Dirichlet series
| L(s) = 1 | − 4·5-s − 4·9-s − 16·11-s + 4·13-s − 5·16-s + 12·17-s + 12·19-s − 4·23-s + 8·25-s + 8·27-s − 8·31-s + 32·37-s + 8·41-s + 16·45-s + 64·55-s + 20·59-s − 4·61-s − 16·65-s − 36·67-s − 28·71-s − 24·73-s − 64·79-s + 20·80-s + 16·81-s − 24·83-s − 48·85-s − 4·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 4/3·9-s − 4.82·11-s + 1.10·13-s − 5/4·16-s + 2.91·17-s + 2.75·19-s − 0.834·23-s + 8/5·25-s + 1.53·27-s − 1.43·31-s + 5.26·37-s + 1.24·41-s + 2.38·45-s + 8.62·55-s + 2.60·59-s − 0.512·61-s − 1.98·65-s − 4.39·67-s − 3.32·71-s − 2.80·73-s − 7.20·79-s + 2.23·80-s + 16/9·81-s − 2.63·83-s − 5.20·85-s − 0.423·89-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.15736\times 10^{12}\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.0005288871927\) |
| \(L(\frac12)\) | \(\approx\) | \(0.0005288871927\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{5} \) |
| 3 | \( 1 + 4 T^{2} - 8 T^{3} - 32 T^{5} + 14 T^{6} + 65 T^{8} + 88 T^{9} - 40 T^{10} + 88 p T^{11} + 65 p^{2} T^{12} + 14 p^{4} T^{14} - 32 p^{5} T^{15} - 8 p^{7} T^{17} + 4 p^{8} T^{18} + p^{10} T^{20} \) | |
| 7 | \( ( 1 + T^{4} )^{5} \) | |
| 13 | \( 1 - 4 T - 6 T^{2} + 124 T^{3} - 409 T^{4} - 680 T^{5} + 8604 T^{6} - 22536 T^{7} - 2384 T^{8} + 230120 T^{9} - 957612 T^{10} + 230120 p T^{11} - 2384 p^{2} T^{12} - 22536 p^{3} T^{13} + 8604 p^{4} T^{14} - 680 p^{5} T^{15} - 409 p^{6} T^{16} + 124 p^{7} T^{17} - 6 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| good | 5 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 51 T^{4} + 184 T^{5} + 456 T^{6} + 1116 T^{7} + 601 p T^{8} + 7736 T^{9} + 18432 T^{10} + 44728 T^{11} + 115112 T^{12} + 272488 T^{13} + 620544 T^{14} + 1466536 T^{15} + 3880418 T^{16} + 9240432 T^{17} + 19833072 T^{18} + 44316568 T^{19} + 97161386 T^{20} + 44316568 p T^{21} + 19833072 p^{2} T^{22} + 9240432 p^{3} T^{23} + 3880418 p^{4} T^{24} + 1466536 p^{5} T^{25} + 620544 p^{6} T^{26} + 272488 p^{7} T^{27} + 115112 p^{8} T^{28} + 44728 p^{9} T^{29} + 18432 p^{10} T^{30} + 7736 p^{11} T^{31} + 601 p^{13} T^{32} + 1116 p^{13} T^{33} + 456 p^{14} T^{34} + 184 p^{15} T^{35} + 51 p^{16} T^{36} + 16 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) |
| 11 | \( 1 + 16 T + 128 T^{2} + 768 T^{3} + 3945 T^{4} + 16992 T^{5} + 61824 T^{6} + 192816 T^{7} + 434432 T^{8} + 134352 T^{9} - 4906624 T^{10} - 35066848 T^{11} - 171458997 T^{12} - 649239840 T^{13} - 2002203264 T^{14} - 4918975376 T^{15} - 6227153949 T^{16} + 1812298560 p T^{17} + 184527052544 T^{18} + 906030317728 T^{19} + 3409593472000 T^{20} + 906030317728 p T^{21} + 184527052544 p^{2} T^{22} + 1812298560 p^{4} T^{23} - 6227153949 p^{4} T^{24} - 4918975376 p^{5} T^{25} - 2002203264 p^{6} T^{26} - 649239840 p^{7} T^{27} - 171458997 p^{8} T^{28} - 35066848 p^{9} T^{29} - 4906624 p^{10} T^{30} + 134352 p^{11} T^{31} + 434432 p^{12} T^{32} + 192816 p^{13} T^{33} + 61824 p^{14} T^{34} + 16992 p^{15} T^{35} + 3945 p^{16} T^{36} + 768 p^{17} T^{37} + 128 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( ( 1 - 6 T + 101 T^{2} - 486 T^{3} + 5245 T^{4} - 22048 T^{5} + 182440 