Dirichlet series
| L(s) = 1 | − 2·3-s − 2·5-s + 5·7-s − 27·9-s + 8·11-s − 3·13-s + 4·15-s + 20·17-s − 9·19-s − 10·21-s − 8·23-s − 50·25-s + 53·27-s − 37·29-s − 9·31-s − 16·33-s − 10·35-s − 11·37-s + 6·39-s − 27·41-s + 17·43-s + 54·45-s + 7·47-s − 62·49-s − 40·51-s − 53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1.88·7-s − 9·9-s + 2.41·11-s − 0.832·13-s + 1.03·15-s + 4.85·17-s − 2.06·19-s − 2.18·21-s − 1.66·23-s − 10·25-s + 10.1·27-s − 6.87·29-s − 1.61·31-s − 2.78·33-s − 1.69·35-s − 1.80·37-s + 0.960·39-s − 4.21·41-s + 2.59·43-s + 8.04·45-s + 1.02·47-s − 8.85·49-s − 5.60·51-s − 0.137·53-s − 2.15·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 17^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 17^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{60} \cdot 17^{20} \cdot 59^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35944\times 10^{36}\) |
| Root analytic conductor: | \(8.00449\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 2^{60} \cdot 17^{20} \cdot 59^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 17 | \( ( 1 - T )^{20} \) | |
| 59 | \( ( 1 + T )^{20} \) | |
| good | 3 | \( 1 + 2 T + 31 T^{2} + 7 p^{2} T^{3} + 485 T^{4} + 991 T^{5} + 5101 T^{6} + 10403 T^{7} + 40541 T^{8} + 82093 T^{9} + 259570 T^{10} + 519386 T^{11} + 1393409 T^{12} + 2739173 T^{13} + 2145794 p T^{14} + 12339202 T^{15} + 26030167 T^{16} + 48198637 T^{17} + 93075263 T^{18} + 6098932 p^{3} T^{19} + 295887040 T^{20} + 6098932 p^{4} T^{21} + 93075263 p^{2} T^{22} + 48198637 p^{3} T^{23} + 26030167 p^{4} T^{24} + 12339202 p^{5} T^{25} + 2145794 p^{7} T^{26} + 2739173 p^{7} T^{27} + 1393409 p^{8} T^{28} + 519386 p^{9} T^{29} + 259570 p^{10} T^{30} + 82093 p^{11} T^{31} + 40541 p^{12} T^{32} + 10403 p^{13} T^{33} + 5101 p^{14} T^{34} + 991 p^{15} T^{35} + 485 p^{16} T^{36} + 7 p^{19} T^{37} + 31 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + 2 T + 54 T^{2} + 104 T^{3} + 1497 T^{4} + 2753 T^{5} + 5627 p T^{6} + 9871 p T^{7} + 400684 T^{8} + 26821 p^{2} T^{9} + 4585859 T^{10} + 7317554 T^{11} + 43684807 T^{12} + 13275039 p T^{13} + 354154473 T^{14} + 102228188 p T^{15} + 2478882106 T^{16} + 3386477826 T^{17} + 15116206484 T^{18} + 19455464801 T^{19} + 80708779699 T^{20} + 19455464801 p T^{21} + 15116206484 p^{2} T^{22} + 3386477826 p^{3} T^{23} + 2478882106 p^{4} T^{24} + 102228188 p^{6} T^{25} + 354154473 p^{6} T^{26} + 13275039 p^{8} T^{27} + 43684807 p^{8} T^{28} + 7317554 p^{9} T^{29} + 4585859 p^{10} T^{30} + 26821 p^{13} T^{31} + 400684 p^{12} T^{32} + 9871 p^{14} T^{33} + 5627 p^{15} T^{34} + 2753 p^{15} T^{35} + 1497 p^{16} T^{36} + 104 p^{17} T^{37} + 54 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 5 T + 87 T^{2} - 53 p T^{3} + 3642 T^{4} - 13749 T^{5} + 99265 T^{6} - 340337 T^{7} + 1999069 T^{8} - 6338498 T^{9} + 31882299 T^{10} - 94656474 T^{11} + 420207773 T^{12} - 1177171122 T^{13} + 4704643947 T^{14} - 12480337120 T^{15} + 45555206555 T^{16} - 114437897121 T^{17} + 385831067544 T^{18} - 915106587109 T^{19} + 