Dirichlet series
| L(s) = 1 | − 5·2-s + 5·4-s + 9·7-s + 17·8-s − 19·11-s + 8·13-s − 45·14-s − 45·16-s − 9·17-s + 17·19-s + 95·22-s + 10·23-s − 25·25-s − 40·26-s + 45·28-s − 27·29-s + 7·31-s + 12·32-s + 45·34-s + 20·37-s − 85·38-s − 19·41-s + 20·43-s − 95·44-s − 50·46-s + 19·47-s − 5·49-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 5/2·4-s + 3.40·7-s + 6.01·8-s − 5.72·11-s + 2.21·13-s − 12.0·14-s − 11.2·16-s − 2.18·17-s + 3.90·19-s + 20.2·22-s + 2.08·23-s − 5·25-s − 7.84·26-s + 8.50·28-s − 5.01·29-s + 1.25·31-s + 2.12·32-s + 7.71·34-s + 3.28·37-s − 13.7·38-s − 2.96·41-s + 3.04·43-s − 14.3·44-s − 7.37·46-s + 2.77·47-s − 5/7·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{38} \cdot 11^{19} \cdot 61^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{38} \cdot 11^{19} \cdot 61^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(3^{38} \cdot 11^{19} \cdot 61^{19}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.58518\times 10^{31}\) |
| Root analytic conductor: | \(6.94418\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((38,\ 3^{38} \cdot 11^{19} \cdot 61^{19} ,\ ( \ : [1/2]^{19} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.991227989\) |
| \(L(\frac12)\) | \(\approx\) | \(5.991227989\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 + T )^{19} \) | |
| 61 | \( ( 1 + T )^{19} \) | |
| good | 2 | \( 1 + 5 T + 5 p^{2} T^{2} + 29 p T^{3} + 75 p T^{4} + 333 T^{5} + 687 T^{6} + 1281 T^{7} + 2245 T^{8} + 3601 T^{9} + 5389 T^{10} + 7263 T^{11} + 8807 T^{12} + 8713 T^{13} + 5731 T^{14} - 1447 p T^{15} - 9351 p T^{16} - 44917 T^{17} - 79669 T^{18} - 122313 T^{19} - 79669 p T^{20} - 44917 p^{2} T^{21} - 9351 p^{4} T^{22} - 1447 p^{5} T^{23} + 5731 p^{5} T^{24} + 8713 p^{6} T^{25} + 8807 p^{7} T^{26} + 7263 p^{8} T^{27} + 5389 p^{9} T^{28} + 3601 p^{10} T^{29} + 2245 p^{11} T^{30} + 1281 p^{12} T^{31} + 687 p^{13} T^{32} + 333 p^{14} T^{33} + 75 p^{16} T^{34} + 29 p^{17} T^{35} + 5 p^{19} T^{36} + 5 p^{18} T^{37} + p^{19} T^{38} \) |
| 5 | \( 1 + p^{2} T^{2} - 4 p T^{3} + 364 T^{4} - 107 p T^{5} + 4067 T^{6} - 7933 T^{7} + 38993 T^{8} - 85681 T^{9} + 330223 T^{10} - 753981 T^{11} + 2490057 T^{12} - 5697247 T^{13} + 16785768 T^{14} - 37904374 T^{15} + 101886947 T^{16} - 224216574 T^{17} + 560432351 T^{18} - 1185201854 T^{19} + 560432351 p T^{20} - 224216574 p^{2} T^{21} + 101886947 p^{3} T^{22} - 37904374 p^{4} T^{23} + 16785768 p^{5} T^{24} - 5697247 p^{6} T^{25} + 2490057 p^{7} T^{26} - 753981 p^{8} T^{27} + 330223 p^{9} T^{28} - 85681 p^{10} T^{29} + 38993 p^{11} T^{30} - 7933 p^{12} T^{31} + 4067 p^{13} T^{32} - 107 p^{15} T^{33} + 364 p^{15} T^{34} - 4 p^{17} T^{35} + p^{19} T^{36} + p^{19} T^{38} \) | |