T^{6} - 673696 T^{7} + 4637682 T^{8} - 15153844 T^{9} + 89906758 T^{10} - 15153844 p T^{11} + 4637682 p^{2} T^{12} - 673696 p^{3} T^{13} + 182440 p^{4} T^{14} - 22048 p^{5} T^{15} + 5245 p^{6} T^{16} - 486 p^{7} T^{17} + 101 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 - 12 T + 72 T^{2} - 320 T^{3} + 859 T^{4} - 368 T^{5} - 328 p T^{6} + 36940 T^{7} - 226435 T^{8} + 1817680 T^{9} - 11908640 T^{10} + 63694960 T^{11} - 233425816 T^{12} + 405518576 T^{13} + 389803040 T^{14} - 6471049008 T^{15} + 33098949186 T^{16} - 171893202840 T^{17} + 1119887657808 T^{18} - 6834168750640 T^{19} + 34492477003866 T^{20} - 6834168750640 p T^{21} + 1119887657808 p^{2} T^{22} - 171893202840 p^{3} T^{23} + 33098949186 p^{4} T^{24} - 6471049008 p^{5} T^{25} + 389803040 p^{6} T^{26} + 405518576 p^{7} T^{27} - 233425816 p^{8} T^{28} + 63694960 p^{9} T^{29} - 11908640 p^{10} T^{30} + 1817680 p^{11} T^{31} - 226435 p^{12} T^{32} + 36940 p^{13} T^{33} - 328 p^{15} T^{34} - 368 p^{15} T^{35} + 859 p^{16} T^{36} - 320 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( ( 1 + 2 T + 141 T^{2} + 338 T^{3} + 9748 T^{4} + 26670 T^{5} + 442675 T^{6} + 1283546 T^{7} + 14815791 T^{8} + 1818436 p T^{9} + 384355304 T^{10} + 1818436 p^{2} T^{11} + 14815791 p^{2} T^{12} + 1283546 p^{3} T^{13} + 442675 p^{4} T^{14} + 26670 p^{5} T^{15} + 9748 p^{6} T^{16} + 338 p^{7} T^{17} + 141 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( 1 - 142 T^{2} + 11155 T^{4} - 609454 T^{6} + 25006869 T^{8} - 778803704 T^{10} + 17632882676 T^{12} - 218473585624 T^{14} - 3218501698822 T^{16} + 301596955581788 T^{18} - 10937141613624014 T^{20} + 301596955581788 p^{2} T^{22} - 3218501698822 p^{4} T^{24} - 218473585624 p^{6} T^{26} + 17632882676 p^{8} T^{28} - 778803704 p^{10} T^{30} + 25006869 p^{12} T^{32} - 609454 p^{14} T^{34} + 11155 p^{16} T^{36} - 142 p^{18} T^{38} + p^{20} T^{40} \) | |
| 31 | \( 1 + 8 T + 32 T^{2} + 328 T^{3} + 4164 T^{4} + 17016 T^{5} + 56672 T^{6} + 518248 T^{7} + 4755782 T^{8} + 16788648 T^{9} + 60899424 T^{10} + 611766056 T^{11} + 7185067846 T^{12} + 31040212792 T^{13} + 115920047712 T^{14} + 1065338262552 T^{15} + 8755047728585 T^{16} + 27067701154896 T^{17} + 92459550623168 T^{18} + 764449640669888 T^{19} + 6324695263350028 T^{20} + 764449640669888 p T^{21} + 92459550623168 p^{2} T^{22} + 27067701154896 p^{3} T^{23} + 8755047728585 p^{4} T^{24} + 1065338262552 p^{5} T^{25} + 115920047712 p^{6} T^{26} + 31040212792 p^{7} T^{27} + 7185067846 p^{8} T^{28} + 611766056 p^{9} T^{29} + 60899424 p^{10} T^{30} + 16788648 p^{11} T^{31} + 4755782 p^{12} T^{32} + 518248 p^{13} T^{33} + 56672 p^{14} T^{34} + 17016 p^{15} T^{35} + 4164 p^{16} T^{36} + 328 p^{17} T^{37} + 32 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 32 T + 512 T^{2} - 4992 T^{3} + 28721 T^{4} - 56512 T^{5} - 436736 T^{6} + 2117856 T^{7} + 28438384 T^{8} - 422502816 T^{9} + 2527598080 T^{10} - 6633071552 T^{11} - 2695654285 T^{12} - 48502279616 T^{13} + 1988598904320 T^{14} - 19824178027168 T^{15} + 124909963886739 T^{16} - 634463992481664 T^{17} + 3085984968326144 T^{18} - 15903284307257920 