2876051138713 T^{20} - 915106587109 p T^{21} + 385831067544 p^{2} T^{22} - 114437897121 p^{3} T^{23} + 45555206555 p^{4} T^{24} - 12480337120 p^{5} T^{25} + 4704643947 p^{6} T^{26} - 1177171122 p^{7} T^{27} + 420207773 p^{8} T^{28} - 94656474 p^{9} T^{29} + 31882299 p^{10} T^{30} - 6338498 p^{11} T^{31} + 1999069 p^{12} T^{32} - 340337 p^{13} T^{33} + 99265 p^{14} T^{34} - 13749 p^{15} T^{35} + 3642 p^{16} T^{36} - 53 p^{18} T^{37} + 87 p^{18} T^{38} - 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 8 T + 185 T^{2} - 1260 T^{3} + 16014 T^{4} - 95217 T^{5} + 871572 T^{6} - 418782 p T^{7} + 33734865 T^{8} - 160665931 T^{9} + 994592393 T^{10} - 4313556960 T^{11} + 23343586226 T^{12} - 92956808840 T^{13} + 449754886246 T^{14} - 1654695808399 T^{15} + 7271406198616 T^{16} - 24820088424290 T^{17} + 100165193299172 T^{18} - 317795048647585 T^{19} + 1186412049216748 T^{20} - 317795048647585 p T^{21} + 100165193299172 p^{2} T^{22} - 24820088424290 p^{3} T^{23} + 7271406198616 p^{4} T^{24} - 1654695808399 p^{5} T^{25} + 449754886246 p^{6} T^{26} - 92956808840 p^{7} T^{27} + 23343586226 p^{8} T^{28} - 4313556960 p^{9} T^{29} + 994592393 p^{10} T^{30} - 160665931 p^{11} T^{31} + 33734865 p^{12} T^{32} - 418782 p^{14} T^{33} + 871572 p^{14} T^{34} - 95217 p^{15} T^{35} + 16014 p^{16} T^{36} - 1260 p^{17} T^{37} + 185 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 3 T + 144 T^{2} + 430 T^{3} + 10116 T^{4} + 30034 T^{5} + 462415 T^{6} + 1362299 T^{7} + 15503099 T^{8} + 45152323 T^{9} + 408194894 T^{10} + 1168407619 T^{11} + 8846340345 T^{12} + 24698786124 T^{13} + 163688504237 T^{14} + 442458494646 T^{15} + 2666700877778 T^{16} + 6944693423113 T^{17} + 3017655928334 p T^{18} + 98310434151789 T^{19} + 530314483862154 T^{20} + 98310434151789 p T^{21} + 3017655928334 p^{3} T^{22} + 6944693423113 p^{3} T^{23} + 2666700877778 p^{4} T^{24} + 442458494646 p^{5} T^{25} + 163688504237 p^{6} T^{26} + 24698786124 p^{7} T^{27} + 8846340345 p^{8} T^{28} + 1168407619 p^{9} T^{29} + 408194894 p^{10} T^{30} + 45152323 p^{11} T^{31} + 15503099 p^{12} T^{32} + 1362299 p^{13} T^{33} + 462415 p^{14} T^{34} + 30034 p^{15} T^{35} + 10116 p^{16} T^{36} + 430 p^{17} T^{37} + 144 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 9 T + 255 T^{2} + 93 p T^{3} + 28723 T^{4} + 156949 T^{5} + 1940613 T^{6} + 8293363 T^{7} + 89338270 T^{8} + 282175396 T^{9} + 2996149453 T^{10} + 5784838460 T^{11} + 75637850468 T^{12} + 23339891944 T^{13} + 1442576482413 T^{14} - 3325075096688 T^{15} + 20247251566389 T^{16} - 152412679509203 T^{17} + 205763095189070 T^{18} - 4087277891912045 T^{19} + 2352864705481153 T^{20} - 4087277891912045 p T^{21} + 205763095189070 p^{2} T^{22} - 152412679509203 p^{3} T^{23} + 20247251566389 p^{4} T^{24} - 3325075096688 p^{5} T^{25} + 1442576482413 p^{6} T^{26} + 23339891944 p^{7} T^{27} + 75637850468 p^{8} T^{28} + 5784838460 p^{9} T^{29} + 2996149453 p^{10} T^{30} + 282175396 p^{11} T^{31} + 89338270 p^{12} T^{32} + 8293363 p^{13} T^{33} + 1940613 