Dirichlet series
| L(s) = 1 | + 2·4-s − 4·8-s − 8·11-s − 3·16-s + 24·17-s − 4·19-s − 16·29-s + 16·37-s − 8·43-s − 16·44-s + 44·49-s + 16·53-s + 16·59-s − 4·61-s − 10·64-s − 8·67-s + 48·68-s − 8·76-s + 56·79-s + 48·83-s + 32·88-s + 56·97-s − 40·101-s − 32·107-s − 44·109-s + 40·113-s − 32·116-s + ⋯ |
| L(s) = 1 | + 4-s − 1.41·8-s − 2.41·11-s − 3/4·16-s + 5.82·17-s − 0.917·19-s − 2.97·29-s + 2.63·37-s − 1.21·43-s − 2.41·44-s + 44/7·49-s + 2.19·53-s + 2.08·59-s − 0.512·61-s − 5/4·64-s − 0.977·67-s + 5.82·68-s − 0.917·76-s + 6.30·79-s + 5.26·83-s + 3.41·88-s + 5.68·97-s − 3.98·101-s − 3.09·107-s − 4.21·109-s + 3.76·113-s − 2.97·116-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 3^{40} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.55665\times 10^{15}\) |
| Root analytic conductor: | \(2.39775\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 3^{40} \cdot 5^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(27.16679843\) |
| \(L(\frac12)\) | \(\approx\) | \(27.16679843\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T^{2} + p^{2} T^{3} + 7 T^{4} - p^{4} T^{5} + 3 p T^{6} + 9 p^{2} T^{7} - 21 p T^{8} - 5 p^{3} T^{9} + 17 p^{3} T^{10} - 5 p^{4} T^{11} - 21 p^{3} T^{12} + 9 p^{5} T^{13} + 3 p^{5} T^{14} - p^{9} T^{15} + 7 p^{6} T^{16} + p^{9} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 + T^{4} )^{5} \) | |
| good | 7 | \( 1 - 44 T^{2} + 1094 T^{4} - 19500 T^{6} + 5613 p^{2} T^{8} - 3248528 T^{10} + 33295592 T^{12} - 304383248 T^{14} + 2534940418 T^{16} - 19551215176 T^{18} + 141185912484 T^{20} - 19551215176 p^{2} T^{22} + 2534940418 p^{4} T^{24} - 304383248 p^{6} T^{26} + 33295592 p^{8} T^{28} - 3248528 p^{10} T^{30} + 5613 p^{14} T^{32} - 19500 p^{14} T^{34} + 1094 p^{16} T^{36} - 44 p^{18} T^{38} + p^{20} T^{40} \) |
| 11 | \( 1 + 8 T + 32 T^{2} + 200 T^{3} + 554 T^{4} - 1000 T^{5} - 5728 T^{6} - 51944 T^{7} - 327075 T^{8} - 617248 T^{9} - 1187072 T^{10} - 2571808 T^{11} + 38679288 T^{12} + 179987360 T^{13} + 438289920 T^{14} + 2262964128 T^{15} + 4544695506 T^{16} - 9223438160 T^{17} - 264826304 p^{2} T^{18} - 20259866224 p T^{19} - 1355202715844 T^{20} - 20259866224 p^{2} T^{21} - 264826304 p^{4} T^{22} - 9223438160 p^{3} T^{23} + 4544695506 p^{4} T^{24} + 2262964128 p^{5} T^{25} + 438289920 p^{6} T^{26} + 179987360 p^{7} T^{27} + 38679288 p^{8} T^{28} - 2571808 p^{9} T^{29} - 1187072 p^{10} T^{30} - 617248 p^{11} T^{31} - 327075 p^{12} T^{32} - 51944 p^{13} T^{33} - 5728 p^{14} T^{34} - 1000 p^{15} T^{35} + 554 p^{16} T^{36} + 200 p^{17} T^{37} + 32 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 - 144 T^{3} - 590 T^{4} + 1424 T^{5} + 10368 T^{6} + 83488 T^{7} + 1665 p T^{8} - 1321888 T^{9} - 4891264 T^{10} - 17160736 T^{11} + 70894648 T^{12} + 437611168 T^{13} + 963639424 T^{14} + 521050016 T^{15} - 27855377150 T^{16} - 79983343392 T^{17} - 71511524992 T^{18} + 576265858496 T^{19} + 5962938206380 T^{20} + 576265858496 p T^{21} - 71511524992 p^{2} T^{22} - 79983343392 p^{3} T^{23} - 27855377150 p^{4} T^{24} + 521050016 p^{5} T^{25} + 963639424 p^{6} T^{26} + 437611168 p^{7} T^{27} + 70894648 p^{8} T^{28} - 17160736 p^{9} T^{29} - 4891264 p^{10} T^{30} - 1321888 p^{11} T^{31} + 1665 p^{13} T^{32} + 83488 p^{13} T^{33} + 10368 p^{14} T^{34} + 1424 p^{15} T^{35} - 590 p^{16} T^{36} - 144 p^{17} T^{37} + p^{20} T^{40} \) | |