T^{19} + 90891228701526496 T^{20} - 15903284307257920 p T^{21} + 3085984968326144 p^{2} T^{22} - 634463992481664 p^{3} T^{23} + 124909963886739 p^{4} T^{24} - 19824178027168 p^{5} T^{25} + 1988598904320 p^{6} T^{26} - 48502279616 p^{7} T^{27} - 2695654285 p^{8} T^{28} - 6633071552 p^{9} T^{29} + 2527598080 p^{10} T^{30} - 422502816 p^{11} T^{31} + 28438384 p^{12} T^{32} + 2117856 p^{13} T^{33} - 436736 p^{14} T^{34} - 56512 p^{15} T^{35} + 28721 p^{16} T^{36} - 4992 p^{17} T^{37} + 512 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 8 T + 32 T^{2} - 416 T^{3} + 2340 T^{4} + 21848 T^{5} - 163136 T^{6} + 1286520 T^{7} - 16214778 T^{8} + 82845456 T^{9} - 58677280 T^{10} - 323103080 T^{11} + 21672431014 T^{12} - 245748738712 T^{13} + 1536548187808 T^{14} - 8984298103920 T^{15} + 35838213321737 T^{16} + 90956584391856 T^{17} - 1501527840551776 T^{18} + 17347674543423816 T^{19} - 147321401600787188 T^{20} + 17347674543423816 p T^{21} - 1501527840551776 p^{2} T^{22} + 90956584391856 p^{3} T^{23} + 35838213321737 p^{4} T^{24} - 8984298103920 p^{5} T^{25} + 1536548187808 p^{6} T^{26} - 245748738712 p^{7} T^{27} + 21672431014 p^{8} T^{28} - 323103080 p^{9} T^{29} - 58677280 p^{10} T^{30} + 82845456 p^{11} T^{31} - 16214778 p^{12} T^{32} + 1286520 p^{13} T^{33} - 163136 p^{14} T^{34} + 21848 p^{15} T^{35} + 2340 p^{16} T^{36} - 416 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 614 T^{2} + 185339 T^{4} - 36590486 T^{6} + 5300972725 T^{8} - 599186423976 T^{10} + 54837086637268 T^{12} - 4161164302373160 T^{14} + 265899920030312506 T^{16} - 14449049663607798980 T^{18} + \)\(67\!\cdots\!10\)\( T^{20} - 14449049663607798980 p^{2} T^{22} + 265899920030312506 p^{4} T^{24} - 4161164302373160 p^{6} T^{26} + 54837086637268 p^{8} T^{28} - 599186423976 p^{10} T^{30} + 5300972725 p^{12} T^{32} - 36590486 p^{14} T^{34} + 185339 p^{16} T^{36} - 614 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 + 192 T^{3} - 2108 T^{4} - 11072 T^{5} + 18432 T^{6} - 483072 T^{7} - 1454874 T^{8} + 13611648 T^{9} + 7399424 T^{10} + 25595328 T^{11} - 14750492698 T^{12} + 5907635520 T^{13} + 13299638272 T^{14} - 2722915072896 T^{15} + 30785616002729 T^{16} + 121885125238016 T^{17} - 197252991643648 T^{18} + 1939746538107264 T^{19} + 38495174143478988 T^{20} + 1939746538107264 p T^{21} - 197252991643648 p^{2} T^{22} + 121885125238016 p^{3} T^{23} + 30785616002729 p^{4} T^{24} - 2722915072896 p^{5} T^{25} + 13299638272 p^{6} T^{26} + 5907635520 p^{7} T^{27} - 14750492698 p^{8} T^{28} + 25595328 p^{9} T^{29} + 7399424 p^{10} T^{30} + 13611648 p^{11} T^{31} - 1454874 p^{12} T^{32} - 483072 p^{13} T^{33} + 18432 p^{14} T^{34} - 11072 p^{15} T^{35} - 2108 p^{16} T^{36} + 192 p^{17} T^{37} + p^{20} T^{40} \) | |