| 7 | \( 1 - 9 T + 86 T^{2} - 76 p T^{3} + 3193 T^{4} - 2245 p T^{5} + 10635 p T^{6} - 312953 T^{7} + 1270779 T^{8} - 4747891 T^{9} + 2459329 p T^{10} - 58598584 T^{11} + 194409144 T^{12} - 87593101 p T^{13} + 38617595 p^{2} T^{14} - 5598730940 T^{15} + 16264737590 T^{16} - 45556518226 T^{17} + 125587504358 T^{18} - 334782347206 T^{19} + 125587504358 p T^{20} - 45556518226 p^{2} T^{21} + 16264737590 p^{3} T^{22} - 5598730940 p^{4} T^{23} + 38617595 p^{7} T^{24} - 87593101 p^{7} T^{25} + 194409144 p^{7} T^{26} - 58598584 p^{8} T^{27} + 2459329 p^{10} T^{28} - 4747891 p^{10} T^{29} + 1270779 p^{11} T^{30} - 312953 p^{12} T^{31} + 10635 p^{14} T^{32} - 2245 p^{15} T^{33} + 3193 p^{15} T^{34} - 76 p^{17} T^{35} + 86 p^{17} T^{36} - 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 - 8 T + 125 T^{2} - 879 T^{3} + 8055 T^{4} - 49937 T^{5} + 352375 T^{6} - 1957698 T^{7} + 11695722 T^{8} - 59256283 T^{9} + 313383086 T^{10} - 1465953928 T^{11} + 7038581274 T^{12} - 30646578609 T^{13} + 135623829486 T^{14} - 552635331582 T^{15} + 2275267941252 T^{16} - 8702950115787 T^{17} + 33531541709632 T^{18} - 120515386387602 T^{19} + 33531541709632 p T^{20} - 8702950115787 p^{2} T^{21} + 2275267941252 p^{3} T^{22} - 552635331582 p^{4} T^{23} + 135623829486 p^{5} T^{24} - 30646578609 p^{6} T^{25} + 7038581274 p^{7} T^{26} - 1465953928 p^{8} T^{27} + 313383086 p^{9} T^{28} - 59256283 p^{10} T^{29} + 11695722 p^{11} T^{30} - 1957698 p^{12} T^{31} + 352375 p^{13} T^{32} - 49937 p^{14} T^{33} + 8055 p^{15} T^{34} - 879 p^{16} T^{35} + 125 p^{17} T^{36} - 8 p^{18} T^{37} + p^{19} T^{38} \) | |
| 17 | \( 1 + 9 T + 155 T^{2} + 1150 T^{3} + 12186 T^{4} + 80063 T^{5} + 660999 T^{6} + 3950590 T^{7} + 27673665 T^{8} + 152720701 T^{9} + 947579608 T^{10} + 4871861803 T^{11} + 27433956192 T^{12} + 132108866795 T^{13} + 685419029810 T^{14} + 3101461999645 T^{15} + 14969290650336 T^{16} + 63739222793853 T^{17} + 287928271918558 T^{18} + 1153564810721111 T^{19} + 287928271918558 p T^{20} + 63739222793853 p^{2} T^{21} + 14969290650336 p^{3} T^{22} + 3101461999645 p^{4} T^{23} + 685419029810 p^{5} T^{24} + 132108866795 p^{6} T^{25} + 27433956192 p^{7} T^{26} + 4871861803 p^{8} T^{27} + 947579608 p^{9} T^{28} + 152720701 p^{10} T^{29} + 27673665 p^{11} T^{30} + 3950590 p^{12} T^{31} + 660999 p^{13} T^{32} + 80063 p^{14} T^{33} + 12186 p^{15} T^{34} + 1150 p^{16} T^{35} + 155 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 - 17 T + 288 T^{2} - 172 p T^{3} + 35007 T^{4} - 308746 T^{5} + 2573440 T^{6} - 18867802 T^{7} + 131543937 T^{8} - 831239199 T^{9} + 5027031863 T^{10} - 28015530644 T^{11} + 150536367591 T^{12} - 754277236101 T^{13} + 3683036143953 T^{14} - 16981481460006 T^{15} + 77564728648267 T^{16} - 339994280939897 T^{17} + 1506214010137031 T^{18} - 6497393486738832 T^{19} + 1506214010137031 p T^{20} - 339994280939897 p^{2} T^{21} + 77564728648267 p^{3} T^{22} - 16981481460006 p^{4} T^{23} + 3683036143953 p^{5} T^{24} - 754277236101 