Dirichlet series
| L(s) = 1 | − 20·2-s + 4·3-s + 210·4-s + 20·5-s − 80·6-s + 11·7-s − 1.54e3·8-s − 9·9-s − 400·10-s + 10·11-s + 840·12-s − 9·13-s − 220·14-s + 80·15-s + 8.85e3·16-s − 11·17-s + 180·18-s + 17·19-s + 4.20e3·20-s + 44·21-s − 200·22-s − 3·23-s − 6.16e3·24-s + 210·25-s + 180·26-s − 62·27-s + 2.31e3·28-s + ⋯ |
| L(s) = 1 | − 14.1·2-s + 2.30·3-s + 105·4-s + 8.94·5-s − 32.6·6-s + 4.15·7-s − 544.·8-s − 3·9-s − 126.·10-s + 3.01·11-s + 242.·12-s − 2.49·13-s − 58.7·14-s + 20.6·15-s + 2.21e3·16-s − 2.66·17-s + 42.4·18-s + 3.90·19-s + 939.·20-s + 9.60·21-s − 42.6·22-s − 0.625·23-s − 1.25e3·24-s + 42·25-s + 35.3·26-s − 11.9·27-s + 436.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 401^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 401^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 5^{20} \cdot 401^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28359\times 10^{30}\) |
| Root analytic conductor: | \(5.65862\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 5^{20} \cdot 401^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(68.52460532\) |
| \(L(\frac12)\) | \(\approx\) | \(68.52460532\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{20} \) |
| 5 | \( ( 1 - T )^{20} \) | |
| 401 | \( ( 1 + T )^{20} \) | |
| good | 3 | \( 1 - 4 T + 25 T^{2} - 74 T^{3} + 280 T^{4} - 694 T^{5} + 2074 T^{6} - 4639 T^{7} + 12079 T^{8} - 8426 p T^{9} + 6649 p^{2} T^{10} - 39742 p T^{11} + 263159 T^{12} - 503756 T^{13} + 1050631 T^{14} - 214849 p^{2} T^{15} + 3835144 T^{16} - 6787892 T^{17} + 4296751 p T^{18} - 21982472 T^{19} + 40146458 T^{20} - 21982472 p T^{21} + 4296751 p^{3} T^{22} - 6787892 p^{3} T^{23} + 3835144 p^{4} T^{24} - 214849 p^{7} T^{25} + 1050631 p^{6} T^{26} - 503756 p^{7} T^{27} + 263159 p^{8} T^{28} - 39742 p^{10} T^{29} + 6649 p^{12} T^{30} - 8426 p^{12} T^{31} + 12079 p^{12} T^{32} - 4639 p^{13} T^{33} + 2074 p^{14} T^{34} - 694 p^{15} T^{35} + 280 p^{16} T^{36} - 74 p^{17} T^{37} + 25 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) |
| 7 | \( 1 - 11 T + 109 T^{2} - 760 T^{3} + 4810 T^{4} - 25911 T^{5} + 129587 T^{6} - 586561 T^{7} + 2504480 T^{8} - 9960560 T^{9} + 37784799 T^{10} - 135712250 T^{11} + 468685321 T^{12} - 1548757706 T^{13} + 4950678598 T^{14} - 15252837261 T^{15} + 45666591933 T^{16} - 132444676281 T^{17} + 374472954091 T^{18} - 1028761385485 T^{19} + 2759849551454 T^{20} - 1028761385485 p T^{21} + 374472954091 p^{2} T^{22} - 132444676281 p^{3} T^{23} + 45666591933 p^{4} T^{24} - 15252837261 p^{5} T^{25} + 4950678598 p^{6} T^{26} - 1548757706 p^{7} T^{27} + 468685321 p^{8} T^{28} - 135712250 p^{9} T^{29} + 37784799 p^{10} T^{30} - 9960560 p^{11} T^{31} + 2504480 p^{12} T^{32} - 586561 p^{13} T^{33} + 129587 p^{14} T^{34} - 25911 p^{15} T^{35} + 4810 p^{16} T^{36} - 760 p^{17} T^{37} + 109 p^{18} T^{38} - 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 10 T + 135 T^{2} - 