Dirichlet series
| L(s) = 1 | + 2·4-s + 12·11-s + 2·16-s + 8·19-s + 12·29-s + 12·37-s + 4·43-s + 24·44-s − 10·49-s − 36·53-s + 8·61-s − 4·64-s − 12·67-s + 16·76-s + 24·79-s − 5·81-s + 40·83-s − 72·97-s + 56·101-s − 60·107-s + 4·109-s − 16·113-s + 24·116-s + 72·121-s + 24·125-s + 127-s + 8·128-s + ⋯ |
| L(s) = 1 | + 4-s + 3.61·11-s + 1/2·16-s + 1.83·19-s + 2.22·29-s + 1.97·37-s + 0.609·43-s + 3.61·44-s − 1.42·49-s − 4.94·53-s + 1.02·61-s − 1/2·64-s − 1.46·67-s + 1.83·76-s + 2.70·79-s − 5/9·81-s + 4.39·83-s − 7.31·97-s + 5.57·101-s − 5.80·107-s + 0.383·109-s − 1.50·113-s + 2.22·116-s + 6.54·121-s + 2.14·125-s + 0.0887·127-s + 0.707·128-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 3^{20} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.73531\times 10^{8}\) |
| Root analytic conductor: | \(1.63797\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 3^{20} \cdot 7^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(21.28855233\) |
| \(L(\frac12)\) | \(\approx\) | \(21.28855233\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T^{2} + p T^{4} + p^{2} T^{6} - p^{3} T^{7} - p^{2} T^{8} - p^{4} T^{12} - p^{6} T^{13} + p^{6} T^{14} + p^{7} T^{16} - p^{9} T^{18} + p^{10} T^{20} \) |
| 3 | \( ( 1 + T^{4} )^{5} \) | |
| 7 | \( ( 1 + T^{2} )^{10} \) | |
| good | 5 | \( 1 - 24 T^{3} + 2 p T^{4} + 24 T^{5} + 288 T^{6} - 64 p T^{7} + 97 p T^{8} - 544 p T^{9} + 5088 T^{10} - 20208 T^{11} + 28392 T^{12} - 21904 T^{13} + 280352 T^{14} - 74912 p T^{15} + 85402 T^{16} - 2818304 T^{17} + 5181408 T^{18} - 7099744 T^{19} + 30439356 T^{20} - 7099744 p T^{21} + 5181408 p^{2} T^{22} - 2818304 p^{3} T^{23} + 85402 p^{4} T^{24} - 74912 p^{6} T^{25} + 280352 p^{6} T^{26} - 21904 p^{7} T^{27} + 28392 p^{8} T^{28} - 20208 p^{9} T^{29} + 5088 p^{10} T^{30} - 544 p^{12} T^{31} + 97 p^{13} T^{32} - 64 p^{14} T^{33} + 288 p^{14} T^{34} + 24 p^{15} T^{35} + 2 p^{17} T^{36} - 24 p^{17} T^{37} + p^{20} T^{40} \) |
| 11 | \( 1 - 12 T + 72 T^{2} - 404 T^{3} + 2214 T^{4} - 9404 T^{5} + 35048 T^{6} - 135156 T^{7} + 41519 p T^{8} - 1318720 T^{9} + 4165952 T^{10} - 13629648 T^{11} + 41111256 T^{12} - 141076784 T^{13} + 543949056 T^{14} - 1968344544 T^{15} + 6732047034 T^{16} - 23311377560 T^{17} + 80768058832 T^{18} - 261514302488 T^{19} + 834968255588 T^{20} - 261514302488 p T^{21} + 80768058832 p^{2} T^{22} - 23311377560 p^{3} T^{23} + 6732047034 p^{4} T^{24} - 1968344544 p^{5} T^{25} + 543949056 p^{6} T^{26} - 141076784 p^{7} T^{27} + 41111256 p^{8} T^{28} - 13629648 p^{9} T^{29} + 4165952 p^{10} T^{30} - 1318720 p^{11} T^{31} + 41519 p^{13} T^{32} - 135156 p^{13} T^{33} + 35048 p^{14} T^{34} - 9404 p^{15} T^{35} + 2214 p^{16} T^{36} - 404 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 - 80 T^{3} - 118 T^{4} + 1616 T^{5} + 3200 T^{6} + 22560 T^{7} - 156451 T^{8} - 297120 T^{9} - 9344 p T^{10} + 6082400 T^{11} + 43092280 T^{12} - 92564192 T^{13} - 225951616 