| 17 | \( ( 1 - 12 T + 8 p T^{2} - 1044 T^{3} + 7705 T^{4} - 46424 T^{5} + 15968 p T^{6} - 1384552 T^{7} + 6875494 T^{8} - 30564448 T^{9} + 132676528 T^{10} - 30564448 p T^{11} + 6875494 p^{2} T^{12} - 1384552 p^{3} T^{13} + 15968 p^{5} T^{14} - 46424 p^{5} T^{15} + 7705 p^{6} T^{16} - 1044 p^{7} T^{17} + 8 p^{9} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 + 4 T + 8 T^{2} + 4 T^{3} - 498 T^{4} - 2836 T^{5} - 7352 T^{6} - 44404 T^{7} - 208307 T^{8} + 45008 T^{9} + 2029024 T^{10} + 24485136 T^{11} + 194865480 T^{12} + 421456624 T^{13} + 562039008 T^{14} - 668922064 T^{15} - 10989456190 T^{16} - 35054077064 T^{17} - 201246114256 T^{18} - 2455736945736 T^{19} - 16911023898412 T^{20} - 2455736945736 p T^{21} - 201246114256 p^{2} T^{22} - 35054077064 p^{3} T^{23} - 10989456190 p^{4} T^{24} - 668922064 p^{5} T^{25} + 562039008 p^{6} T^{26} + 421456624 p^{7} T^{27} + 194865480 p^{8} T^{28} + 24485136 p^{9} T^{29} + 2029024 p^{10} T^{30} + 45008 p^{11} T^{31} - 208307 p^{12} T^{32} - 44404 p^{13} T^{33} - 7352 p^{14} T^{34} - 2836 p^{15} T^{35} - 498 p^{16} T^{36} + 4 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 112 T^{2} + 6418 T^{4} - 268432 T^{6} + 9866397 T^{8} - 339283616 T^{10} + 10672587960 T^{12} - 305013893728 T^{14} + 8125187770498 T^{16} - 205354952908032 T^{18} + 4888712773662060 T^{20} - 205354952908032 p^{2} T^{22} + 8125187770498 p^{4} T^{24} - 305013893728 p^{6} T^{26} + 10672587960 p^{8} T^{28} - 339283616 p^{10} T^{30} + 9866397 p^{12} T^{32} - 268432 p^{14} T^{34} + 6418 p^{16} T^{36} - 112 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 8730 T^{4} + 57168 T^{5} + 338048 T^{6} + 2005456 T^{7} + 11372221 T^{8} + 62917440 T^{9} + 319640064 T^{10} + 1428493120 T^{11} + 5593603768 T^{12} + 13988519744 T^{13} - 33555351040 T^{14} - 921633865408 T^{15} - 8608796954606 T^{16} - 56965068181280 T^{17} - 357222177718528 T^{18} - 2152325935683360 T^{19} - 11865126440588068 T^{20} - 2152325935683360 p T^{21} - 357222177718528 p^{2} T^{22} - 56965068181280 p^{3} T^{23} - 8608796954606 p^{4} T^{24} - 921633865408 p^{5} T^{25} - 33555351040 p^{6} T^{26} + 13988519744 p^{7} T^{27} + 5593603768 p^{8} T^{28} + 1428493120 p^{9} T^{29} + 319640064 p^{10} T^{30} + 62917440 p^{11} T^{31} + 11372221 p^{12} T^{32} + 2005456 p^{13} T^{33} + 338048 p^{14} T^{34} + 57168 p^{15} T^{35} + 8730 p^{16} T^{36} + 1040 p^{17} T^{37} + 128 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( ( 1 + 106 T^{2} + 168 T^{3} + 7585 T^{4} + 16920 T^{5} + 394928 T^{6} + 