8 p^{2} T^{3} + 8005 T^{4} - 46606 T^{5} + 297499 T^{6} - 1488356 T^{7} + 7976885 T^{8} - 35418536 T^{9} + 1375563 p^{2} T^{10} - 670802614 T^{11} + 2842273407 T^{12} - 10592077576 T^{13} + 41364010581 T^{14} - 145097041890 T^{15} + 533021349843 T^{16} - 1791638233136 T^{17} + 6314223167762 T^{18} - 20648559894180 T^{19} + 70901564095950 T^{20} - 20648559894180 p T^{21} + 6314223167762 p^{2} T^{22} - 1791638233136 p^{3} T^{23} + 533021349843 p^{4} T^{24} - 145097041890 p^{5} T^{25} + 41364010581 p^{6} T^{26} - 10592077576 p^{7} T^{27} + 2842273407 p^{8} T^{28} - 670802614 p^{9} T^{29} + 1375563 p^{12} T^{30} - 35418536 p^{11} T^{31} + 7976885 p^{12} T^{32} - 1488356 p^{13} T^{33} + 297499 p^{14} T^{34} - 46606 p^{15} T^{35} + 8005 p^{16} T^{36} - 8 p^{19} T^{37} + 135 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 9 T + 125 T^{2} + 803 T^{3} + 6936 T^{4} + 35787 T^{5} + 245019 T^{6} + 1073173 T^{7} + 6422048 T^{8} + 24590846 T^{9} + 135968394 T^{10} + 462566134 T^{11} + 2458241730 T^{12} + 7521270846 T^{13} + 39614362494 T^{14} + 110604235870 T^{15} + 588518940767 T^{16} + 1528601024818 T^{17} + 8236033085448 T^{18} + 1566957565122 p T^{19} + 109643739132620 T^{20} + 1566957565122 p^{2} T^{21} + 8236033085448 p^{2} T^{22} + 1528601024818 p^{3} T^{23} + 588518940767 p^{4} T^{24} + 110604235870 p^{5} T^{25} + 39614362494 p^{6} T^{26} + 7521270846 p^{7} T^{27} + 2458241730 p^{8} T^{28} + 462566134 p^{9} T^{29} + 135968394 p^{10} T^{30} + 24590846 p^{11} T^{31} + 6422048 p^{12} T^{32} + 1073173 p^{13} T^{33} + 245019 p^{14} T^{34} + 35787 p^{15} T^{35} + 6936 p^{16} T^{36} + 803 p^{17} T^{37} + 125 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 11 T + 11 p T^{2} + 1466 T^{3} + 885 p T^{4} + 95975 T^{5} + 766879 T^{6} + 4276345 T^{7} + 29230872 T^{8} + 148160148 T^{9} + 904326022 T^{10} + 4257130832 T^{11} + 23760103160 T^{12} + 105394177764 T^{13} + 547114384798 T^{14} + 2312777899509 T^{15} + 11321989575037 T^{16} + 45992275060073 T^{17} + 214715852192474 T^{18} + 842066173843817 T^{19} + 3779929420654682 T^{20} + 842066173843817 p T^{21} + 214715852192474 p^{2} T^{22} + 45992275060073 p^{3} T^{23} + 11321989575037 p^{4} T^{24} + 2312777899509 p^{5} T^{25} + 547114384798 p^{6} T^{26} + 105394177764 p^{7} T^{27} + 23760103160 p^{8} T^{28} + 4257130832 p^{9} T^{29} + 904326022 p^{10} T^{30} + 148160148 p^{11} T^{31} + 29230872 p^{12} T^{32} + 4276345 p^{13} T^{33} + 766879 p^{14} T^{34} + 95975 p^{15} T^{35} + 885 p^{17} T^{36} + 1466 p^{17} T^{37} + 11 p^{19} T^{38} + 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 17 T + 328 T^{2} - 3718 T^{3} + 2274 p T^{4} - 378002 T^{5} + 3319937 T^{6} - 23893056 T^{7} + 171910607 T^{8} - 1049674263 T^{9} + 6420497259 T^{10} - 33673266646 T^{11} + 178116297884 T^{12} - 799930391474 T^{13} + 3679058705569 T^{14} - 13840172383731 T^{15} + 55520795289792 T^{16} - 166885253979295 