T^{14} - 2287891296 T^{15} + 291408890 p T^{16} + 30959844064 T^{17} + 31244658048 T^{18} + 7310121920 T^{19} - 2235843760900 T^{20} + 7310121920 p T^{21} + 31244658048 p^{2} T^{22} + 30959844064 p^{3} T^{23} + 291408890 p^{5} T^{24} - 2287891296 p^{5} T^{25} - 225951616 p^{6} T^{26} - 92564192 p^{7} T^{27} + 43092280 p^{8} T^{28} + 6082400 p^{9} T^{29} - 9344 p^{11} T^{30} - 297120 p^{11} T^{31} - 156451 p^{12} T^{32} + 22560 p^{13} T^{33} + 3200 p^{14} T^{34} + 1616 p^{15} T^{35} - 118 p^{16} T^{36} - 80 p^{17} T^{37} + p^{20} T^{40} \) | |
| 17 | \( ( 1 + 94 T^{2} + 88 T^{3} + 4293 T^{4} + 6432 T^{5} + 134312 T^{6} + 219984 T^{7} + 3234206 T^{8} + 4965576 T^{9} + 61650188 T^{10} + 4965576 p T^{11} + 3234206 p^{2} T^{12} + 219984 p^{3} T^{13} + 134312 p^{4} T^{14} + 6432 p^{5} T^{15} + 4293 p^{6} T^{16} + 88 p^{7} T^{17} + 94 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 170 T^{4} + 664 T^{5} - 3552 T^{6} + 30376 T^{7} - 137251 T^{8} + 141472 T^{9} + 439424 T^{10} - 1977376 T^{11} + 92609976 T^{12} - 734762848 T^{13} + 3144939136 T^{14} - 11850068256 T^{15} + 18020441650 T^{16} + 30523407824 T^{17} - 82150980672 T^{18} + 1163681013680 T^{19} - 8113217288772 T^{20} + 1163681013680 p T^{21} - 82150980672 p^{2} T^{22} + 30523407824 p^{3} T^{23} + 18020441650 p^{4} T^{24} - 11850068256 p^{5} T^{25} + 3144939136 p^{6} T^{26} - 734762848 p^{7} T^{27} + 92609976 p^{8} T^{28} - 1977376 p^{9} T^{29} + 439424 p^{10} T^{30} + 141472 p^{11} T^{31} - 137251 p^{12} T^{32} + 30376 p^{13} T^{33} - 3552 p^{14} T^{34} + 664 p^{15} T^{35} + 170 p^{16} T^{36} - 120 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 228 T^{2} + 24654 T^{4} - 1671172 T^{6} + 79227605 T^{8} - 2786970288 T^{10} + 76106467832 T^{12} - 1708777858416 T^{14} + 34629617977562 T^{16} - 710121266972024 T^{18} + 15689370725592020 T^{20} - 710121266972024 p^{2} T^{22} + 34629617977562 p^{4} T^{24} - 1708777858416 p^{6} T^{26} + 76106467832 p^{8} T^{28} - 2786970288 p^{10} T^{30} + 79227605 p^{12} T^{32} - 1671172 p^{14} T^{34} + 24654 p^{16} T^{36} - 228 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5062 T^{4} - 19812 T^{5} + 91080 T^{6} - 552988 T^{7} + 898845 T^{8} + 3230928 T^{9} - 3305184 T^{10} + 180759600 T^{11} - 2401479032 T^{12} + 11590609840 T^{13} - 61731406432 T^{14} + 542524821712 T^{15} - 2180038368078 T^{16} + 2872412612888 T^{17} - 25762237441424 T^{18} + 2546561760168 T^{19} + 1284480158865124 T^{20} + 2546561760168 p T^{21} - 25762237441424 p^{2} T^{22} + 2872412612888 p^{3} T^{23} - 2180038368078 p^{4} T^{24} + 542524821712 p^{5} T^{25} - 61731406432 p^{6} T^{26} + 11590609840 p^{7} T^{27} - 2401479032 p^{8} T^{28} + 180759600 p^{9} T^{29} - 3305184 p^{10} T^{30} + 3230928 p^{11} T^{31} + 898845 p^{12} T^{32} - 552988 p^{13} T^{33} + 91080 p^{14} T^{34} - 19812 p^{15} T^{35} + 5062 p^{16} T^{36} - 660 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( ( 1 + 134 T^{2} - 64 T^{3} + 7637 T^{4} + 64 T^{5} + 250776 T^{6} + 647232 T^{7} + 5787898 T^{8} + 46731200 T^{9} + 141742500 T^{10} + 46731200 p T^{11} + 5787898 p^{2} T^{12} + 647232 p^{3} T^{13} + 250776 p^{4} T^{14} + 64 p^{5} T^{15} + 7637 p^{6} T^{16} - 64 p^{7} T^{17} + 134 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 37 | \( 1 - 12 T + 72 T^{2} - 692 T^{3} + 4646 T^{4} - 25508 T^{5} + 211016 T^{6} - 1455420 T^{7} + 12729021 T^{8} - 102763568 T^{9} + 668772640 T^{10} - 132634896 p T^{11} + 30685175816 T^{12} - 181058288208 T^{13} + 1246622676896 T^{14} - 7968453274800 T^{15} + 52782260071730 T^{16} - 346974533424104 T^{17} + 2138919388551792 T^{18} - 13427579452955416 T^{19} + 82729716000045604 T^{20} - 13427579452955416 p T^{21} + 2138919388551792 p^{2} T^{22} - 346974533424104 p^{3} T^{23} + 52782260071730 p^{4} T^{24} - 7968453274800 p^{5} T^{25} + 1246622676896 p^{6} T^{26} - 181058288208 p^{7} T^{27} + 30685175816 p^{8} T^{28} - 132634896 p^{10} T^{29} + 668772640 p^{10} T^{30} - 102763568 p^{11} T^{31} + 12729021 p^{12} T^{32} - 1455420 p^{13} T^{33} + 211016 p^{14} T^{34} - 25508 p^{15} T^{35} + 4646 p^{16} T^{36} - 692 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 444 T^{2} + 98574 T^{4} - 14512124 T^{6} + 1588488149 T^{8} - 3357926832 p T^{10} + 9846636632440 T^{12} - 599501624736304 T^{14} + 31880388630283994 T^{16} - 1512585137889846120 T^{18} + 64990434418739739988 T^{20} - 1512585137889846120 p^{2} T^{22} + 31880388630283994 p^{4} T^{24} - 599501624736304 p^{6} T^{26} + 9846636632440 p^{8} T^{28} - 3357926832 p^{11} T^{30} + 1588488149 p^{12} T^{32} - 14512124 p^{14} T^{34} + 98574 p^{16} T^{36} - 444 p^{18} T^{38} + p^{20} T^{40} \) | |
| 43 | \( 1 - 4 T + 8 T^{2} + 300 T^{3} + 7078 T^{4} - 44284 T^{5} + 165512 T^{6} + 1626836 T^{7} + 19994909 T^{8} - 198352400 T^{9} + 930543520 T^{10} + 1629993648 T^{11} + 32879446344 T^{12} - 523812434096 T^{13} + 2532563214112 T^{14} - 8844397053040 T^{15} + 52614175288210 T^{16} - 1034528298704760 T^{17} + 4610401791613808 T^{18} - 37116758465801176 T^{19} + 100269344691342500 T^{20} - 37116758465801176 p T^{21} + 4610401791613808 p^{2} T^{22} - 1034528298704760 p^{3} T^{23} + 52614175288210 p^{4} T^{24} - 8844397053040 p^{5} T^{25} + 2532563214112 p^{6} T^{26} - 523812434096 p^{7} T^{27} + 32879446344 p^{8} T^{28} + 1629993648 p^{9} T^{29} + 930543520 p^{10} T^{30} - 198352400 p^{11} T^{31} + 19994909 p^{12} T^{32} + 1626836 p^{13} T^{33} + 165512 p^{14} T^{34} - 44284 p^{15} T^{35} + 7078 p^{16} T^{36} + 300 p^{17} T^{37} + 8 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( ( 1 + 254 T^{2} - 128 T^{3} + 33997 T^{4} - 26752 T^{5} + 3110696 T^{6} - 2769024 T^{7} + 212743186 T^{8} - 186077312 T^{9} + 11293892148 T^{10} - 186077312 p T^{11} + 212743186 p^{2} T^{12} - 2769024 p^{3} T^{13} + 3110696 p^{4} T^{14} - 26752 p^{5} T^{15} + 33997 p^{6} T^{16} - 128 p^{7} T^{17} + 