p^{6} T^{25} + 150536367591 p^{7} T^{26} - 28015530644 p^{8} T^{27} + 5027031863 p^{9} T^{28} - 831239199 p^{10} T^{29} + 131543937 p^{11} T^{30} - 18867802 p^{12} T^{31} + 2573440 p^{13} T^{32} - 308746 p^{14} T^{33} + 35007 p^{15} T^{34} - 172 p^{17} T^{35} + 288 p^{17} T^{36} - 17 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 - 10 T + 10 p T^{2} - 2089 T^{3} + 27593 T^{4} - 227939 T^{5} + 2262592 T^{6} - 17086886 T^{7} + 141160625 T^{8} - 979720939 T^{9} + 7085381547 T^{10} - 45421660648 T^{11} + 295739078543 T^{12} - 1758638785657 T^{13} + 10479904969397 T^{14} - 58000348761914 T^{15} + 319464719445611 T^{16} - 1648981055585327 T^{17} + 8443403994806727 T^{18} - 40675585999222302 T^{19} + 8443403994806727 p T^{20} - 1648981055585327 p^{2} T^{21} + 319464719445611 p^{3} T^{22} - 58000348761914 p^{4} T^{23} + 10479904969397 p^{5} T^{24} - 1758638785657 p^{6} T^{25} + 295739078543 p^{7} T^{26} - 45421660648 p^{8} T^{27} + 7085381547 p^{9} T^{28} - 979720939 p^{10} T^{29} + 141160625 p^{11} T^{30} - 17086886 p^{12} T^{31} + 2262592 p^{13} T^{32} - 227939 p^{14} T^{33} + 27593 p^{15} T^{34} - 2089 p^{16} T^{35} + 10 p^{18} T^{36} - 10 p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 + 27 T + 543 T^{2} + 7731 T^{3} + 94409 T^{4} + 972228 T^{5} + 9099203 T^{6} + 76790998 T^{7} + 609906611 T^{8} + 4538969769 T^{9} + 32519557765 T^{10} + 223195611792 T^{11} + 1489413751209 T^{12} + 9601114071190 T^{13} + 60213588863043 T^{14} + 365598701408931 T^{15} + 2159271329728813 T^{16} + 12384125184864807 T^{17} + 69251480492257511 T^{18} + 377365268591971366 T^{19} + 69251480492257511 p T^{20} + 12384125184864807 p^{2} T^{21} + 2159271329728813 p^{3} T^{22} + 365598701408931 p^{4} T^{23} + 60213588863043 p^{5} T^{24} + 9601114071190 p^{6} T^{25} + 1489413751209 p^{7} T^{26} + 223195611792 p^{8} T^{27} + 32519557765 p^{9} T^{28} + 4538969769 p^{10} T^{29} + 609906611 p^{11} T^{30} + 76790998 p^{12} T^{31} + 9099203 p^{13} T^{32} + 972228 p^{14} T^{33} + 94409 p^{15} T^{34} + 7731 p^{16} T^{35} + 543 p^{17} T^{36} + 27 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 - 7 T + 313 T^{2} - 1745 T^{3} + 48034 T^{4} - 228546 T^{5} + 4976968 T^{6} - 21203546 T^{7} + 393833617 T^{8} - 1547203197 T^{9} + 25251760587 T^{10} - 93132965616 T^{11} + 1358073204047 T^{12} - 153246677545 p T^{13} + 62639119135757 T^{14} - 208507084444934 T^{15} + 81061924821629 p T^{16} - 7940015937146499 T^{17} + 88435740818992781 T^{18} - 263486053627797950 T^{19} + 88435740818992781 p T^{20} - 7940015937146499 p^{2} T^{21} + 81061924821629 p^{4} T^{22} - 208507084444934 p^{4} T^{23} + 62639119135757 p^{5} T^{24} - 153246677545 p^{7} T^{25} + 1358073204047 p^{7} T^{26} - 93132965616 p^{8} T^{27} + 25251760587 p^{9} T^{28} - 1547203197 p^{10} T^{29} + 393833617 p^{11} T^{30} - 21203546 p^{12} T^{31} + 