T^{17} + 618842889946139 T^{18} - 1519199049734360 T^{19} + 7761996180389628 T^{20} - 1519199049734360 p T^{21} + 618842889946139 p^{2} T^{22} - 166885253979295 p^{3} T^{23} + 55520795289792 p^{4} T^{24} - 13840172383731 p^{5} T^{25} + 3679058705569 p^{6} T^{26} - 799930391474 p^{7} T^{27} + 178116297884 p^{8} T^{28} - 33673266646 p^{9} T^{29} + 6420497259 p^{10} T^{30} - 1049674263 p^{11} T^{31} + 171910607 p^{12} T^{32} - 23893056 p^{13} T^{33} + 3319937 p^{14} T^{34} - 378002 p^{15} T^{35} + 2274 p^{17} T^{36} - 3718 p^{17} T^{37} + 328 p^{18} T^{38} - 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 3 T + 111 T^{2} + 81 T^{3} + 6554 T^{4} - 8790 T^{5} + 281777 T^{6} - 990877 T^{7} + 453929 p T^{8} - 54806504 T^{9} + 368391954 T^{10} - 2188486123 T^{11} + 12502551884 T^{12} - 72140341875 T^{13} + 394333527374 T^{14} - 2106944090532 T^{15} + 11261617726288 T^{16} - 56751441150884 T^{17} + 291511876182384 T^{18} - 1416598262006791 T^{19} + 6953049529150452 T^{20} - 1416598262006791 p T^{21} + 291511876182384 p^{2} T^{22} - 56751441150884 p^{3} T^{23} + 11261617726288 p^{4} T^{24} - 2106944090532 p^{5} T^{25} + 394333527374 p^{6} T^{26} - 72140341875 p^{7} T^{27} + 12502551884 p^{8} T^{28} - 2188486123 p^{9} T^{29} + 368391954 p^{10} T^{30} - 54806504 p^{11} T^{31} + 453929 p^{13} T^{32} - 990877 p^{13} T^{33} + 281777 p^{14} T^{34} - 8790 p^{15} T^{35} + 6554 p^{16} T^{36} + 81 p^{17} T^{37} + 111 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 6 T + 279 T^{2} - 1754 T^{3} + 39553 T^{4} - 243622 T^{5} + 3749498 T^{6} - 21733423 T^{7} + 264103463 T^{8} - 1412924349 T^{9} + 14664372197 T^{10} - 71922469734 T^{11} + 670676080400 T^{12} - 3016767543462 T^{13} + 26260811561223 T^{14} - 109030716579071 T^{15} + 912551573568488 T^{16} - 3544260981938023 T^{17} + 29048131149675435 T^{18} - 107588090610467364 T^{19} + 866530819130155742 T^{20} - 107588090610467364 p T^{21} + 29048131149675435 p^{2} T^{22} - 3544260981938023 p^{3} T^{23} + 912551573568488 p^{4} T^{24} - 109030716579071 p^{5} T^{25} + 26260811561223 p^{6} T^{26} - 3016767543462 p^{7} T^{27} + 670676080400 p^{8} T^{28} - 71922469734 p^{9} T^{29} + 14664372197 p^{10} T^{30} - 1412924349 p^{11} T^{31} + 264103463 p^{12} T^{32} - 21733423 p^{13} T^{33} + 3749498 p^{14} T^{34} - 243622 p^{15} T^{35} + 39553 p^{16} T^{36} - 1754 p^{17} T^{37} + 279 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 28 T + 672 T^{2} - 11309 T^{3} + 167975 T^{4} - 2095701 T^{5} + 23693151 T^{6} - 237991181 T^{7} + 2201384919 T^{8} - 18525865649 T^{9} + 144838949648 T^{10} - 1040741709740 T^{11} + 6970732125174 T^{12} - 42986376020677 T^{13} + 246787658105407 T^{14} - 41875163832447 p T^{15} + 6342038871927391 T^{16} - 28318552383402343 T^{17} + 121028837362334126 T^{18} - 513350287388893799 T^{19} + 2580102329863498336 T^{20} - 513350287388893799 p T^{21} + 121028837362334126 p^{2} T^{22} - 