254 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 + 36 T + 648 T^{2} + 8284 T^{3} + 81222 T^{4} + 595308 T^{5} + 3111560 T^{6} + 8277492 T^{7} - 42534915 T^{8} - 754029808 T^{9} - 6543023328 T^{10} - 50633847440 T^{11} - 420252241016 T^{12} - 3590853938064 T^{13} - 513815867104 p T^{14} - 163759336672624 T^{15} - 644108178966222 T^{16} - 225724213191624 T^{17} + 21359012626695920 T^{18} + 241077022660459848 T^{19} + 1941342482920284900 T^{20} + 241077022660459848 p T^{21} + 21359012626695920 p^{2} T^{22} - 225724213191624 p^{3} T^{23} - 644108178966222 p^{4} T^{24} - 163759336672624 p^{5} T^{25} - 513815867104 p^{7} T^{26} - 3590853938064 p^{7} T^{27} - 420252241016 p^{8} T^{28} - 50633847440 p^{9} T^{29} - 6543023328 p^{10} T^{30} - 754029808 p^{11} T^{31} - 42534915 p^{12} T^{32} + 8277492 p^{13} T^{33} + 3111560 p^{14} T^{34} + 595308 p^{15} T^{35} + 81222 p^{16} T^{36} + 8284 p^{17} T^{37} + 648 p^{18} T^{38} + 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 64 T^{3} + 17498 T^{4} + 16320 T^{5} + 2048 T^{6} + 964352 T^{7} + 139305821 T^{8} + 265011968 T^{9} + 159053824 T^{10} + 7117236224 T^{11} + 675196772280 T^{12} + 1865453307392 T^{13} + 2177063692288 T^{14} + 46993822767872 T^{15} + 2378495821379602 T^{16} + 7365185499789056 T^{17} + 13796808199690240 T^{18} + 308926250030895744 T^{19} + 7830691115365441372 T^{20} + 308926250030895744 p T^{21} + 13796808199690240 p^{2} T^{22} + 7365185499789056 p^{3} T^{23} + 2378495821379602 p^{4} T^{24} + 46993822767872 p^{5} T^{25} + 2177063692288 p^{6} T^{26} + 1865453307392 p^{7} T^{27} + 675196772280 p^{8} T^{28} + 7117236224 p^{9} T^{29} + 159053824 p^{10} T^{30} + 265011968 p^{11} T^{31} + 139305821 p^{12} T^{32} + 964352 p^{13} T^{33} + 2048 p^{14} T^{34} + 16320 p^{15} T^{35} + 17498 p^{16} T^{36} + 64 p^{17} T^{37} + p^{20} T^{40} \) | |
| 61 | \( 1 - 8 T + 32 T^{2} - 24 T^{3} - 22 T^{4} + 28776 T^{5} - 229216 T^{6} + 1701688 T^{7} - 7327587 T^{8} + 193356480 T^{9} - 944223232 T^{10} + 7441249856 T^{11} + 25030907320 T^{12} - 252322194176 T^{13} + 6350522522880 T^{14} - 30700611174144 T^{15} + 72405460654514 T^{16} + 1378721064691056 T^{17} + 10443615781913408 T^{18} + 4859962566983248 T^{19} + 374225284282585404 T^{20} + 4859962566983248 p T^{21} + 10443615781913408 p^{2} T^{22} + 1378721064691056 p^{3} T^{23} + 72405460654514 p^{4} T^{24} - 30700611174144 p^{5} T^{25} + 6350522522880 p^{6} T^{26} - 252322194176 p^{7} T^{27} + 25030907320 p^{8} T^{28} + 7441249856 p^{9} T^{29} - 944223232 p^{10} T^{30} + 193356480 p^{11} T^{31} - 7327587 p^{12} T^{32} + 1701688 p^{13} T^{33} - 229216 p^{14} T^{34} + 28776 p^{15} T^{35} - 22 p^{16} T^{36} - 24 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 4102 T^{4} + 54484 T^{5} + 457032 T^{6} + 4150628 T^{7} - 7647939 T^{8} - 292442192 T^{9} - 1506268000 T^{10} - 8570912144 T^{11} - 68934819128 T^{12} - 1058425781488 T^{13} - 