| 53 | \( 1 - 532 T^{2} + 142902 T^{4} - 25796020 T^{6} + 3522129901 T^{8} - 388470223472 T^{10} + 36074945395112 T^{12} - 2897776025110128 T^{14} + 204789422003363906 T^{16} - 12858347212947144760 T^{18} + \)\(72\!\cdots\!40\)\( T^{20} - 12858347212947144760 p^{2} T^{22} + 204789422003363906 p^{4} T^{24} - 2897776025110128 p^{6} T^{26} + 36074945395112 p^{8} T^{28} - 388470223472 p^{10} T^{30} + 3522129901 p^{12} T^{32} - 25796020 p^{14} T^{34} + 142902 p^{16} T^{36} - 532 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( 1 - 20 T + 200 T^{2} - 1396 T^{3} + 14194 T^{4} - 182444 T^{5} + 1784488 T^{6} - 12396172 T^{7} + 72393329 T^{8} - 567168528 T^{9} + 5483644768 T^{10} - 37653132528 T^{11} + 59668033792 T^{12} + 1118176257872 T^{13} - 8444204575072 T^{14} + 35923019381296 T^{15} - 757763425480898 T^{16} + 13130959687375256 T^{17} - 121223966521886896 T^{18} + 728436408016060856 T^{19} - 4318904921944677476 T^{20} + 728436408016060856 p T^{21} - 121223966521886896 p^{2} T^{22} + 13130959687375256 p^{3} T^{23} - 757763425480898 p^{4} T^{24} + 35923019381296 p^{5} T^{25} - 8444204575072 p^{6} T^{26} + 1118176257872 p^{7} T^{27} + 59668033792 p^{8} T^{28} - 37653132528 p^{9} T^{29} + 5483644768 p^{10} T^{30} - 567168528 p^{11} T^{31} + 72393329 p^{12} T^{32} - 12396172 p^{13} T^{33} + 1784488 p^{14} T^{34} - 182444 p^{15} T^{35} + 14194 p^{16} T^{36} - 1396 p^{17} T^{37} + 200 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( ( 1 + 2 T + 333 T^{2} + 206 T^{3} + 55250 T^{4} - 29270 T^{5} + 6168109 T^{6} - 8166814 T^{7} + 520667899 T^{8} - 867784984 T^{9} + 35168612650 T^{10} - 867784984 p T^{11} + 520667899 p^{2} T^{12} - 8166814 p^{3} T^{13} + 6168109 p^{4} T^{14} - 29270 p^{5} T^{15} + 55250 p^{6} T^{16} + 206 p^{7} T^{17} + 333 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 36 T + 648 T^{2} + 7836 T^{3} + 80176 T^{4} + 844340 T^{5} + 9143640 T^{6} + 90019604 T^{7} + 777755694 T^{8} + 6194159748 T^{9} + 47752215320 T^{10} + 346895477484 T^{11} + 2149168377554 T^{12} + 8574997980788 T^{13} - 24046515379752 T^{14} - 987252407982708 T^{15} - 13396825804084015 T^{16} - 143809230866785800 T^{17} - 1443510152534391248 T^{18} - 13844992803433421376 T^{19} - \)\(12\!\cdots\!00\)\( T^{20} - 13844992803433421376 p T^{21} - 1443510152534391248 p^{2} T^{22} - 143809230866785800 p^{3} T^{23} - 13396825804084015 p^{4} T^{24} - 987252407982708 p^{5} T^{25} - 24046515379752 p^{6} T^{26} + 8574997980788 p^{7} T^{27} + 2149168377554 p^{8} T^{28} + 346895477484 p^{9} T^{29} + 47752215320 p^{10} T^{30} + 6194159748 p^{11} T^{31} + 777755694 p^{12} T^{32} + 90019604 p^{13} T^{33} + 9143640 p^{14} T^{34} + 844340 p^{15} T^{35} + 80176 p^{16} T^{36} + 7836 p^{17} T^{37} + 648 p^{18} T^{38} + 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 28 T + 392 T^{2} + 5308 T^{3} + 75046 T^{4} + 11612 p T^{5} + 7754056 T^{6} + 77484260 T^{7} + 683350253 T^{8} + 4691744432 T^{9} + 32731660576 T^{10} + 226546353712 T^{11} + 830563363976 T^{12} + 190433514896 T^{13} - 2833629395808 T^{14} - 137138954740720 T^{15} - 1474577711404686 T^{16} + 10882009358406664 T^{17} + 277039106139187952 T^{18} + 3000724489500509128 T^{19} + 27528905378062536100 T^{20} + 3000724489500509128 p T^{21} + 277039106139187952 p^{2} T^{22} + 10882009358406664 p^{3} T^{23} - 1474577711404686 p^{4} T^{24} - 137138954740720 p^{5} T^{25} - 2833629395808 p^{6} T^{26} + 190433514896 p^{7} T^{27} + 830563363976 p^{8} T^{28} + 226546353712 