + 44575775518757750 p^{12} T^{22} - 1611236897501190 p^{14} T^{23} + 48452285556185 p^{16} T^{24} - 1393921802456 p^{18} T^{25} + 40153103477 p^{20} T^{26} - 1004410316 p^{22} T^{27} + 20688750 p^{24} T^{28} - 411940 p^{26} T^{29} + 8456 p^{28} T^{30} - 132 p^{30} T^{31} + p^{32} T^{32} \) | |
| 37 | \( 1 - 138 T + 9322 T^{2} - 464686 T^{3} + 20283478 T^{4} - 839430968 T^{5} + 32904810991 T^{6} - 1143914384480 T^{7} + 35859949664529 T^{8} - 1196883873178162 T^{9} + 49065705117423912 T^{10} - 2160094201360248294 T^{11} + 93774140138122933405 T^{12} - \)\(40\!\cdots\!96\)\( T^{13} + \)\(16\!\cdots\!63\)\( T^{14} - \)\(69\!\cdots\!12\)\( T^{15} + \)\(27\!\cdots\!98\)\( T^{16} - \)\(69\!\cdots\!12\)\( p^{2} T^{17} + \)\(16\!\cdots\!63\)\( p^{4} T^{18} - \)\(40\!\cdots\!96\)\( p^{6} T^{19} + 93774140138122933405 p^{8} T^{20} - 2160094201360248294 p^{10} T^{21} + 49065705117423912 p^{12} T^{22} - 1196883873178162 p^{14} T^{23} + 35859949664529 p^{16} T^{24} - 1143914384480 p^{18} T^{25} + 32904810991 p^{20} T^{26} - 839430968 p^{22} T^{27} + 20283478 p^{24} T^{28} - 464686 p^{26} T^{29} + 9322 p^{28} T^{30} - 138 p^{30} T^{31} + p^{32} T^{32} \) | |
| 41 | \( 1 + 88 T - 729 T^{2} - 305630 T^{3} - 11402010 T^{4} - 55280686 T^{5} + 30491909032 T^{6} + 2260665289574 T^{7} + 17831893998760 T^{8} - 4007688064709390 T^{9} - 197693216778478025 T^{10} - 3630202554625654320 T^{11} + \)\(25\!\cdots\!25\)\( T^{12} + \)\(21\!\cdots\!60\)\( T^{13} + \)\(32\!\cdots\!10\)\( T^{14} - \)\(21\!\cdots\!60\)\( T^{15} - \)\(13\!\cdots\!00\)\( T^{16} - \)\(21\!\cdots\!60\)\( p^{2} T^{17} + \)\(32\!\cdots\!10\)\( p^{4} T^{18} + \)\(21\!\cdots\!60\)\( p^{6} T^{19} + \)\(25\!\cdots\!25\)\( p^{8} T^{20} - 3630202554625654320 p^{10} T^{21} - 197693216778478025 p^{12} T^{22} - 4007688064709390 p^{14} T^{23} + 17831893998760 p^{16} T^{24} + 2260665289574 p^{18} T^{25} + 30491909032 p^{20} T^{26} - 55280686 p^{22} T^{27} - 11402010 p^{24} T^{28} - 305630 p^{26} T^{29} - 729 p^{28} T^{30} + 88 p^{30} T^{31} + p^{32} T^{32} \) | |
| 43 | \( 1 + 78 T + 3042 T^{2} + 171206 T^{3} + 2782983 T^{4} - 109413082 T^{5} - 2344307464 T^{6} + 93685920380 T^{7} + 32233223405099 T^{8} + 1192300168601032 T^{9} + 23495318312602092 T^{10} - 8210642199302082 p T^{11} - \)\(20\!\cdots\!35\)\( T^{12} - \)\(77\!\cdots\!74\)\( T^{13} - \)\(14\!\cdots\!82\)\( T^{14} - \)\(12\!\cdots\!38\)\( T^{15} + \)\(25\!\cdots\!28\)\( T^{16} - \)\(12\!\cdots\!38\)\( p^{2} T^{17} - \)\(14\!\cdots\!82\)\( p^{4} T^{18} - \)\(77\!\cdots\!74\)\( p^{6} T^{19} - \)\(20\!\cdots\!35\)\( p^{8} T^{20} - 8210642199302082 p^{11} T^{21} + 23495318312602092 p^{12} T^{22} + 1192300168601032 p^{14} T^{23} + 32233223405099 p^{16} T^{24} + 93685920380 p^{18} T^{25} - 2344307464 p^{20} T^{26} - 109413082 p^{22} T^{27} + 2782983 p^{24} T^{28} + 171206 p^{26} T^{29} + 3042 p^{28} T^{30} + 78 p^{30} T^{31} + p^{32} T^{32} \) | |