p^{14} T^{34} + 156949 p^{15} T^{35} + 28723 p^{16} T^{36} + 93 p^{18} T^{37} + 255 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 8 T + 292 T^{2} + 2156 T^{3} + 41227 T^{4} + 280610 T^{5} + 3741932 T^{6} + 23441848 T^{7} + 244999461 T^{8} + 1410400335 T^{9} + 12326171430 T^{10} + 65140566450 T^{11} + 497103257765 T^{12} + 2413736462326 T^{13} + 16640951889186 T^{14} + 74612885218025 T^{15} + 478999248558686 T^{16} + 2006866126632746 T^{17} + 12315220441241672 T^{18} + 49115102111106258 T^{19} + 12720297325886888 p T^{20} + 49115102111106258 p T^{21} + 12315220441241672 p^{2} T^{22} + 2006866126632746 p^{3} T^{23} + 478999248558686 p^{4} T^{24} + 74612885218025 p^{5} T^{25} + 16640951889186 p^{6} T^{26} + 2413736462326 p^{7} T^{27} + 497103257765 p^{8} T^{28} + 65140566450 p^{9} T^{29} + 12326171430 p^{10} T^{30} + 1410400335 p^{11} T^{31} + 244999461 p^{12} T^{32} + 23441848 p^{13} T^{33} + 3741932 p^{14} T^{34} + 280610 p^{15} T^{35} + 41227 p^{16} T^{36} + 2156 p^{17} T^{37} + 292 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 37 T + 943 T^{2} + 17737 T^{3} + 278854 T^{4} + 3754350 T^{5} + 44967303 T^{6} + 485197767 T^{7} + 4800171001 T^{8} + 1512901782 p T^{9} + 374281676705 T^{10} + 2995492656385 T^{11} + 780922721385 p T^{12} + 162370612622668 T^{13} + 1109804278664006 T^{14} + 7254352245241430 T^{15} + 45540590615513526 T^{16} + 275244650668164141 T^{17} + 1606763367596003762 T^{18} + 9072135706053980015 T^{19} + 49634065273387283504 T^{20} + 9072135706053980015 p T^{21} + 1606763367596003762 p^{2} T^{22} + 275244650668164141 p^{3} T^{23} + 45540590615513526 p^{4} T^{24} + 7254352245241430 p^{5} T^{25} + 1109804278664006 p^{6} T^{26} + 162370612622668 p^{7} T^{27} + 780922721385 p^{9} T^{28} + 2995492656385 p^{9} T^{29} + 374281676705 p^{10} T^{30} + 1512901782 p^{12} T^{31} + 4800171001 p^{12} T^{32} + 485197767 p^{13} T^{33} + 44967303 p^{14} T^{34} + 3754350 p^{15} T^{35} + 278854 p^{16} T^{36} + 17737 p^{17} T^{37} + 943 p^{18} T^{38} + 37 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 + 9 T + 396 T^{2} + 2866 T^{3} + 74093 T^{4} + 447902 T^{5} + 8974647 T^{6} + 1501310 p T^{7} + 804483764 T^{8} + 3649471448 T^{9} + 57312116355 T^{10} + 230855289157 T^{11} + 3382021421319 T^{12} + 12247066081883 T^{13} + 169560331926222 T^{14} + 558281642096262 T^{15} + 7339264983570021 T^{16} + 22209830934347631 T^{17} + 276982888020947212 T^{18} + 778280410524654378 T^{19} + 9163256163177548772 T^{20} + 778280410524654378 p T^{21} + 276982888020947212 p^{2} T^{22} + 22209830934347631 p^{3} T^{23} + 7339264983570021 p^{4} T^{24} + 558281642096262 p^{5} T^{25} + 169560331926222 p^{6} T^{26} + 12247066081883 p^{7} T^{27} + 3382021421319 p^{8} T^{28} + 230855289157 p^{9} T^{29} + 57312116355 p^{10} T^{30} + 3649471448 p^{11} T^{31} + 804483764 p^{12} T^{32} + 1501310 p^{14} T^{33} + 8974647 p^{14} T^{34} + 447902 p^{15} T^{35} + 74093 p^{16} T^{36} + 2866 p^{17} T^{37} + 396 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + 11 T + 420 