4976968 p^{13} T^{32} - 228546 p^{14} T^{33} + 48034 p^{15} T^{34} - 1745 p^{16} T^{35} + 313 p^{17} T^{36} - 7 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 - 20 T + 529 T^{2} - 7579 T^{3} + 121179 T^{4} - 1399089 T^{5} + 17203375 T^{6} - 170179590 T^{7} + 1761811754 T^{8} - 418380331 p T^{9} + 141219691014 T^{10} - 1126765129536 T^{11} + 9296912563794 T^{12} - 68317644444629 T^{13} + 518158068049246 T^{14} - 3539692761737466 T^{15} + 24938736417249380 T^{16} - 4305659250143987 p T^{17} + 1049317852548917248 T^{18} - 6285947908952657866 T^{19} + 1049317852548917248 p T^{20} - 4305659250143987 p^{3} T^{21} + 24938736417249380 p^{3} T^{22} - 3539692761737466 p^{4} T^{23} + 518158068049246 p^{5} T^{24} - 68317644444629 p^{6} T^{25} + 9296912563794 p^{7} T^{26} - 1126765129536 p^{8} T^{27} + 141219691014 p^{9} T^{28} - 418380331 p^{11} T^{29} + 1761811754 p^{11} T^{30} - 170179590 p^{12} T^{31} + 17203375 p^{13} T^{32} - 1399089 p^{14} T^{33} + 121179 p^{15} T^{34} - 7579 p^{16} T^{35} + 529 p^{17} T^{36} - 20 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 + 19 T + 506 T^{2} + 6636 T^{3} + 107341 T^{4} + 1105926 T^{5} + 13788708 T^{6} + 118113822 T^{7} + 1248252525 T^{8} + 9179930965 T^{9} + 87252857649 T^{10} + 564463578812 T^{11} + 5060173753283 T^{12} + 29512057711503 T^{13} + 258914623693343 T^{14} + 34071985404082 p T^{15} + 12193461557429751 T^{16} + 62175477043936019 T^{17} + 536335147097237381 T^{18} + 2622017764078053616 T^{19} + 536335147097237381 p T^{20} + 62175477043936019 p^{2} T^{21} + 12193461557429751 p^{3} T^{22} + 34071985404082 p^{5} T^{23} + 258914623693343 p^{5} T^{24} + 29512057711503 p^{6} T^{25} + 5060173753283 p^{7} T^{26} + 564463578812 p^{8} T^{27} + 87252857649 p^{9} T^{28} + 9179930965 p^{10} T^{29} + 1248252525 p^{11} T^{30} + 118113822 p^{12} T^{31} + 13788708 p^{13} T^{32} + 1105926 p^{14} T^{33} + 107341 p^{15} T^{34} + 6636 p^{16} T^{35} + 506 p^{17} T^{36} + 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 43 | \( 1 - 20 T + 572 T^{2} - 8077 T^{3} + 135512 T^{4} - 1497988 T^{5} + 18790148 T^{6} - 3996555 p T^{7} + 1774200657 T^{8} - 13927589453 T^{9} + 125276923181 T^{10} - 868765034041 T^{11} + 7102666891208 T^{12} - 44614823605178 T^{13} + 343538447765116 T^{14} - 2004023717345733 T^{15} + 15033040387500544 T^{16} - 84102416638225978 T^{17} + 634888635999058653 T^{18} - 82417794194082922 p T^{19} + 634888635999058653 p T^{20} - 84102416638225978 p^{2} T^{21} + 15033040387500544 p^{3} T^{22} - 2004023717345733 p^{4} T^{23} + 343538447765116 p^{5} T^{24} - 44614823605178 p^{6} T^{25} + 7102666891208 p^{7} T^{26} - 868765034041 p^{8} T^{27} + 125276923181 p^{9} T^{28} - 13927589453 p^{10} T^{29} + 1774200657 p^{11} T^{30} - 3996555 p^{13} T^{31} + 18790148 p^{13} T^{32} - 1497988 p^{14} T^{33} + 135512 p^{15} T^{34} - 8077 