28318552383402343 p^{3} T^{23} + 6342038871927391 p^{4} T^{24} - 41875163832447 p^{6} T^{25} + 246787658105407 p^{6} T^{26} - 42986376020677 p^{7} T^{27} + 6970732125174 p^{8} T^{28} - 1040741709740 p^{9} T^{29} + 144838949648 p^{10} T^{30} - 18525865649 p^{11} T^{31} + 2201384919 p^{12} T^{32} - 237991181 p^{13} T^{33} + 23693151 p^{14} T^{34} - 2095701 p^{15} T^{35} + 167975 p^{16} T^{36} - 11309 p^{17} T^{37} + 672 p^{18} T^{38} - 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 33 T + 920 T^{2} - 18109 T^{3} + 313072 T^{4} - 4577405 T^{5} + 60573576 T^{6} - 718614854 T^{7} + 7868440417 T^{8} - 79393277341 T^{9} + 749405759288 T^{10} - 6625840847018 T^{11} + 55371470318800 T^{12} - 438534234155610 T^{13} + 3312305751623064 T^{14} - 23944987027564343 T^{15} + 166488942952783662 T^{16} - 1117799717799449857 T^{17} + 7273115895541291536 T^{18} - 46011991044893902286 T^{19} + \)\(28\!\cdots\!76\)\( T^{20} - 46011991044893902286 p T^{21} + 7273115895541291536 p^{2} T^{22} - 1117799717799449857 p^{3} T^{23} + 166488942952783662 p^{4} T^{24} - 23944987027564343 p^{5} T^{25} + 3312305751623064 p^{6} T^{26} - 438534234155610 p^{7} T^{27} + 55371470318800 p^{8} T^{28} - 6625840847018 p^{9} T^{29} + 749405759288 p^{10} T^{30} - 79393277341 p^{11} T^{31} + 7868440417 p^{12} T^{32} - 718614854 p^{13} T^{33} + 60573576 p^{14} T^{34} - 4577405 p^{15} T^{35} + 313072 p^{16} T^{36} - 18109 p^{17} T^{37} + 920 p^{18} T^{38} - 33 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 32 T + 942 T^{2} - 19053 T^{3} + 349697 T^{4} - 5380853 T^{5} + 76242290 T^{6} - 965917263 T^{7} + 11420272429 T^{8} - 124364941580 T^{9} + 1277017218691 T^{10} - 12280763635141 T^{11} + 112253680031242 T^{12} - 971318643399844 T^{13} + 8039228992738246 T^{14} - 63459953289904227 T^{15} + 481541267720648344 T^{16} - 3503326536622076286 T^{17} + 24588161616267071735 T^{18} - \)\(16\!\cdots\!69\)\( T^{19} + \)\(10\!\cdots\!26\)\( T^{20} - \)\(16\!\cdots\!69\)\( p T^{21} + 24588161616267071735 p^{2} T^{22} - 3503326536622076286 p^{3} T^{23} + 481541267720648344 p^{4} T^{24} - 63459953289904227 p^{5} T^{25} + 8039228992738246 p^{6} T^{26} - 971318643399844 p^{7} T^{27} + 112253680031242 p^{8} T^{28} - 12280763635141 p^{9} T^{29} + 1277017218691 p^{10} T^{30} - 124364941580 p^{11} T^{31} + 11420272429 p^{12} T^{32} - 965917263 p^{13} T^{33} + 76242290 p^{14} T^{34} - 5380853 p^{15} T^{35} + 349697 p^{16} T^{36} - 19053 p^{17} T^{37} + 942 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 30 T + 856 T^{2} - 16665 T^{3} + 299663 T^{4} - 4514387 T^{5} + 63414770 T^{6} - 797741034 T^{7} + 9455911046 T^{8} - 103474312852 T^{9} + 1075697205290 T^{10} - 10496289515336 T^{11} + 97896045691830 T^{12} - 865628308270888 T^{13} + 7348985142939734 T^{14} - 59527585557919532 T^{15} + 464424940702494377 T^{16} - 3470866238516529340 T^{17} + 25035376828074798390 T^{18} - \)\(17\!