983384 T^{7} + 16954718 T^{8} + 40225768 T^{9} + 576843852 T^{10} + 40225768 p T^{11} + 16954718 p^{2} T^{12} + 983384 p^{3} T^{13} + 394928 p^{4} T^{14} + 16920 p^{5} T^{15} + 7585 p^{6} T^{16} + 168 p^{7} T^{17} + 106 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 16 T + 128 T^{2} - 672 T^{3} + 6162 T^{4} - 70784 T^{5} + 569600 T^{6} - 3413488 T^{7} + 19861869 T^{8} - 123126816 T^{9} + 716885888 T^{10} - 4034578080 T^{11} + 21526450616 T^{12} - 48238904928 T^{13} - 461739447680 T^{14} + 6000672782496 T^{15} - 40743757680254 T^{16} + 299639706763712 T^{17} - 2499774907942784 T^{18} + 18995663482828448 T^{19} - 123848268007123988 T^{20} + 18995663482828448 p T^{21} - 2499774907942784 p^{2} T^{22} + 299639706763712 p^{3} T^{23} - 40743757680254 p^{4} T^{24} + 6000672782496 p^{5} T^{25} - 461739447680 p^{6} T^{26} - 48238904928 p^{7} T^{27} + 21526450616 p^{8} T^{28} - 4034578080 p^{9} T^{29} + 716885888 p^{10} T^{30} - 123126816 p^{11} T^{31} + 19861869 p^{12} T^{32} - 3413488 p^{13} T^{33} + 569600 p^{14} T^{34} - 70784 p^{15} T^{35} + 6162 p^{16} T^{36} - 672 p^{17} T^{37} + 128 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 476 T^{2} + 114110 T^{4} - 18274748 T^{6} + 2190915981 T^{8} - 209008861744 T^{10} + 16471748141672 T^{12} - 1099136114820592 T^{14} + 63139453400466834 T^{16} - 3156343211287957256 T^{18} + \)\(13\!\cdots\!40\)\( T^{20} - 3156343211287957256 p^{2} T^{22} + 63139453400466834 p^{4} T^{24} - 1099136114820592 p^{6} T^{26} + 16471748141672 p^{8} T^{28} - 209008861744 p^{10} T^{30} + 2190915981 p^{12} T^{32} - 18274748 p^{14} T^{34} + 114110 p^{16} T^{36} - 476 p^{18} T^{38} + p^{20} T^{40} \) | |
| 43 | \( 1 + 8 T + 32 T^{2} + 1048 T^{3} + 9178 T^{4} + 8776 T^{5} + 325664 T^{6} + 4217688 T^{7} - 8931811 T^{8} - 89066208 T^{9} + 1042990208 T^{10} - 5701991584 T^{11} - 104515354952 T^{12} + 162647516832 T^{13} + 118990472832 T^{14} - 36795608260128 T^{15} + 27913512242834 T^{16} + 1218511170422768 T^{17} - 6101787245330496 T^{18} - 6117903329494064 T^{19} + 531886492206818396 T^{20} - 6117903329494064 p T^{21} - 6101787245330496 p^{2} T^{22} + 1218511170422768 p^{3} T^{23} + 27913512242834 p^{4} T^{24} - 36795608260128 p^{5} T^{25} + 118990472832 p^{6} T^{26} + 162647516832 p^{7} T^{27} - 104515354952 p^{8} T^{28} - 5701991584 p^{9} T^{29} + 1042990208 p^{10} T^{30} - 89066208 p^{11} T^{31} - 8931811 p^{12} T^{32} + 4217688 p^{13} T^{33} + 325664 p^{14} T^{34} + 8776 p^{15} T^{35} + 9178 p^{16} T^{36} + 1048 p^{17} T^{37} + 32 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( ( 1 + 304 T^{2} - 160 T^{3} + 44969 T^{4} - 41120 T^{5} + 4344240 T^{6} - 4750112 T^{7} + 305975190 T^{8} - 332022304 T^{9} + 16398279936 T^{10} - 332022304 p T^{11} + 305975190 p^{2} T^{12} - 4750112 p^{3} T^{13} + 4344240 p^{4} T^{14} - 41120 p^{5} T^{15} + 44969 p^{6} T^{16} - 160 p^{7} T^{17} + 304 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 - 16 T + 128 T^{2} - 1360 T^{3} + 7802 T^{4} + 22000 T^{5} - 425856 T^{6} + 6433968 T^{7} - 83599203 T^{8} + 473355712 T^{9} - 2058661376 T^{10} + 8722880704 T^{11} + 145119466552 T^{12} - 2040942293824 T^{13} + 13788918864384 T^{14} - 124310453507136 T^{15} + 630832808996754 T^{16} + 876994252593184 T^{17} - 20345070641070336 T^{18} + 297723719191012000 T^{19} - 3358726545562167268 T^{20} + 297723719191012000 p T^{21} - 20345070641070336 p^{2} T^{22} + 876994252593184 p^{3} T^{23} + 630832808996754 p^{4} T^{24} - 124310453507136 p^{5} T^{25} + 13788918864384 p^{6} T^{26} - 2040942293824 p^{7} T^{27} + 145119466552 p^{8} T^{28} + 8722880704 p^{9} T^{29} - 2058661376 p^{10} T^{30} + 473355712 p^{11} T^{31} - 83599203 p^{12} T^{32} + 6433968 p^{13} T^{33} - 425856 p^{14} T^{34} + 22000 p^{15} T^{35} + 7802 p^{16} T^{36} - 1360 p^{17} T^{37} + 128 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 16 T + 128 T^{2} - 832 T^{3} - 3830 T^{4} + 112768 T^{5} - 967936 T^{6} + 6092944 T^{7} + 19688605 T^{8} - 531165952 T^{9} + 3560300416 T^{10} - 9919139776 T^{11} - 261052956296 T^{12} + 2908838413888 T^{13} - 13517190541696 T^{14} - 125929737088 p T^{15} + 1333257994894034 T^{16} - 9932172677825952 T^{17} + 28544397136988800 T^{18} + 231913105379115712 T^{19} - 4508030391091702148 T^{20} + 231913105379115712 p T^{21} + 28544397136988800 p^{2} T^{22} - 9932172677825952 p^{3} T^{23} + 1333257994894034 p^{4} T^{24} - 125929737088 p^{6} T^{25} - 13517190541696 p^{6} T^{26} + 2908838413888 p^{7} T^{27} - 261052956296 p^{8} T^{28} - 9919139776 p^{9} T^{29} + 3560300416 p^{10} T^{30} - 531165952 p^{11} T^{31} + 19688605 p^{12} T^{32} + 6092944 p^{13} T^{33} - 967936 p^{14} T^{34} + 112768 p^{15} T^{35} - 3830 p^{16} T^{36} - 832 p^{17} T^{37} + 128 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 4 T + 8 T^{2} + 1084 T^{3} + 16342 T^{4} - 2900 T^{5} + 445192 T^{6} + 14185012 T^{7} + 77733053 T^{8} - 309073008 T^{9} + 6247273888 T^{10} + 81831472240 T^{11} + 96905034120 T^{12} - 1462773185296 T^{13} + 46883494128416 T^{14} + 291053402071440 T^{15} - 616912164726638 T^{16} + 18752754555896 T^{17} + 238366637020041584 T^{18} + 599779334934785160 T^{19} - 4018461695447802172 T^{20} + 599779334934785160 p T^{21} + 238366637020041584 p^{2} T^{22} + 18752754555896 p^{3} T^{23} - 616912164726638 p^{4} T^{24} + 291053402071440 p^{5} T^{25} + 46883494128416 p^{6} T^{26} - 1462773185296 p^{7} T^{27} + 96905034120 p^{8} T^{28} + 81831472240 p^{9} T^{29} + 6247273888 p^{10} T^{30} - 309073008 p^{11} T^{31} + 77733053 p^{12} T^{32} + 14185012 p^{13} T^{33} + 445192 p^{14} T^{34} - 2900 p^{15} T^{35} + 16342 p^{16} T^{36} + 1084 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 8 T + 32 T^{2} - 424 T^{3} - 838 T^{4} - 22200 T^{5} - 60896 T^{6} - 4075112 T^{7} - 28419011 T^{8} - 229446880 T^{9} + 997069952 T^{10} + 6834249312 T^{11} + 95953352760 T^{12} + 695664143520 T^{13} + 