9188787191264 T^{14} - 93290480178608 T^{15} + 105617726671378 T^{16} + 7399255991086056 T^{17} + 46962879527876336 T^{18} + 365051212101087880 T^{19} + 2665710348810285028 T^{20} + 365051212101087880 p T^{21} + 46962879527876336 p^{2} T^{22} + 7399255991086056 p^{3} T^{23} + 105617726671378 p^{4} T^{24} - 93290480178608 p^{5} T^{25} - 9188787191264 p^{6} T^{26} - 1058425781488 p^{7} T^{27} - 68934819128 p^{8} T^{28} - 8570912144 p^{9} T^{29} - 1506268000 p^{10} T^{30} - 292442192 p^{11} T^{31} - 7647939 p^{12} T^{32} + 4150628 p^{13} T^{33} + 457032 p^{14} T^{34} + 54484 p^{15} T^{35} + 4102 p^{16} T^{36} + 444 p^{17} T^{37} + 72 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 588 T^{2} + 187310 T^{4} - 41853292 T^{6} + 7286062485 T^{8} - 1044227363664 T^{10} + 127464606182264 T^{12} - 13544991508155920 T^{14} + 1271322412181875034 T^{16} - \)\(10\!\cdots\!40\)\( T^{18} + \)\(79\!\cdots\!36\)\( T^{20} - \)\(10\!\cdots\!40\)\( p^{2} T^{22} + 1271322412181875034 p^{4} T^{24} - 13544991508155920 p^{6} T^{26} + 127464606182264 p^{8} T^{28} - 1044227363664 p^{10} T^{30} + 7286062485 p^{12} T^{32} - 41853292 p^{14} T^{34} + 187310 p^{16} T^{36} - 588 p^{18} T^{38} + p^{20} T^{40} \) | |
| 73 | \( 1 - 588 T^{2} + 185438 T^{4} - 41247404 T^{6} + 7184467213 T^{8} - 1034846312688 T^{10} + 127434000595944 T^{12} - 13711360420506544 T^{14} + 1308031080629694098 T^{16} - \)\(11\!\cdots\!16\)\( T^{18} + \)\(85\!\cdots\!04\)\( T^{20} - \)\(11\!\cdots\!16\)\( p^{2} T^{22} + 1308031080629694098 p^{4} T^{24} - 13711360420506544 p^{6} T^{26} + 127434000595944 p^{8} T^{28} - 1034846312688 p^{10} T^{30} + 7184467213 p^{12} T^{32} - 41247404 p^{14} T^{34} + 185438 p^{16} T^{36} - 588 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 12 T + 410 T^{2} - 3636 T^{3} + 71661 T^{4} - 457104 T^{5} + 6962296 T^{6} - 26461552 T^{7} + 422059634 T^{8} - 425195336 T^{9} + 24696310492 T^{10} - 425195336 p T^{11} + 422059634 p^{2} T^{12} - 26461552 p^{3} T^{13} + 6962296 p^{4} T^{14} - 457104 p^{5} T^{15} + 71661 p^{6} T^{16} - 3636 p^{7} T^{17} + 410 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 - 40 T + 800 T^{2} - 11352 T^{3} + 130714 T^{4} - 1275720 T^{5} + 10891552 T^{6} - 81158136 T^{7} + 533032573 T^{8} - 3591219744 T^{9} + 28582322304 T^{10} - 250649734880 T^{11} + 2277897381816 T^{12} - 18560960241440 T^{13} + 98547001707136 T^{14} + 326327140527904 T^{15} - 18693002705618542 T^{16} + 320247589539147216 T^{17} - 4017303386407089728 T^{18} + 43457553116792444976 T^{19} - \)\(41\!\cdots\!72\)\( T^{20} + 43457553116792444976 p T^{21} - 4017303386407089728 p^{2} T^{22} + 320247589539147216 p^{3} T^{23} - 18693002705618542 p^{4} T^{24} + 326327140527904 p^{5} T^{25} + 98547001707136 p^{6} T^{26} - 18560960241440 p^{7} T^{27} + 2277897381816 p^{8} T^{28} - 250649734880 p^{9} T^{29} + 28582322304 p^{10} T^{30} - 3591219744 p^{11} T^{31} + 533032573 p^{12} T^{32} - 81158136 p^{13} T^{33} + 10891552 p^{14} T^{34} - 1275720 p^{15} T^{35} + 130714 p^{16} T^{36} - 11352 p^{17} T^{37} + 800 p^{18} T^{38} - 40 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 700 T^{2} + 256046 T^{4} - 64249660 T^{6} + 12305003221 T^{8} - 1904804480368 T^{10} + 247648665045240 T^{12} - 27922948227608560 T^{14} + 2822449388027483546 T^{16} - \)\(26\!