T^{2} + 3528 T^{3} + 77910 T^{4} + 514721 T^{5} + 8845306 T^{6} + 46676807 T^{7} + 720939262 T^{8} + 3056427762 T^{9} + 46674084169 T^{10} + 157259725988 T^{11} + 2546545804966 T^{12} + 177025728901 p T^{13} + 120999815465686 T^{14} + 221609509005091 T^{15} + 5163521500789543 T^{16} + 6346729534034029 T^{17} + 204894397555576259 T^{18} + 181385306424030326 T^{19} + 7742501906680923900 T^{20} + 181385306424030326 p T^{21} + 204894397555576259 p^{2} T^{22} + 6346729534034029 p^{3} T^{23} + 5163521500789543 p^{4} T^{24} + 221609509005091 p^{5} T^{25} + 120999815465686 p^{6} T^{26} + 177025728901 p^{8} T^{27} + 2546545804966 p^{8} T^{28} + 157259725988 p^{9} T^{29} + 46674084169 p^{10} T^{30} + 3056427762 p^{11} T^{31} + 720939262 p^{12} T^{32} + 46676807 p^{13} T^{33} + 8845306 p^{14} T^{34} + 514721 p^{15} T^{35} + 77910 p^{16} T^{36} + 3528 p^{17} T^{37} + 420 p^{18} T^{38} + 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 27 T + 658 T^{2} + 11933 T^{3} + 191310 T^{4} + 2702083 T^{5} + 34838769 T^{6} + 413557489 T^{7} + 4582181819 T^{8} + 47677599091 T^{9} + 469099238700 T^{10} + 4383643527563 T^{11} + 39054475136130 T^{12} + 332739018951659 T^{13} + 2717401691032347 T^{14} + 21315122665758159 T^{15} + 3922447703984509 p T^{16} + 1168617579432505796 T^{17} + 8185008585472393757 T^{18} + 1348692458632325718 p T^{19} + \)\(36\!\cdots\!20\)\( T^{20} + 1348692458632325718 p^{2} T^{21} + 8185008585472393757 p^{2} T^{22} + 1168617579432505796 p^{3} T^{23} + 3922447703984509 p^{5} T^{24} + 21315122665758159 p^{5} T^{25} + 2717401691032347 p^{6} T^{26} + 332739018951659 p^{7} T^{27} + 39054475136130 p^{8} T^{28} + 4383643527563 p^{9} T^{29} + 469099238700 p^{10} T^{30} + 47677599091 p^{11} T^{31} + 4582181819 p^{12} T^{32} + 413557489 p^{13} T^{33} + 34838769 p^{14} T^{34} + 2702083 p^{15} T^{35} + 191310 p^{16} T^{36} + 11933 p^{17} T^{37} + 658 p^{18} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 17 T + 610 T^{2} - 8834 T^{3} + 179157 T^{4} - 2265690 T^{5} + 33973653 T^{6} - 382216687 T^{7} + 4696518257 T^{8} - 47676652964 T^{9} + 505824617704 T^{10} - 4684247923967 T^{11} + 1028665962331 p T^{12} - 376805945725294 T^{13} + 3227318386091311 T^{14} - 25446008253226253 T^{15} + 200100480885307440 T^{16} - 1466291825987948587 T^{17} + 10669104034447346946 T^{18} - 72816425556089665355 T^{19} + \)\(49\!\cdots\!12\)\( T^{20} - 72816425556089665355 p T^{21} + 10669104034447346946 p^{2} T^{22} - 1466291825987948587 p^{3} T^{23} + 200100480885307440 p^{4} T^{24} - 25446008253226253 p^{5} T^{25} + 3227318386091311 p^{6} T^{26} - 376805945725294 p^{7} T^{27} + 1028665962331 p^{9} T^{28} - 4684247923967 p^{9} T^{29} + 505824617704 p^{10} T^{30} - 47676652964 p^{11} T^{31} + 4696518257 p^{12} T^{32} - 382216687 p^{13} T^{33} + 33973653 p^{14} T^{34} - 2265690 p^{15} T^{35} + 179157 p^{16} T^{36} - 8834 p^{17} T^{37} + 610 p^{18} T^{38} - 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 7 T + 514 T^{2} - 2676 T^{3} + 127007 T^{4} - 494022 T^{5} + 20701324 T^{6} - 60175260 T^{7} + 2548348224 T^{8} - 5562152951 T^{9} + 254332687352 T^{10} - 421800582482 T^{11} + 21399445469682 T^{12} - 27481061134261 T^{13} + 1553207193496961 T^{14} - 1589717399909578 T^{15} + 98682618601105253 T^{16} - 84092794643740564 T^{17} + 5539361413778898969 T^{18} - 4166288414026580369 T^{19} + \)\(27\!