12099014319744 T^{14} + 101449895918816 T^{15} + 321671739837202 T^{16} - 3133814523337232 T^{17} - 51032016425390144 T^{18} - 247805543009044144 T^{19} - 1680983801129963364 T^{20} - 247805543009044144 p T^{21} - 51032016425390144 p^{2} T^{22} - 3133814523337232 p^{3} T^{23} + 321671739837202 p^{4} T^{24} + 101449895918816 p^{5} T^{25} + 12099014319744 p^{6} T^{26} + 695664143520 p^{7} T^{27} + 95953352760 p^{8} T^{28} + 6834249312 p^{9} T^{29} + 997069952 p^{10} T^{30} - 229446880 p^{11} T^{31} - 28419011 p^{12} T^{32} - 4075112 p^{13} T^{33} - 60896 p^{14} T^{34} - 22200 p^{15} T^{35} - 838 p^{16} T^{36} - 424 p^{17} T^{37} + 32 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 732 T^{2} + 262846 T^{4} - 62004284 T^{6} + 10869894349 T^{8} - 1519546281008 T^{10} + 177450055097448 T^{12} - 17897817661276656 T^{14} + 1599396438554740882 T^{16} - \)\(12\!\cdots\!96\)\( T^{18} + \)\(95\!\cdots\!44\)\( T^{20} - \)\(12\!\cdots\!96\)\( p^{2} T^{22} + 1599396438554740882 p^{4} T^{24} - 17897817661276656 p^{6} T^{26} + 177450055097448 p^{8} T^{28} - 1519546281008 p^{10} T^{30} + 10869894349 p^{12} T^{32} - 62004284 p^{14} T^{34} + 262846 p^{16} T^{36} - 732 p^{18} T^{38} + p^{20} T^{40} \) | |
| 73 | \( 1 - 572 T^{2} + 177950 T^{4} - 39374364 T^{6} + 6851292685 T^{8} - 988314282928 T^{10} + 122021228586728 T^{12} - 13165992559394160 T^{14} + 1259088031400101266 T^{16} - \)\(10\!\cdots\!48\)\( T^{18} + \)\(82\!\cdots\!68\)\( T^{20} - \)\(10\!\cdots\!48\)\( p^{2} T^{22} + 1259088031400101266 p^{4} T^{24} - 13165992559394160 p^{6} T^{26} + 122021228586728 p^{8} T^{28} - 988314282928 p^{10} T^{30} + 6851292685 p^{12} T^{32} - 39374364 p^{14} T^{34} + 177950 p^{16} T^{36} - 572 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 28 T + 686 T^{2} - 10556 T^{3} + 144753 T^{4} - 1385704 T^{5} + 11031760 T^{6} - 35498472 T^{7} - 257006930 T^{8} + 8134369440 T^{9} - 82806904444 T^{10} + 8134369440 p T^{11} - 257006930 p^{2} T^{12} - 35498472 p^{3} T^{13} + 11031760 p^{4} T^{14} - 1385704 p^{5} T^{15} + 144753 p^{6} T^{16} - 10556 p^{7} T^{17} + 686 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 - 48 T + 1152 T^{2} - 240 p T^{3} + 278202 T^{4} - 3128304 T^{5} + 28073088 T^{6} - 189233296 T^{7} + 617163773 T^{8} + 5269732160 T^{9} - 111238823424 T^{10} + 1032238880960 T^{11} - 3509898117576 T^{12} - 56130753000128 T^{13} + 1215946473548288 T^{14} - 13758625118911296 T^{15} + 95573734332580882 T^{16} - 1394119349449696 p T^{17} - 7422316285142264064 T^{18} + \)\(13\!\cdots\!64\)\( T^{19} - \)\(14\!\cdots\!44\)\( T^{20} + \)\(13\!