\cdots\!02\)\( T^{19} + \)\(11\!\cdots\!90\)\( T^{20} - \)\(17\!\cdots\!02\)\( p T^{21} + 25035376828074798390 p^{2} T^{22} - 3470866238516529340 p^{3} T^{23} + 464424940702494377 p^{4} T^{24} - 59527585557919532 p^{5} T^{25} + 7348985142939734 p^{6} T^{26} - 865628308270888 p^{7} T^{27} + 97896045691830 p^{8} T^{28} - 10496289515336 p^{9} T^{29} + 1075697205290 p^{10} T^{30} - 103474312852 p^{11} T^{31} + 9455911046 p^{12} T^{32} - 797741034 p^{13} T^{33} + 63414770 p^{14} T^{34} - 4514387 p^{15} T^{35} + 299663 p^{16} T^{36} - 16665 p^{17} T^{37} + 856 p^{18} T^{38} - 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 13 T + 545 T^{2} - 6712 T^{3} + 151015 T^{4} - 1726403 T^{5} + 28006418 T^{6} - 294963660 T^{7} + 3878399311 T^{8} - 37634958770 T^{9} + 425669221389 T^{10} - 3820578959723 T^{11} + 38456235420320 T^{12} - 320862713801166 T^{13} + 2935213631999119 T^{14} - 22868067052871623 T^{15} + 192742167062395496 T^{16} - 1406526534033716800 T^{17} + 11021027114470315793 T^{18} - 75435176263696806234 T^{19} + \)\(55\!\cdots\!54\)\( T^{20} - 75435176263696806234 p T^{21} + 11021027114470315793 p^{2} T^{22} - 1406526534033716800 p^{3} T^{23} + 192742167062395496 p^{4} T^{24} - 22868067052871623 p^{5} T^{25} + 2935213631999119 p^{6} T^{26} - 320862713801166 p^{7} T^{27} + 38456235420320 p^{8} T^{28} - 3820578959723 p^{9} T^{29} + 425669221389 p^{10} T^{30} - 37634958770 p^{11} T^{31} + 3878399311 p^{12} T^{32} - 294963660 p^{13} T^{33} + 28006418 p^{14} T^{34} - 1726403 p^{15} T^{35} + 151015 p^{16} T^{36} - 6712 p^{17} T^{37} + 545 p^{18} T^{38} - 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 580 T^{2} - 94 T^{3} + 168300 T^{4} - 37522 T^{5} + 32529045 T^{6} - 6734733 T^{7} + 4709063900 T^{8} - 650592727 T^{9} + 544673756091 T^{10} - 22341386100 T^{11} + 52434444393602 T^{12} + 3172847169196 T^{13} + 4318179455645649 T^{14} + 632502433537527 T^{15} + 309916584861821571 T^{16} + 63053809635564805 T^{17} + 19616370967515961547 T^{18} + 4421096949156420672 T^{19} + \)\(11\!\cdots\!80\)\( T^{20} + 4421096949156420672 p T^{21} + 19616370967515961547 p^{2} T^{22} + 63053809635564805 p^{3} T^{23} + 309916584861821571 p^{4} T^{24} + 632502433537527 p^{5} T^{25} + 4318179455645649 p^{6} T^{26} + 3172847169196 p^{7} T^{27} + 52434444393602 p^{8} T^{28} - 22341386100 p^{9} T^{29} + 544673756091 p^{10} T^{30} - 650592727 p^{11} T^{31} + 4709063900 p^{12} T^{32} - 6734733 p^{13} T^{33} + 32529045 p^{14} T^{34} - 37522 p^{15} T^{35} + 168300 p^{16} T^{36} - 94 p^{17} T^{37} + 580 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( 1 - 52 T + 1878 T^{2} - 50022 T^{3} + 1116871 T^{4} - 21390055 T^{5} + 365572667 T^{6} - 5645806566 T^{7} + 80247743447 T^{8} - 1057776840220 T^{9} + 13063792894135 T^{10} - 151898601180455 T^{11} + 1673407854431448 T^{12} - 17520989353594035 T^{13} + 175071493851574117 T^{14} - 1672684747762174640 T^{15} + 15322812599928299672 T^{16} - \)\(13\!