\cdots\!22\)\( T^{20} - 4166288414026580369 p T^{21} + 5539361413778898969 p^{2} T^{22} - 84092794643740564 p^{3} T^{23} + 98682618601105253 p^{4} T^{24} - 1589717399909578 p^{5} T^{25} + 1553207193496961 p^{6} T^{26} - 27481061134261 p^{7} T^{27} + 21399445469682 p^{8} T^{28} - 421800582482 p^{9} T^{29} + 254332687352 p^{10} T^{30} - 5562152951 p^{11} T^{31} + 2548348224 p^{12} T^{32} - 60175260 p^{13} T^{33} + 20701324 p^{14} T^{34} - 494022 p^{15} T^{35} + 127007 p^{16} T^{36} - 2676 p^{17} T^{37} + 514 p^{18} T^{38} - 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + T + 541 T^{2} - 38 T^{3} + 149475 T^{4} - 127957 T^{5} + 28176307 T^{6} - 39958288 T^{7} + 4061678248 T^{8} - 7317908510 T^{9} + 474844636326 T^{10} - 970616577652 T^{11} + 46591671278930 T^{12} - 101379184562821 T^{13} + 3919581633976250 T^{14} - 8705952454677363 T^{15} + 286582453949162595 T^{16} - 629706986159469104 T^{17} + 18366125645137633549 T^{18} - 38874051053117478903 T^{19} + \)\(10\!\cdots\!37\)\( T^{20} - 38874051053117478903 p T^{21} + 18366125645137633549 p^{2} T^{22} - 629706986159469104 p^{3} T^{23} + 286582453949162595 p^{4} T^{24} - 8705952454677363 p^{5} T^{25} + 3919581633976250 p^{6} T^{26} - 101379184562821 p^{7} T^{27} + 46591671278930 p^{8} T^{28} - 970616577652 p^{9} T^{29} + 474844636326 p^{10} T^{30} - 7317908510 p^{11} T^{31} + 4061678248 p^{12} T^{32} - 39958288 p^{13} T^{33} + 28176307 p^{14} T^{34} - 127957 p^{15} T^{35} + 149475 p^{16} T^{36} - 38 p^{17} T^{37} + 541 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 12 T + 655 T^{2} + 7684 T^{3} + 213309 T^{4} + 2431661 T^{5} + 46242009 T^{6} + 507753100 T^{7} + 7522844602 T^{8} + 78897034283 T^{9} + 980093817284 T^{10} + 9759405357429 T^{11} + 106476108052207 T^{12} + 1003714694726075 T^{13} + 9915103171279701 T^{14} + 88429016682198543 T^{15} + 807117830351519727 T^{16} + 6814213847461090146 T^{17} + 58223268121987140479 T^{18} + \)\(46\!\cdots\!23\)\( T^{19} + \)\(37\!\cdots\!20\)\( T^{20} + \)\(46\!\cdots\!23\)\( p T^{21} + 58223268121987140479 p^{2} T^{22} + 6814213847461090146 p^{3} T^{23} + 807117830351519727 p^{4} T^{24} + 88429016682198543 p^{5} T^{25} + 9915103171279701 p^{6} T^{26} + 1003714694726075 p^{7} T^{27} + 106476108052207 p^{8} T^{28} + 9759405357429 p^{9} T^{29} + 980093817284 p^{10} T^{30} + 78897034283 p^{11} T^{31} + 7522844602 p^{12} T^{32} + 507753100 p^{13} T^{33} + 46242009 p^{14} T^{34} + 2431661 p^{15} T^{35} + 213309 p^{16} T^{36} + 7684 p^{17} T^{37} + 655 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 26 T + 953 T^{2} + 19916 T^{3} + 444653 T^{4} + 7747054 T^{5} + 134669597 T^{6} + 2025561425 T^{7} + 29804561945 T^{8} + 397042326939 T^{9} + 5143341072052 T^{10} + 61750976484353 T^{11} + 719870079467870 T^{12} + 7881909854009983 T^{13} + 83799008796162203 T^{14} + 843447520370875407 T^{15} + 8247192552919314904 T^{16} + 76702151514555129443 T^{17} + \)\(69\!