p^{9} T^{29} + 32731660576 p^{10} T^{30} + 4691744432 p^{11} T^{31} + 683350253 p^{12} T^{32} + 77484260 p^{13} T^{33} + 7754056 p^{14} T^{34} + 11612 p^{16} T^{35} + 75046 p^{16} T^{36} + 5308 p^{17} T^{37} + 392 p^{18} T^{38} + 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 24 T + 288 T^{2} + 3712 T^{3} + 48033 T^{4} + 487280 T^{5} + 4750688 T^{6} + 48087256 T^{7} + 448168640 T^{8} + 4244161816 T^{9} + 41818312032 T^{10} + 393072087536 T^{11} + 3790127967491 T^{12} + 36738117526832 T^{13} + 335760457588064 T^{14} + 3062977445485272 T^{15} + 27548853022306627 T^{16} + 234121290503577312 T^{17} + 1989622804343385280 T^{18} + 17038485758718985008 T^{19} + \)\(14\!\cdots\!12\)\( T^{20} + 17038485758718985008 p T^{21} + 1989622804343385280 p^{2} T^{22} + 234121290503577312 p^{3} T^{23} + 27548853022306627 p^{4} T^{24} + 3062977445485272 p^{5} T^{25} + 335760457588064 p^{6} T^{26} + 36738117526832 p^{7} T^{27} + 3790127967491 p^{8} T^{28} + 393072087536 p^{9} T^{29} + 41818312032 p^{10} T^{30} + 4244161816 p^{11} T^{31} + 448168640 p^{12} T^{32} + 48087256 p^{13} T^{33} + 4750688 p^{14} T^{34} + 487280 p^{15} T^{35} + 48033 p^{16} T^{36} + 3712 p^{17} T^{37} + 288 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( ( 1 + 32 T + 942 T^{2} + 17636 T^{3} + 309926 T^{4} + 4277856 T^{5} + 56767528 T^{6} + 636367560 T^{7} + 6977220269 T^{8} + 66766447900 T^{9} + 631832249076 T^{10} + 66766447900 p T^{11} + 6977220269 p^{2} T^{12} + 636367560 p^{3} T^{13} + 56767528 p^{4} T^{14} + 4277856 p^{5} T^{15} + 309926 p^{6} T^{16} + 17636 p^{7} T^{17} + 942 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 + 24 T + 288 T^{2} + 3848 T^{3} + 55694 T^{4} + 666904 T^{5} + 7369376 T^{6} + 84156456 T^{7} + 896666129 T^{8} + 9153341536 T^{9} + 97224840064 T^{10} + 973218096512 T^{11} + 9203605288960 T^{12} + 91555741754752 T^{13} + 914487247646336 T^{14} + 8606271371788768 T^{15} + 82912782795025854 T^{16} + 799643955888729712 T^{17} + 7348311005882756928 T^{18} + 68674702688343386480 T^{19} + \)\(64\!\cdots\!76\)\( T^{20} + 68674702688343386480 p T^{21} + 7348311005882756928 p^{2} T^{22} + 799643955888729712 p^{3} T^{23} + 82912782795025854 p^{4} T^{24} + 8606271371788768 p^{5} T^{25} + 914487247646336 p^{6} T^{26} + 91555741754752 p^{7} T^{27} + 9203605288960 p^{8} T^{28} + 973218096512 p^{9} T^{29} + 97224840064 p^{10} T^{30} + 9153341536 p^{11} T^{31} + 896666129 p^{12} T^{32} + 84156456 p^{13} T^{33} + 7369376 p^{14} T^{34} + 666904 p^{15} T^{35} + 55694 p^{16} T^{36} + 3848 p^{17} T^{37} + 288 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 4 T + 8 T^{2} - 900 T^{3} - 5082 T^{4} + 108284 T^{5} + 878792 T^{6} + 6613956 T^{7} - 66715283 T^{8} - 606309232 T^{9} + 2484658208 T^{10} + 20118316784 T^{11} + 372343264392 T^{12} - 1841321820496 T^{13} - 976010404448 T^{14} + 183031757127376 T^{15} - 2814184473905294 T^{16} - 41730315066515080 T^{17} - 358062644864379152 T^{18} + 313461260045854984 T^{19} + 42432684078688840612 T^{20} + 313461260045854984 p T^{21} - 358062644864379152 p^{2} T^{22} - 41730315066515080 p^{3} T^{23} - 2814184473905294 