| 47 | \( 1 + 22 T + 4367 T^{2} - 127346 T^{3} + 8648108 T^{4} - 400470628 T^{5} + 35915341176 T^{6} - 1154920715120 T^{7} + 38632424913304 T^{8} - 3635939093091602 T^{9} + 132651930309092597 T^{10} - 463998061160947094 T^{11} + \)\(25\!\cdots\!95\)\( T^{12} - \)\(69\!\cdots\!16\)\( T^{13} + \)\(12\!\cdots\!88\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!28\)\( T^{16} + \)\(13\!\cdots\!08\)\( p^{2} T^{17} + \)\(12\!\cdots\!88\)\( p^{4} T^{18} - \)\(69\!\cdots\!16\)\( p^{6} T^{19} + \)\(25\!\cdots\!95\)\( p^{8} T^{20} - 463998061160947094 p^{10} T^{21} + 132651930309092597 p^{12} T^{22} - 3635939093091602 p^{14} T^{23} + 38632424913304 p^{16} T^{24} - 1154920715120 p^{18} T^{25} + 35915341176 p^{20} T^{26} - 400470628 p^{22} T^{27} + 8648108 p^{24} T^{28} - 127346 p^{26} T^{29} + 4367 p^{28} T^{30} + 22 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 - 182 T + 16412 T^{2} - 998154 T^{3} + 46385108 T^{4} - 1572071772 T^{5} + 17705084191 T^{6} + 2952895003890 T^{7} - 335391733779871 T^{8} + 23981885595946802 T^{9} - 1385927946983394108 T^{10} + 61800061722625112674 T^{11} - \)\(13\!\cdots\!45\)\( T^{12} - \)\(89\!\cdots\!84\)\( T^{13} + \)\(12\!\cdots\!13\)\( T^{14} - \)\(95\!\cdots\!38\)\( T^{15} + \)\(55\!\cdots\!98\)\( T^{16} - \)\(95\!\cdots\!38\)\( p^{2} T^{17} + \)\(12\!\cdots\!13\)\( p^{4} T^{18} - \)\(89\!\cdots\!84\)\( p^{6} T^{19} - \)\(13\!\cdots\!45\)\( p^{8} T^{20} + 61800061722625112674 p^{10} T^{21} - 1385927946983394108 p^{12} T^{22} + 23981885595946802 p^{14} T^{23} - 335391733779871 p^{16} T^{24} + 2952895003890 p^{18} T^{25} + 17705084191 p^{20} T^{26} - 1572071772 p^{22} T^{27} + 46385108 p^{24} T^{28} - 998154 p^{26} T^{29} + 16412 p^{28} T^{30} - 182 p^{30} T^{31} + p^{32} T^{32} \) | |
| 59 | \( 1 + 350 T + 60619 T^{2} + 7103790 T^{3} + 656250000 T^{4} + 52567889660 T^{5} + 3852205646200 T^{6} + 261978047059060 T^{7} + 16449560689167700 T^{8} + 961620997883533190 T^{9} + 54367201752689575297 T^{10} + \)\(31\!\cdots\!50\)\( T^{11} + \)\(18\!\cdots\!43\)\( T^{12} + \)\(11\!\cdots\!20\)\( T^{13} + \)\(69\!\cdots\!00\)\( T^{14} + \)\(42\!\cdots\!80\)\( T^{15} + \)\(25\!\cdots\!00\)\( T^{16} + \)\(42\!\cdots\!80\)\( p^{2} T^{17} + \)\(69\!\cdots\!00\)\( p^{4} T^{18} + \)\(11\!\cdots\!20\)\( p^{6} T^{19} + \)\(18\!\cdots\!43\)\( p^{8} T^{20} + \)\(31\!\cdots\!50\)\( p^{10} T^{21} + 54367201752689575297 p^{12} T^{22} + 961620997883533190 p^{14} T^{23} + 16449560689167700 p^{16} T^{24} + 261978047059060 p^{18} T^{25} + 3852205646200 p^{20} T^{26} + 52567889660 p^{22} T^{27} + 656250000 p^{24} T^{28} + 7103790 p^{26} T^{29} + 60619 p^{28} T^{30} + 350 p^{30} T^{31} + p^{32} T^{32} \) | |