p^{16} T^{35} + 572 p^{17} T^{36} - 20 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 - 19 T + 486 T^{2} - 6712 T^{3} + 102965 T^{4} - 1146923 T^{5} + 13578502 T^{6} - 130762400 T^{7} + 28217871 p T^{8} - 11621437031 T^{9} + 106840792116 T^{10} - 879080625727 T^{11} + 7526347922476 T^{12} - 58855151532759 T^{13} + 474007528803564 T^{14} - 3531745422259027 T^{15} + 26901966093952584 T^{16} - 191280615501341947 T^{17} + 1385568946871706288 T^{18} - 9415498032408347691 T^{19} + 1385568946871706288 p T^{20} - 191280615501341947 p^{2} T^{21} + 26901966093952584 p^{3} T^{22} - 3531745422259027 p^{4} T^{23} + 474007528803564 p^{5} T^{24} - 58855151532759 p^{6} T^{25} + 7526347922476 p^{7} T^{26} - 879080625727 p^{8} T^{27} + 106840792116 p^{9} T^{28} - 11621437031 p^{10} T^{29} + 28217871 p^{12} T^{30} - 130762400 p^{12} T^{31} + 13578502 p^{13} T^{32} - 1146923 p^{14} T^{33} + 102965 p^{15} T^{34} - 6712 p^{16} T^{35} + 486 p^{17} T^{36} - 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 + 3 T + 553 T^{2} + 1479 T^{3} + 152698 T^{4} + 391018 T^{5} + 28234280 T^{6} + 72193850 T^{7} + 3938669069 T^{8} + 10194471857 T^{9} + 441688867147 T^{10} + 1151757712728 T^{11} + 41342815285789 T^{12} + 107148447751419 T^{13} + 3305250065811811 T^{14} + 8375541396988102 T^{15} + 228888609419105435 T^{16} + 557600895340413919 T^{17} + 13840952306811604433 T^{18} + 31860554916814458306 T^{19} + 13840952306811604433 p T^{20} + 557600895340413919 p^{2} T^{21} + 228888609419105435 p^{3} T^{22} + 8375541396988102 p^{4} T^{23} + 3305250065811811 p^{5} T^{24} + 107148447751419 p^{6} T^{25} + 41342815285789 p^{7} T^{26} + 1151757712728 p^{8} T^{27} + 441688867147 p^{9} T^{28} + 10194471857 p^{10} T^{29} + 3938669069 p^{11} T^{30} + 72193850 p^{12} T^{31} + 28234280 p^{13} T^{32} + 391018 p^{14} T^{33} + 152698 p^{15} T^{34} + 1479 p^{16} T^{35} + 553 p^{17} T^{36} + 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 - 28 T + 710 T^{2} - 12043 T^{3} + 193793 T^{4} - 2523471 T^{5} + 31692672 T^{6} - 342644362 T^{7} + 3627986625 T^{8} - 33994128649 T^{9} + 317195915191 T^{10} - 2654252058056 T^{11} + 22606553903935 T^{12} - 173800732721867 T^{13} + 1400519610931769 T^{14} - 10218195396510582 T^{15} + 80843054411446731 T^{16} - 579240599508755665 T^{17} + 4631677843792297891 T^{18} - 33493722583958564698 T^{19} + 4631677843792297891 p T^{20} - 579240599508755665 p^{2} T^{21} + 80843054411446731 p^{3} T^{22} - 10218195396510582 p^{4} T^{23} + 1400519610931769 p^{5} T^{24} - 173800732721867 p^{6} T^{25} + 22606553903935 p^{7} T^{26} - 2654252058056 p^{8} T^{27} + 317195915191 p^{9} T^{28} - 33994128649 p^{10} T^{29} + 3627986625 p^{11} T^{30} - 342644362 p^{12} T^{31} + 31692672 p^{13} T^{32} - 2523471 p^{14} T^{33} + 193793 p^{15} T^{34} - 12043 p^{16} T^{35} + 710 p^{17} T^{36} - 28 