p^{4} T^{24} + 183031757127376 p^{5} T^{25} - 976010404448 p^{6} T^{26} - 1841321820496 p^{7} T^{27} + 372343264392 p^{8} T^{28} + 20118316784 p^{9} T^{29} + 2484658208 p^{10} T^{30} - 606309232 p^{11} T^{31} - 66715283 p^{12} T^{32} + 6613956 p^{13} T^{33} + 878792 p^{14} T^{34} + 108284 p^{15} T^{35} - 5082 p^{16} T^{36} - 900 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 32 T + 512 T^{2} - 6464 T^{3} + 76964 T^{4} - 878880 T^{5} + 9610240 T^{6} - 101251488 T^{7} + 1094074502 T^{8} - 12373831680 T^{9} + 136858603008 T^{10} - 1475912774944 T^{11} + 16881594839494 T^{12} - 195057756361824 T^{13} + 2092312857012736 T^{14} - 21028443398422528 T^{15} + 203054845466041993 T^{16} - 1974015332522436160 T^{17} + 19556395996742148608 T^{18} - \)\(19\!\cdots\!20\)\( T^{19} + \)\(18\!\cdots\!32\)\( T^{20} - \)\(19\!\cdots\!20\)\( p T^{21} + 19556395996742148608 p^{2} T^{22} - 1974015332522436160 p^{3} T^{23} + 203054845466041993 p^{4} T^{24} - 21028443398422528 p^{5} T^{25} + 2092312857012736 p^{6} T^{26} - 195057756361824 p^{7} T^{27} + 16881594839494 p^{8} T^{28} - 1475912774944 p^{9} T^{29} + 136858603008 p^{10} T^{30} - 12373831680 p^{11} T^{31} + 1094074502 p^{12} T^{32} - 101251488 p^{13} T^{33} + 9610240 p^{14} T^{34} - 878880 p^{15} T^{35} + 76964 p^{16} T^{36} - 6464 p^{17} T^{37} + 512 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.55845027347166772133016630204, −2.53699426612400211211341772119, −2.33245740099804142203600963811, −2.27592070043299868185912835367, −2.22643909446892652218646148661, −2.21201275067507488468171290073, −2.16210963548957773130978972429, −2.13009656458528767253806291517, −2.04720580251779214286725564249, −1.84014862623174512449303415469, −1.75783261071597281411913537212, −1.62244702656999144573228098075, −1.52546437375092169635505502676, −1.38271731331638374575734185294, −1.32279090456245854732054618603, −1.12153499312664704917105288152, −1.11973554140540108762840973935, −1.08057655745310155074739966356, −1.07406411149926278383208740382, −0.907192332753740321577904903345, −0.853258555836415953098221298151, −0.71401887210069700645622363376, −0.12103646265725557721954828456, −0.04668420959033085122899467663, −0.04198805029354520586007903800, 0.04198805029354520586007903800, 0.04668420959033085122899467663, 0.12103646265725557721954828456, 0.71401887210069700645622363376, 0.853258555836415953098221298151, 0.907192332753740321577904903345, 1.07406411149926278383208740382, 1.08057655745310155074739966356, 1.11973554140540108762840973935, 1.12153499312664704917105288152, 1.32279090456245854732054618603, 1.38271731331638374575734185294, 1.52546437375092169635505502676, 1.62244702656999144573228098075, 1.75783261071597281411913537212, 1.84014862623174512449303415469, 2.04720580251779214286725564249, 2.13009656458528767253806291517, 2.16210963548957773130978972429, 2.21201275067507488468171290073, 2.22643909446892652218646148661, 2.27592070043299868185912835367, 2.33245740099804142203600963811, 2.53699426612400211211341772119, 2.55845027347166772133016630204
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.