| 61 | \( 1 - 372 T + 55256 T^{2} - 3292110 T^{3} - 103754130 T^{4} + 28533017544 T^{5} - 1370324489333 T^{6} - 51462531369216 T^{7} + 9478004505604285 T^{8} - 467252329185897240 T^{9} + 3176765215186111750 T^{10} + \)\(11\!\cdots\!30\)\( T^{11} - \)\(92\!\cdots\!75\)\( T^{12} + \)\(33\!\cdots\!60\)\( T^{13} + \)\(68\!\cdots\!35\)\( T^{14} - \)\(17\!\cdots\!60\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(17\!\cdots\!60\)\( p^{2} T^{17} + \)\(68\!\cdots\!35\)\( p^{4} T^{18} + \)\(33\!\cdots\!60\)\( p^{6} T^{19} - \)\(92\!\cdots\!75\)\( p^{8} T^{20} + \)\(11\!\cdots\!30\)\( p^{10} T^{21} + 3176765215186111750 p^{12} T^{22} - 467252329185897240 p^{14} T^{23} + 9478004505604285 p^{16} T^{24} - 51462531369216 p^{18} T^{25} - 1370324489333 p^{20} T^{26} + 28533017544 p^{22} T^{27} - 103754130 p^{24} T^{28} - 3292110 p^{26} T^{29} + 55256 p^{28} T^{30} - 372 p^{30} T^{31} + p^{32} T^{32} \) | |
| 67 | \( 1 + 112 T + 14817 T^{2} + 1145984 T^{3} + 123515638 T^{4} + 8898190012 T^{5} + 691494911216 T^{6} + 39024436975080 T^{7} + 2606207534094024 T^{8} + 113590052308880948 T^{9} + 4289661257101412517 T^{10} - \)\(22\!\cdots\!44\)\( T^{11} - \)\(32\!\cdots\!35\)\( T^{12} - \)\(44\!\cdots\!76\)\( T^{13} - \)\(35\!\cdots\!02\)\( T^{14} - \)\(32\!\cdots\!32\)\( T^{15} - \)\(20\!\cdots\!52\)\( T^{16} - \)\(32\!\cdots\!32\)\( p^{2} T^{17} - \)\(35\!\cdots\!02\)\( p^{4} T^{18} - \)\(44\!\cdots\!76\)\( p^{6} T^{19} - \)\(32\!\cdots\!35\)\( p^{8} T^{20} - \)\(22\!\cdots\!44\)\( p^{10} T^{21} + 4289661257101412517 p^{12} T^{22} + 113590052308880948 p^{14} T^{23} + 2606207534094024 p^{16} T^{24} + 39024436975080 p^{18} T^{25} + 691494911216 p^{20} T^{26} + 8898190012 p^{22} T^{27} + 123515638 p^{24} T^{28} + 1145984 p^{26} T^{29} + 14817 p^{28} T^{30} + 112 p^{30} T^{31} + p^{32} T^{32} \) | |
| 71 | \( 1 - 122 T + 2651 T^{2} + 765930 T^{3} - 43210520 T^{4} - 2144604456 T^{5} + 206118207592 T^{6} + 11552582096064 T^{7} - 389449688521040 T^{8} - 190759168399140190 T^{9} + 13117500305265709925 T^{10} + \)\(75\!\cdots\!30\)\( T^{11} - \)\(10\!\cdots\!25\)\( T^{12} - \)\(34\!\cdots\!40\)\( T^{13} + \)\(45\!\cdots\!60\)\( T^{14} - \)\(91\!\cdots\!60\)\( T^{15} - \)\(12\!\cdots\!00\)\( T^{16} - \)\(91\!\cdots\!60\)\( p^{2} T^{17} + \)\(45\!\cdots\!60\)\( p^{4} T^{18} - \)\(34\!\cdots\!40\)\( p^{6} T^{19} - \)\(10\!\cdots\!25\)\( p^{8} T^{20} + \)\(75\!\cdots\!30\)\( p^{10} T^{21} + 13117500305265709925 p^{12} T^{22} - 190759168399140190 p^{14} T^{23} - 389449688521040 p^{16} T^{24} + 11552582096064 p^{18} T^{25} + 206118207592 p^{20} T^{26} - 2144604456 p^{22} T^{27} - 43210520 p^{24} T^{28} + 765930 p^{26} T^{29} + 2651 p^{28} T^{30} - 122 p^{30} T^{31} + p^{32} T^{32} \) | |
| 73 | \( 1 + 248 T + 40082 T^{2} + 3730146 T^{3} + 203798938 T^{4} - 4122838392 T^{5} - 1583596343969 T^{6} - 136528685812380 T^{7} - 1710173106894911 T^{8} + 574375277885875492 T^{9} + 58705220561600607932 T^{10} + \)\(39\!\cdots\!44\)\( T^{11} - \)\(33\!\cdots\!95\)\( T^{12} - \)\(34\!\cdots\!74\)\( T^{13} - \)\(80\!\cdots\!37\)\( T^{14} + \)\(13\!\cdots\!12\)\( T^{15} + \)\(16\!\cdots\!18\)\( T^{16} + \)\(13\!\cdots\!12\)\( p^{2} T^{17} - \)\(80\!\cdots\!37\)\( p^{4} T^{18} - \)\(34\!\cdots\!74\)\( p^{6} T^{19} - \)\(33\!\cdots\!95\)\( p^{8} T^{20} + \)\(39\!\cdots\!44\)\( p^{10} T^{21} + 58705220561600607932 p^{12} T^{22} + 574375277885875492 p^{14} T^{23} - 1710173106894911 p^{16} T^{24} - 136528685812380 p^{18} T^{25} - 1583596343969 p^{20} T^{26} - 4122838392 p^{22} T^{27} + 203798938 p^{24} T^{28} + 3730146 p^{26} T^{29} + 40082 p^{28} T^{30} + 248 p^{30} T^{31} + p^{32} T^{32} \) | |