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 - 3 T + 617 T^{2} - 1296 T^{3} + 195240 T^{4} - 274406 T^{5} + 42305846 T^{6} - 35902906 T^{7} + 7029083151 T^{8} - 2751947733 T^{9} + 949648979305 T^{10} - 21362463620 T^{11} + 108075690044113 T^{12} + 27914267730953 T^{13} + 10597462643039983 T^{14} + 4749868698195514 T^{15} + 908526944174305233 T^{16} + 495817426944109421 T^{17} + 68710270553343799099 T^{18} + 38074969022944131304 T^{19} + 68710270553343799099 p T^{20} + 495817426944109421 p^{2} T^{21} + 908526944174305233 p^{3} T^{22} + 4749868698195514 p^{4} T^{23} + 10597462643039983 p^{5} T^{24} + 27914267730953 p^{6} T^{25} + 108075690044113 p^{7} T^{26} - 21362463620 p^{8} T^{27} + 949648979305 p^{9} T^{28} - 2751947733 p^{10} T^{29} + 7029083151 p^{11} T^{30} - 35902906 p^{12} T^{31} + 42305846 p^{13} T^{32} - 274406 p^{14} T^{33} + 195240 p^{15} T^{34} - 1296 p^{16} T^{35} + 617 p^{17} T^{36} - 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 71 | \( 1 - 19 T + 878 T^{2} - 14241 T^{3} + 374920 T^{4} - 5352616 T^{5} + 104752181 T^{6} - 1343370758 T^{7} + 21623226530 T^{8} - 252588529195 T^{9} + 3518873427046 T^{10} - 37800903314792 T^{11} + 469312958702142 T^{12} - 4665621917976129 T^{13} + 52560369889705106 T^{14} - 485419890861740250 T^{15} + 5019201466897587468 T^{16} - 43139697670195372617 T^{17} + \)\(41\!\cdots\!54\)\( T^{18} - \)\(32\!\cdots\!34\)\( T^{19} + \)\(41\!\cdots\!54\)\( p T^{20} - 43139697670195372617 p^{2} T^{21} + 5019201466897587468 p^{3} T^{22} - 485419890861740250 p^{4} T^{23} + 52560369889705106 p^{5} T^{24} - 4665621917976129 p^{6} T^{25} + 469312958702142 p^{7} T^{26} - 37800903314792 p^{8} T^{27} + 3518873427046 p^{9} T^{28} - 252588529195 p^{10} T^{29} + 21623226530 p^{11} T^{30} - 1343370758 p^{12} T^{31} + 104752181 p^{13} T^{32} - 5352616 p^{14} T^{33} + 374920 p^{15} T^{34} - 14241 p^{16} T^{35} + 878 p^{17} T^{36} - 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 - 20 T + 983 T^{2} - 15447 T^{3} + 442119 T^{4} - 5776019 T^{5} + 124649027 T^{6} - 1395944110 T^{7} + 25117566350 T^{8} - 245527957621 T^{9} + 3892497812088 T^{10} - 33616760931648 T^{11} + 487253061211176 T^{12} - 3757469846070647 T^{13} + 51174778307022026 T^{14} - 357004941641166770 T^{15} + 4656335452940103068 T^{16} - 29922965328739857229 T^{17} + \)\(37\!\cdots\!70\)\( T^{18} - \)\(22\!\cdots\!34\)\( T^{19} + \)\(37\!\cdots\!70\)\( p T^{20} - 29922965328739857229 p^{2} T^{21} + 4656335452940103068 p^{3} T^{22} - 357004941641166770 p^{4} T^{23} + 51174778307022026 p^{5} T^{24} - 3757469846070647 p^{6} T^{25} + 487253061211176 p^{7} T^{26} - 33616760931648 p^{8} T^{27} + 3892497812088 p^{9} T^{28} - 245527957621 p^{10} T^{29} + 25117566350 p^{11} T^{30} - 1395944110 p^{12} T^{31} + 124649027 p^{13} T^{32} - 5776019 p^{14} T^{33} + 442119 p^{15} T^{34} - 15447 