\cdots\!75\)\( T^{18} + \)\(59\!\cdots\!80\)\( T^{19} + \)\(50\!\cdots\!02\)\( T^{20} + \)\(59\!\cdots\!80\)\( p T^{21} + \)\(69\!\cdots\!75\)\( p^{2} T^{22} + 76702151514555129443 p^{3} T^{23} + 8247192552919314904 p^{4} T^{24} + 843447520370875407 p^{5} T^{25} + 83799008796162203 p^{6} T^{26} + 7881909854009983 p^{7} T^{27} + 719870079467870 p^{8} T^{28} + 61750976484353 p^{9} T^{29} + 5143341072052 p^{10} T^{30} + 397042326939 p^{11} T^{31} + 29804561945 p^{12} T^{32} + 2025561425 p^{13} T^{33} + 134669597 p^{14} T^{34} + 7747054 p^{15} T^{35} + 444653 p^{16} T^{36} + 19916 p^{17} T^{37} + 953 p^{18} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 3 T + 706 T^{2} + 2690 T^{3} + 250669 T^{4} + 1115794 T^{5} + 60127511 T^{6} + 293362369 T^{7} + 10993037406 T^{8} + 56024907546 T^{9} + 1632209969577 T^{10} + 8395614447344 T^{11} + 204309175552611 T^{12} + 1036577548737460 T^{13} + 22081498438669700 T^{14} + 108876848524266762 T^{15} + 2094017150878584927 T^{16} + 9933353120671214630 T^{17} + \)\(17\!\cdots\!02\)\( T^{18} + \)\(79\!\cdots\!88\)\( T^{19} + \)\(13\!\cdots\!72\)\( T^{20} + \)\(79\!\cdots\!88\)\( p T^{21} + \)\(17\!\cdots\!02\)\( p^{2} T^{22} + 9933353120671214630 p^{3} T^{23} + 2094017150878584927 p^{4} T^{24} + 108876848524266762 p^{5} T^{25} + 22081498438669700 p^{6} T^{26} + 1036577548737460 p^{7} T^{27} + 204309175552611 p^{8} T^{28} + 8395614447344 p^{9} T^{29} + 1632209969577 p^{10} T^{30} + 56024907546 p^{11} T^{31} + 10993037406 p^{12} T^{32} + 293362369 p^{13} T^{33} + 60127511 p^{14} T^{34} + 1115794 p^{15} T^{35} + 250669 p^{16} T^{36} + 2690 p^{17} T^{37} + 706 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 45 T + 1756 T^{2} + 47975 T^{3} + 1168570 T^{4} + 24007296 T^{5} + 451344915 T^{6} + 7604469636 T^{7} + 119333799866 T^{8} + 1726708586923 T^{9} + 23555000545453 T^{10} + 301146645769363 T^{11} + 3661733540774130 T^{12} + 42169291416841067 T^{13} + 6368592043265812 p T^{14} + 4889820949948147927 T^{15} + 49477354397440645955 T^{16} + \)\(48\!\cdots\!33\)\( T^{17} + \)\(44\!\cdots\!96\)\( T^{18} + \)\(40\!\cdots\!91\)\( T^{19} + \)\(35\!\cdots\!20\)\( T^{20} + \)\(40\!\cdots\!91\)\( p T^{21} + \)\(44\!\cdots\!96\)\( p^{2} T^{22} + \)\(48\!\cdots\!33\)\( p^{3} T^{23} + 49477354397440645955 p^{4} T^{24} + 4889820949948147927 p^{5} T^{25} + 6368592043265812 p^{7} T^{26} + 42169291416841067 p^{7} T^{27} + 3661733540774130 p^{8} T^{28} + 301146645769363 p^{9} T^{29} + 23555000545453 p^{10} T^{30} + 1726708586923 p^{11} T^{31} + 119333799866 p^{12} T^{32} + 7604469636 p^{13} T^{33} + 451344915 p^{14} T^{34} + 24007296 p^{15} T^{35} + 1168570 p^{16} T^{36} + 47975 p^{17} T^{37} + 1756 p^{18} T^{38} + 45 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 5 T + 1056 T^{2} - 5353 T^{3} + 553503 T^{4} - 2823710 T^{5} + 191447956 T^{6} - 974349567 T^{7} + 48990962452 T^{8} - 246393301734 T^{9} + 9854516572571 T^{10} - 48500569979900 T^{11} + 1615884271716892 T^{12} - 7705743697634840 T^{13} + 221061967427757181 T^{14} - 1011227368500988576 T^{15} + 25613964920194473443 T^{16} - 1408072677520866830 p T^{17} + \)\(25\!