\cdots\!64\)\( p T^{21} - 7422316285142264064 p^{2} T^{22} - 1394119349449696 p^{4} T^{23} + 95573734332580882 p^{4} T^{24} - 13758625118911296 p^{5} T^{25} + 1215946473548288 p^{6} T^{26} - 56130753000128 p^{7} T^{27} - 3509898117576 p^{8} T^{28} + 1032238880960 p^{9} T^{29} - 111238823424 p^{10} T^{30} + 5269732160 p^{11} T^{31} + 617163773 p^{12} T^{32} - 189233296 p^{13} T^{33} + 28073088 p^{14} T^{34} - 3128304 p^{15} T^{35} + 278202 p^{16} T^{36} - 240 p^{18} T^{37} + 1152 p^{18} T^{38} - 48 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 684 T^{2} + 249630 T^{4} - 64152460 T^{6} + 12994670157 T^{8} - 2197503836272 T^{10} + 320683254270696 T^{12} - 41222336702768944 T^{14} + 4731155078141965714 T^{16} - \)\(48\!\cdots\!68\)\( T^{18} + \)\(45\!\cdots\!84\)\( T^{20} - \)\(48\!\cdots\!68\)\( p^{2} T^{22} + 4731155078141965714 p^{4} T^{24} - 41222336702768944 p^{6} T^{26} + 320683254270696 p^{8} T^{28} - 2197503836272 p^{10} T^{30} + 12994670157 p^{12} T^{32} - 64152460 p^{14} T^{34} + 249630 p^{16} T^{36} - 684 p^{18} T^{38} + p^{20} T^{40} \) | |
| 97 | \( ( 1 - 28 T + 926 T^{2} - 18364 T^{3} + 366781 T^{4} - 5748592 T^{5} + 87017992 T^{6} - 1129849648 T^{7} + 13959653890 T^{8} - 153268750856 T^{9} + 16426299188 p T^{10} - 153268750856 p T^{11} + 13959653890 p^{2} T^{12} - 1129849648 p^{3} T^{13} + 87017992 p^{4} T^{14} - 5748592 p^{5} T^{15} + 366781 p^{6} T^{16} - 18364 p^{7} T^{17} + 926 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 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
−2.37284885772500456606795308453, −2.32334202825989837201461102126, −2.30050106721583968809051761280, −2.18889045957673829820896266224, −2.12448734105544034120426337444, −2.11975890457177564839479082262, −2.06391524688856767100796786526, −2.00507166374953494753721010873, −1.91993732789269722009632674654, −1.71367045861928441117873627186, −1.59615042558229627364336348413, −1.56079164239271494737924116737, −1.40838470193206943029200852829, −1.26652640414397549154034285868, −1.21481106841268343624948487505, −1.20578843682141625512310803716, −1.20242830088662974497012369484, −0.871422843744498332303595097428, −0.867138565895150028694800821094, −0.76910358289905036605869069879, −0.74740466293142742329523861445, −0.67413938707029312910982257992, −0.45876338124953100182592005182, −0.29713453782763036588827828406, −0.25508049756672141514650214960, 0.25508049756672141514650214960, 0.29713453782763036588827828406, 0.45876338124953100182592005182, 0.67413938707029312910982257992, 0.74740466293142742329523861445, 0.76910358289905036605869069879, 0.867138565895150028694800821094, 0.871422843744498332303595097428, 1.20242830088662974497012369484, 1.20578843682141625512310803716, 1.21481106841268343624948487505, 1.26652640414397549154034285868, 1.40838470193206943029200852829, 1.56079164239271494737924116737, 1.59615042558229627364336348413, 1.71367045861928441117873627186, 1.91993732789269722009632674654, 2.00507166374953494753721010873, 2.06391524688856767100796786526, 2.11975890457177564839479082262, 2.12448734105544034120426337444, 2.18889045957673829820896266224, 2.30050106721583968809051761280, 2.32334202825989837201461102126, 2.37284885772500456606795308453
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.