\cdots\!68\)\( T^{18} + \)\(23\!\cdots\!96\)\( T^{20} - \)\(26\!\cdots\!68\)\( p^{2} T^{22} + 2822449388027483546 p^{4} T^{24} - 27922948227608560 p^{6} T^{26} + 247648665045240 p^{8} T^{28} - 1904804480368 p^{10} T^{30} + 12305003221 p^{12} T^{32} - 64249660 p^{14} T^{34} + 256046 p^{16} T^{36} - 700 p^{18} T^{38} + p^{20} T^{40} \) | |
| 97 | \( ( 1 + 36 T + 1006 T^{2} + 17860 T^{3} + 282157 T^{4} + 3451280 T^{5} + 41738920 T^{6} + 425104976 T^{7} + 4638925202 T^{8} + 43768470968 T^{9} + 461725986516 T^{10} + 43768470968 p T^{11} + 4638925202 p^{2} T^{12} + 425104976 p^{3} T^{13} + 41738920 p^{4} T^{14} + 3451280 p^{5} T^{15} + 282157 p^{6} T^{16} + 17860 p^{7} T^{17} + 1006 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 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
−2.75130397097781454086169659485, −2.74291656102580372261498846493, −2.72682071521651496184353921414, −2.62693808301076452211713336389, −2.53406660453667324057119922466, −2.47741392447796330022017102514, −2.37457743423297268665153542121, −2.19054963234387882456663301459, −2.16828113426192447865831003835, −2.08874307305412467159998135181, −2.02506042371329113269890698322, −1.78436902924377403352887631387, −1.65392547015318276539361546446, −1.61361768227473784890985972680, −1.57726347634880584619816736734, −1.57699224816045814618366530492, −1.54566265925130213618501228240, −1.35168992143907027914991541249, −1.33817026722389206414507000964, −1.03399427121434499818018534039, −0.932689144796110782966526587845, −0.830420926934566288300599966069, −0.75058173803746508909755794102, −0.61517340935940583558693657130, −0.29770401145800972161529147810, 0.29770401145800972161529147810, 0.61517340935940583558693657130, 0.75058173803746508909755794102, 0.830420926934566288300599966069, 0.932689144796110782966526587845, 1.03399427121434499818018534039, 1.33817026722389206414507000964, 1.35168992143907027914991541249, 1.54566265925130213618501228240, 1.57699224816045814618366530492, 1.57726347634880584619816736734, 1.61361768227473784890985972680, 1.65392547015318276539361546446, 1.78436902924377403352887631387, 2.02506042371329113269890698322, 2.08874307305412467159998135181, 2.16828113426192447865831003835, 2.19054963234387882456663301459, 2.37457743423297268665153542121, 2.47741392447796330022017102514, 2.53406660453667324057119922466, 2.62693808301076452211713336389, 2.72682071521651496184353921414, 2.74291656102580372261498846493, 2.75130397097781454086169659485
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.