\cdots\!75\)\( T^{18} - \)\(10\!\cdots\!87\)\( T^{19} + \)\(21\!\cdots\!27\)\( T^{20} - \)\(10\!\cdots\!87\)\( p T^{21} + \)\(25\!\cdots\!75\)\( p^{2} T^{22} - 1408072677520866830 p^{4} T^{23} + 25613964920194473443 p^{4} T^{24} - 1011227368500988576 p^{5} T^{25} + 221061967427757181 p^{6} T^{26} - 7705743697634840 p^{7} T^{27} + 1615884271716892 p^{8} T^{28} - 48500569979900 p^{9} T^{29} + 9854516572571 p^{10} T^{30} - 246393301734 p^{11} T^{31} + 48990962452 p^{12} T^{32} - 974349567 p^{13} T^{33} + 191447956 p^{14} T^{34} - 2823710 p^{15} T^{35} + 553503 p^{16} T^{36} - 5353 p^{17} T^{37} + 1056 p^{18} T^{38} - 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 17 T + 771 T^{2} + 12956 T^{3} + 317646 T^{4} + 5006291 T^{5} + 89578738 T^{6} + 1299876866 T^{7} + 19128079798 T^{8} + 253782805442 T^{9} + 3267135064632 T^{10} + 39686576467582 T^{11} + 462983323675942 T^{12} + 5186137055367824 T^{13} + 56021500032626775 T^{14} + 585169629158358676 T^{15} + 5937721098348293007 T^{16} + 58558232238442045723 T^{17} + \)\(56\!\cdots\!92\)\( T^{18} + \)\(53\!\cdots\!21\)\( T^{19} + \)\(48\!\cdots\!28\)\( T^{20} + \)\(53\!\cdots\!21\)\( p T^{21} + \)\(56\!\cdots\!92\)\( p^{2} T^{22} + 58558232238442045723 p^{3} T^{23} + 5937721098348293007 p^{4} T^{24} + 585169629158358676 p^{5} T^{25} + 56021500032626775 p^{6} T^{26} + 5186137055367824 p^{7} T^{27} + 462983323675942 p^{8} T^{28} + 39686576467582 p^{9} T^{29} + 3267135064632 p^{10} T^{30} + 253782805442 p^{11} T^{31} + 19128079798 p^{12} T^{32} + 1299876866 p^{13} T^{33} + 89578738 p^{14} T^{34} + 5006291 p^{15} T^{35} + 317646 p^{16} T^{36} + 12956 p^{17} T^{37} + 771 p^{18} T^{38} + 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 40 T + 1544 T^{2} + 38886 T^{3} + 937791 T^{4} + 18343325 T^{5} + 346594730 T^{6} + 5727644106 T^{7} + 92025685223 T^{8} + 1340377901949 T^{9} + 19069469337523 T^{10} + 250950914983794 T^{11} + 3236974286492076 T^{12} + 39101354166811511 T^{13} + 464144437061931660 T^{14} + 5200111482381364698 T^{15} + 57354738065356198030 T^{16} + \)\(59\!\cdots\!79\)\( T^{17} + \)\(61\!\cdots\!87\)\( T^{18} + \)\(60\!\cdots\!08\)\( T^{19} + \)\(58\!\cdots\!14\)\( T^{20} + \)\(60\!\cdots\!08\)\( p T^{21} + \)\(61\!\cdots\!87\)\( p^{2} T^{22} + \)\(59\!