p^{16} T^{35} + 983 p^{17} T^{36} - 20 p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 - 69 T + 2783 T^{2} - 80271 T^{3} + 1833295 T^{4} - 34810424 T^{5} + 569306503 T^{6} - 8213895354 T^{7} + 106756573323 T^{8} - 1272070190155 T^{9} + 14143909234770 T^{10} - 149273308631457 T^{11} + 1522063396688776 T^{12} - 15229257243053839 T^{13} + 151241951360093818 T^{14} - 1495740235635165111 T^{15} + 14669808097880173198 T^{16} - \)\(14\!\cdots\!01\)\( T^{17} + \)\(13\!\cdots\!22\)\( T^{18} - \)\(11\!\cdots\!37\)\( T^{19} + \)\(13\!\cdots\!22\)\( p T^{20} - \)\(14\!\cdots\!01\)\( p^{2} T^{21} + 14669808097880173198 p^{3} T^{22} - 1495740235635165111 p^{4} T^{23} + 151241951360093818 p^{5} T^{24} - 15229257243053839 p^{6} T^{25} + 1522063396688776 p^{7} T^{26} - 149273308631457 p^{8} T^{27} + 14143909234770 p^{9} T^{28} - 1272070190155 p^{10} T^{29} + 106756573323 p^{11} T^{30} - 8213895354 p^{12} T^{31} + 569306503 p^{13} T^{32} - 34810424 p^{14} T^{33} + 1833295 p^{15} T^{34} - 80271 p^{16} T^{35} + 2783 p^{17} T^{36} - 69 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 + T + 546 T^{2} - 622 T^{3} + 151951 T^{4} - 457032 T^{5} + 29392894 T^{6} - 133790922 T^{7} + 4497136511 T^{8} - 25906914375 T^{9} + 583122186185 T^{10} - 3883388716012 T^{11} + 66936907499529 T^{12} - 486267239143469 T^{13} + 6998628514481439 T^{14} - 53000991615274038 T^{15} + 676682691789200105 T^{16} - 5139164953933873773 T^{17} + 60700141269974555373 T^{18} - \)\(44\!\cdots\!16\)\( T^{19} + 60700141269974555373 p T^{20} - 5139164953933873773 p^{2} T^{21} + 676682691789200105 p^{3} T^{22} - 53000991615274038 p^{4} T^{23} + 6998628514481439 p^{5} T^{24} - 486267239143469 p^{6} T^{25} + 66936907499529 p^{7} T^{26} - 3883388716012 p^{8} T^{27} + 583122186185 p^{9} T^{28} - 25906914375 p^{10} T^{29} + 4497136511 p^{11} T^{30} - 133790922 p^{12} T^{31} + 29392894 p^{13} T^{32} - 457032 p^{14} T^{33} + 151951 p^{15} T^{34} - 622 p^{16} T^{35} + 546 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 + 8 p T^{2} - 987 T^{3} + 245239 T^{4} - 620559 T^{5} + 55410092 T^{6} - 193333154 T^{7} + 9368692181 T^{8} - 41204661989 T^{9} + 1279635571133 T^{10} - 6988180468952 T^{11} + 149026463314535 T^{12} - 1016065923599399 T^{13} + 15504500517386055 T^{14} - 129196593580182910 T^{15} + 1501138729702439567 T^{16} - 14299173695878752965 T^{17} + \)\(13\!\cdots\!37\)\( T^{18} - \)\(13\!\cdots\!42\)\( T^{19} + \)\(13\!\cdots\!37\)\( p T^{20} - 14299173695878752965 p^{2} T^{21} + 1501138729702439567 p^{3} T^{22} - 129196593580182910 p^{4} T^{23} + 15504500517386055 p^{5} T^{24} - 1016065923599399 p^{6} T^{25} + 149026463314535 p^{7} T^{26} - 6988180468952 p^{8} T^{27} + 1279635571133 p^{9} T^{28} - 41204661989 p^{10} T^{29} + 9368692181 p^{11} T^{30} - 193333154 p^{12} T^{31} + 55410092 p^{13} T^{32} - 620559 p^{14} T^{33} + 245239 p^{15} T^{34} - 987 p^{16} T^{35} + 8 p^{18} T^{36} + p^{19} T^{38} \) | |