| 79 | \( 1 - 760 T + 308414 T^{2} - 87710920 T^{3} + 19483985735 T^{4} - 3582525490520 T^{5} + 565339446558720 T^{6} - 78496071116628320 T^{7} + 9767991672120119935 T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!52\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!13\)\( T^{12} - \)\(88\!\cdots\!60\)\( T^{13} + \)\(73\!\cdots\!30\)\( T^{14} - \)\(59\!\cdots\!60\)\( T^{15} + \)\(47\!\cdots\!60\)\( T^{16} - \)\(59\!\cdots\!60\)\( p^{2} T^{17} + \)\(73\!\cdots\!30\)\( p^{4} T^{18} - \)\(88\!\cdots\!60\)\( p^{6} T^{19} + \)\(10\!\cdots\!13\)\( p^{8} T^{20} - \)\(11\!\cdots\!40\)\( p^{10} T^{21} + \)\(11\!\cdots\!52\)\( p^{12} T^{22} - \)\(11\!\cdots\!20\)\( p^{14} T^{23} + 9767991672120119935 p^{16} T^{24} - 78496071116628320 p^{18} T^{25} + 565339446558720 p^{20} T^{26} - 3582525490520 p^{22} T^{27} + 19483985735 p^{24} T^{28} - 87710920 p^{26} T^{29} + 308414 p^{28} T^{30} - 760 p^{30} T^{31} + p^{32} T^{32} \) | |
| 83 | \( 1 - 132 T + 8712 T^{2} + 1825536 T^{3} - 313208892 T^{4} + 24609282648 T^{5} + 714210319341 T^{6} - 266724647925570 T^{7} + 27578072396649989 T^{8} - 893715050762853198 T^{9} - 53051694029048326428 T^{10} + \)\(14\!\cdots\!74\)\( T^{11} - \)\(12\!\cdots\!85\)\( T^{12} + \)\(77\!\cdots\!06\)\( T^{13} + \)\(49\!\cdots\!63\)\( T^{14} - \)\(58\!\cdots\!68\)\( T^{15} + \)\(74\!\cdots\!98\)\( T^{16} - \)\(58\!\cdots\!68\)\( p^{2} T^{17} + \)\(49\!\cdots\!63\)\( p^{4} T^{18} + \)\(77\!\cdots\!06\)\( p^{6} T^{19} - \)\(12\!\cdots\!85\)\( p^{8} T^{20} + \)\(14\!\cdots\!74\)\( p^{10} T^{21} - 53051694029048326428 p^{12} T^{22} - 893715050762853198 p^{14} T^{23} + 27578072396649989 p^{16} T^{24} - 266724647925570 p^{18} T^{25} + 714210319341 p^{20} T^{26} + 24609282648 p^{22} T^{27} - 313208892 p^{24} T^{28} + 1825536 p^{26} T^{29} + 8712 p^{28} T^{30} - 132 p^{30} T^{31} + p^{32} T^{32} \) | |
| 89 | \( 1 - 550 T + 165449 T^{2} - 36759850 T^{3} + 6686592900 T^{4} - 1045187387100 T^{5} + 144246945765810 T^{6} - 17845461659641600 T^{7} + 1995990471244691550 T^{8} - \)\(20\!\cdots\!50\)\( T^{9} + \)\(18\!\cdots\!67\)\( T^{10} - \)\(15\!\cdots\!50\)\( T^{11} + \)\(10\!\cdots\!23\)\( T^{12} - \)\(67\!\cdots\!00\)\( T^{13} + \)\(33\!\cdots\!50\)\( T^{14} - \)\(12\!\cdots\!00\)\( T^{15} + \)\(51\!\cdots\!80\)\( T^{16} - \)\(12\!\cdots\!00\)\( p^{2} T^{17} + \)\(33\!\cdots\!50\)\( p^{4} T^{18} - \)\(67\!\cdots\!00\)\( p^{6} T^{19} + \)\(10\!\cdots\!23\)\( p^{8} T^{20} - \)\(15\!\cdots\!50\)\( p^{10} T^{21} + \)\(18\!\cdots\!67\)\( p^{12} T^{22} - \)\(20\!\cdots\!50\)\( p^{14} T^{23} + 1995990471244691550 p^{16} T^{24} - 17845461659641600 p^{18} T^{25} + 144246945765810 p^{20} T^{26} - 1045187387100 p^{22} T^{27} + 6686592900 p^{24} T^{28} - 36759850 p^{26} T^{29} + 165449 p^{28} T^{30} - 550 p^{30} T^{31} + p^{32} T^{32} \) | |