\cdots\!79\)\( p^{3} T^{23} + 57354738065356198030 p^{4} T^{24} + 5200111482381364698 p^{5} T^{25} + 464144437061931660 p^{6} T^{26} + 39101354166811511 p^{7} T^{27} + 3236974286492076 p^{8} T^{28} + 250950914983794 p^{9} T^{29} + 19069469337523 p^{10} T^{30} + 1340377901949 p^{11} T^{31} + 92025685223 p^{12} T^{32} + 5727644106 p^{13} T^{33} + 346594730 p^{14} T^{34} + 18343325 p^{15} T^{35} + 937791 p^{16} T^{36} + 38886 p^{17} T^{37} + 1544 p^{18} T^{38} + 40 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 77 T + 3724 T^{2} + 133061 T^{3} + 3889410 T^{4} + 96909001 T^{5} + 2124523217 T^{6} + 41747163663 T^{7} + 746722029464 T^{8} + 12291243438535 T^{9} + 187989586339593 T^{10} + 2692365896599708 T^{11} + 36367996329508302 T^{12} + 466191455761961328 T^{13} + 5703833926348538170 T^{14} + 66934523397695589207 T^{15} + \)\(75\!\cdots\!97\)\( T^{16} + \)\(82\!\cdots\!94\)\( T^{17} + \)\(87\!\cdots\!72\)\( T^{18} + \)\(89\!\cdots\!78\)\( T^{19} + \)\(89\!\cdots\!04\)\( T^{20} + \)\(89\!\cdots\!78\)\( p T^{21} + \)\(87\!\cdots\!72\)\( p^{2} T^{22} + \)\(82\!\cdots\!94\)\( p^{3} T^{23} + \)\(75\!\cdots\!97\)\( p^{4} T^{24} + 66934523397695589207 p^{5} T^{25} + 5703833926348538170 p^{6} T^{26} + 466191455761961328 p^{7} T^{27} + 36367996329508302 p^{8} T^{28} + 2692365896599708 p^{9} T^{29} + 187989586339593 p^{10} T^{30} + 12291243438535 p^{11} T^{31} + 746722029464 p^{12} T^{32} + 41747163663 p^{13} T^{33} + 2124523217 p^{14} T^{34} + 96909001 p^{15} T^{35} + 3889410 p^{16} T^{36} + 133061 p^{17} T^{37} + 3724 p^{18} T^{38} + 77 p^{19} T^{39} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.05200862934112303125822250809, −2.03259118962573640868960114684, −1.91033445301323909218349129050, −1.81647721257277098583872416829, −1.79081236798950943580421182870, −1.67178815842616349396668764136, −1.61241798454744726494844724035, −1.59975682930849567935026295555, −1.58395416122430930810296498854, −1.57163672524889472013323019674, −1.44480060487019275954392333239, −1.39697163311368880374967168263, −1.38326510729422059299519059727, −1.26466718560735549068699678280, −1.26399514532145984493693866031, −1.25588452758572444542148152400, −1.25305632475229947209172411764, −1.23805975387049355993750119745, −1.22874045859818616702458760091, −1.14006203272547792063975895127, −1.11945270341112800554162570350, −1.04050658863622218193582187863, −1.01625410394691307967769642913, −0.875863805228129409629602699193, −0.853428705521966237550270205990, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.853428705521966237550270205990, 0.875863805228129409629602699193, 1.01625410394691307967769642913, 1.04050658863622218193582187863, 1.11945270341112800554162570350, 1.14006203272547792063975895127, 1.22874045859818616702458760091, 1.23805975387049355993750119745, 1.25305632475229947209172411764, 1.25588452758572444542148152400, 1.26399514532145984493693866031, 1.26466718560735549068699678280, 1.38326510729422059299519059727, 1.39697163311368880374967168263, 1.44480060487019275954392333239, 1.57163672524889472013323019674, 1.58395416122430930810296498854, 1.59975682930849567935026295555, 1.61241798454744726494844724035, 1.67178815842616349396668764136, 1.79081236798950943580421182870, 1.81647721257277098583872416829, 1.91033445301323909218349129050, 2.03259118962573640868960114684, 2.05200862934112303125822250809
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.