\cdots\!30\)\( T^{17} + \)\(11\!\cdots\!87\)\( T^{18} - \)\(92\!\cdots\!13\)\( T^{19} + \)\(72\!\cdots\!98\)\( T^{20} - \)\(92\!\cdots\!13\)\( p T^{21} + \)\(11\!\cdots\!87\)\( p^{2} T^{22} - \)\(13\!\cdots\!30\)\( p^{3} T^{23} + 15322812599928299672 p^{4} T^{24} - 1672684747762174640 p^{5} T^{25} + 175071493851574117 p^{6} T^{26} - 17520989353594035 p^{7} T^{27} + 1673407854431448 p^{8} T^{28} - 151898601180455 p^{9} T^{29} + 13063792894135 p^{10} T^{30} - 1057776840220 p^{11} T^{31} + 80247743447 p^{12} T^{32} - 5645806566 p^{13} T^{33} + 365572667 p^{14} T^{34} - 21390055 p^{15} T^{35} + 1116871 p^{16} T^{36} - 50022 p^{17} T^{37} + 1878 p^{18} T^{38} - 52 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 25 T + 809 T^{2} - 13066 T^{3} + 249287 T^{4} - 3013197 T^{5} + 43802961 T^{6} - 425295829 T^{7} + 5280074813 T^{8} - 42617209014 T^{9} + 488390063830 T^{10} - 3333662338553 T^{11} + 38095080824968 T^{12} - 226541564136203 T^{13} + 2776119292967903 T^{14} - 15273161628830562 T^{15} + 201717876443814384 T^{16} - 1077035925100860262 T^{17} + 14266608901344242433 T^{18} - 73582750774492550021 T^{19} + \)\(92\!\cdots\!02\)\( T^{20} - 73582750774492550021 p T^{21} + 14266608901344242433 p^{2} T^{22} - 1077035925100860262 p^{3} T^{23} + 201717876443814384 p^{4} T^{24} - 15273161628830562 p^{5} T^{25} + 2776119292967903 p^{6} T^{26} - 226541564136203 p^{7} T^{27} + 38095080824968 p^{8} T^{28} - 3333662338553 p^{9} T^{29} + 488390063830 p^{10} T^{30} - 42617209014 p^{11} T^{31} + 5280074813 p^{12} T^{32} - 425295829 p^{13} T^{33} + 43802961 p^{14} T^{34} - 3013197 p^{15} T^{35} + 249287 p^{16} T^{36} - 13066 p^{17} T^{37} + 809 p^{18} T^{38} - 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 40 T + 1284 T^{2} - 28673 T^{3} + 562703 T^{4} - 9190003 T^{5} + 137172557 T^{6} - 1807316403 T^{7} + 22182723544 T^{8} - 246076933121 T^{9} + 2563472310635 T^{10} - 24267573215354 T^{11} + 215479037907315 T^{12} - 1714952393890307 T^{13} + 12561359328564502 T^{14} - 77177585048238639 T^{15} + 389509735111653839 T^{16} - 808108512380630575 T^{17} - 8673410536585896382 T^{18} + \)\(17\!\cdots\!67\)\( T^{19} - \)\(16\!\cdots\!88\)\( T^{20} + \)\(17\!\cdots\!67\)\( p T^{21} - 8673410536585896382 p^{2} T^{22} - 808108512380630575 p^{3} T^{23} + 389509735111653839 p^{4} T^{24} - 77177585048238639 p^{5} T^{25} + 12561359328564502 p^{6} T^{26} - 1714952393890307 p^{7} T^{27} + 215479037907315 p^{8} T^{28} - 24267573215354 p^{9} T^{29} + 2563472310635 p^{10} T^{30} - 246076933121 p^{11} T^{31} + 22182723544 p^{12} T^{32} - 1807316403 p^{13} T^{33} + 137172557 p^{14} T^{34} - 9190003 p^{15} T^{35} + 562703 p^{16} T^{36} - 28673 p^{17} T^{37} + 1284 p^{18} T^{38} - 40 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 25 T + 1179 T^{2} - 23627 T^{3} + 642303 T^{4} - 10843765 T^{5} + 219746522 T^{6} - 3227326843 T^{7} + 53701111244 T^{8} - 701374155508 T^{9} + 10064549875983 T^{10} - 118735729776174 T^{11} + 1512292614445371 T^{12} - 16295701656776826 T^{13} + 187615065116936644 T^{14} - 1860880499290935897 T^{15} + 19599269848882745602 T^{16} - \)\(17\!