| 97 | \( 1 + 292 T + 47817 T^{2} + 6560784 T^{3} + 823036208 T^{4} + 86769191932 T^{5} + 7612555610156 T^{6} + 647037231537420 T^{7} + 50219387629801124 T^{8} + 3427143960160694768 T^{9} + \)\(27\!\cdots\!87\)\( T^{10} + \)\(29\!\cdots\!96\)\( T^{11} + \)\(26\!\cdots\!55\)\( T^{12} + \)\(26\!\cdots\!24\)\( T^{13} + \)\(31\!\cdots\!68\)\( T^{14} + \)\(27\!\cdots\!88\)\( T^{15} + \)\(22\!\cdots\!68\)\( T^{16} + \)\(27\!\cdots\!88\)\( p^{2} T^{17} + \)\(31\!\cdots\!68\)\( p^{4} T^{18} + \)\(26\!\cdots\!24\)\( p^{6} T^{19} + \)\(26\!\cdots\!55\)\( p^{8} T^{20} + \)\(29\!\cdots\!96\)\( p^{10} T^{21} + \)\(27\!\cdots\!87\)\( p^{12} T^{22} + 3427143960160694768 p^{14} T^{23} + 50219387629801124 p^{16} T^{24} + 647037231537420 p^{18} T^{25} + 7612555610156 p^{20} T^{26} + 86769191932 p^{22} T^{27} + 823036208 p^{24} T^{28} + 6560784 p^{26} T^{29} + 47817 p^{28} T^{30} + 292 p^{30} T^{31} + p^{32} T^{32} \) | |
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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
−4.89887582646527228188531339257, −4.80874172120921449497613091236, −4.64358051762364519638352131128, −4.46690544175616041891758364281, −4.22349041380162787760010430807, −4.14726094046601676739992841250, −4.06449505979950582610284848831, −3.95402090322453076425028077168, −3.85259891495301111643346244014, −3.79573225803279643190333185569, −3.47205549079042952999339578717, −3.41088024024501624743657621412, −3.15023739410463514149654973669, −2.95117800749499345894057129686, −2.94257835163018886009128878601, −2.78977283012405076010246361053, −2.54716488410991714957692759486, −2.48563933555529104470691546681, −2.33443674933814887616750232085, −2.05943713332830541585227747716, −1.97603508183990244614847532532, −1.58800842353009429692309085554, −1.04903744021631734678930036286, −0.897829304873237536834996009248, −0.69890253451324621373353811518, 0.69890253451324621373353811518, 0.897829304873237536834996009248, 1.04903744021631734678930036286, 1.58800842353009429692309085554, 1.97603508183990244614847532532, 2.05943713332830541585227747716, 2.33443674933814887616750232085, 2.48563933555529104470691546681, 2.54716488410991714957692759486, 2.78977283012405076010246361053, 2.94257835163018886009128878601, 2.95117800749499345894057129686, 3.15023739410463514149654973669, 3.41088024024501624743657621412, 3.47205549079042952999339578717, 3.79573225803279643190333185569, 3.85259891495301111643346244014, 3.95402090322453076425028077168, 4.06449505979950582610284848831, 4.14726094046601676739992841250, 4.22349041380162787760010430807, 4.46690544175616041891758364281, 4.64358051762364519638352131128, 4.80874172120921449497613091236, 4.89887582646527228188531339257
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.