| 97 | \( 1 - 21 T + 847 T^{2} - 11527 T^{3} + 298607 T^{4} - 2986298 T^{5} + 68446913 T^{6} - 534920010 T^{7} + 12571255465 T^{8} - 79993220799 T^{9} + 2032794671226 T^{10} - 10864472031073 T^{11} + 294914282878728 T^{12} - 1337284301131743 T^{13} + 38226363126660730 T^{14} - 149144666063131807 T^{15} + 4466890173200617866 T^{16} - 160101889524293865 p T^{17} + \)\(47\!\cdots\!92\)\( T^{18} - \)\(15\!\cdots\!23\)\( T^{19} + \)\(47\!\cdots\!92\)\( p T^{20} - 160101889524293865 p^{3} T^{21} + 4466890173200617866 p^{3} T^{22} - 149144666063131807 p^{4} T^{23} + 38226363126660730 p^{5} T^{24} - 1337284301131743 p^{6} T^{25} + 294914282878728 p^{7} T^{26} - 10864472031073 p^{8} T^{27} + 2032794671226 p^{9} T^{28} - 79993220799 p^{10} T^{29} + 12571255465 p^{11} T^{30} - 534920010 p^{12} T^{31} + 68446913 p^{13} T^{32} - 2986298 p^{14} T^{33} + 298607 p^{15} T^{34} - 11527 p^{16} T^{35} + 847 p^{17} T^{36} - 21 p^{18} T^{37} + p^{19} T^{38} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.66154632528790067410397152694, −1.63248615343005734937822863836, −1.54559316996896868576097479895, −1.48960031230042014437758154119, −1.45172823314523371725593453620, −1.41060211886554269871241498241, −1.28931020500917756085555491963, −1.28369004472879288217664476007, −1.07860184271669167793380428294, −1.02265820738104332026473935008, −0.872999110720427619735919697770, −0.77100916121701519444734504293, −0.69258282761572338907842898009, −0.69009158803648905573931301033, −0.68823313423112780019199067420, −0.68771374618533636755562818673, −0.65922018938889865794484106532, −0.63281220579478125170188668711, −0.52789755650064710048276365649, −0.49804726852667436000945183617, −0.41203094428738351808325993555, −0.36985154384118916324762814411, −0.28176972484968532322531081022, −0.22093709887265402937075593745, −0.12772768030861900085244819836, 0.12772768030861900085244819836, 0.22093709887265402937075593745, 0.28176972484968532322531081022, 0.36985154384118916324762814411, 0.41203094428738351808325993555, 0.49804726852667436000945183617, 0.52789755650064710048276365649, 0.63281220579478125170188668711, 0.65922018938889865794484106532, 0.68771374618533636755562818673, 0.68823313423112780019199067420, 0.69009158803648905573931301033, 0.69258282761572338907842898009, 0.77100916121701519444734504293, 0.872999110720427619735919697770, 1.02265820738104332026473935008, 1.07860184271669167793380428294, 1.28369004472879288217664476007, 1.28931020500917756085555491963, 1.41060211886554269871241498241, 1.45172823314523371725593453620, 1.48960031230042014437758154119, 1.54559316996896868576097479895, 1.63248615343005734937822863836, 1.66154632528790067410397152694
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.