\cdots\!41\)\( T^{17} + \)\(17\!\cdots\!56\)\( T^{18} - \)\(14\!\cdots\!40\)\( T^{19} + \)\(13\!\cdots\!14\)\( T^{20} - \)\(14\!\cdots\!40\)\( p T^{21} + \)\(17\!\cdots\!56\)\( p^{2} T^{22} - \)\(17\!\cdots\!41\)\( p^{3} T^{23} + 19599269848882745602 p^{4} T^{24} - 1860880499290935897 p^{5} T^{25} + 187615065116936644 p^{6} T^{26} - 16295701656776826 p^{7} T^{27} + 1512292614445371 p^{8} T^{28} - 118735729776174 p^{9} T^{29} + 10064549875983 p^{10} T^{30} - 701374155508 p^{11} T^{31} + 53701111244 p^{12} T^{32} - 3227326843 p^{13} T^{33} + 219746522 p^{14} T^{34} - 10843765 p^{15} T^{35} + 642303 p^{16} T^{36} - 23627 p^{17} T^{37} + 1179 p^{18} T^{38} - 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 5 T + 804 T^{2} + 3251 T^{3} + 323509 T^{4} + 1074418 T^{5} + 87431704 T^{6} + 242424263 T^{7} + 17867342220 T^{8} + 42008911292 T^{9} + 2936371625912 T^{10} + 5940568696023 T^{11} + 402422635461418 T^{12} + 711191350050545 T^{13} + 47060618620907168 T^{14} + 73860683416004584 T^{15} + 4766499495805964611 T^{16} + 6763244830835519256 T^{17} + \)\(42\!\cdots\!56\)\( T^{18} + \)\(55\!\cdots\!99\)\( T^{19} + \)\(32\!\cdots\!18\)\( T^{20} + \)\(55\!\cdots\!99\)\( p T^{21} + \)\(42\!\cdots\!56\)\( p^{2} T^{22} + 6763244830835519256 p^{3} T^{23} + 4766499495805964611 p^{4} T^{24} + 73860683416004584 p^{5} T^{25} + 47060618620907168 p^{6} T^{26} + 711191350050545 p^{7} T^{27} + 402422635461418 p^{8} T^{28} + 5940568696023 p^{9} T^{29} + 2936371625912 p^{10} T^{30} + 42008911292 p^{11} T^{31} + 17867342220 p^{12} T^{32} + 242424263 p^{13} T^{33} + 87431704 p^{14} T^{34} + 1074418 p^{15} T^{35} + 323509 p^{16} T^{36} + 3251 p^{17} T^{37} + 804 p^{18} T^{38} + 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 40 T + 1811 T^{2} - 48477 T^{3} + 1319282 T^{4} - 27369850 T^{5} + 564440865 T^{6} - 9727465556 T^{7} + 165765052229 T^{8} - 2469431623547 T^{9} + 36363007841360 T^{10} - 480299833255876 T^{11} + 79475945000583 p T^{12} - 74807860002705982 T^{13} + 883599841612642101 T^{14} - 9611050821905146937 T^{15} + \)\(10\!\cdots\!80\)\( T^{16} - \)\(10\!\cdots\!14\)\( T^{17} + \)\(10\!\cdots\!83\)\( T^{18} - \)\(95\!\cdots\!23\)\( T^{19} + \)\(88\!\cdots\!94\)\( T^{20} - \)\(95\!\cdots\!23\)\( p T^{21} + \)\(10\!\cdots\!83\)\( p^{2} T^{22} - \)\(10\!\cdots\!14\)\( p^{3} T^{23} + \)\(10\!\cdots\!80\)\( p^{4} T^{24} - 9611050821905146937 p^{5} T^{25} + 883599841612642101 p^{6} T^{26} - 74807860002705982 p^{7} T^{27} + 79475945000583 p^{9} T^{28} - 480299833255876 p^{9} T^{29} + 36363007841360 p^{10} T^{30} - 2469431623547 p^{11} T^{31} + 165765052229 p^{12} T^{32} - 9727465556 p^{13} T^{33} + 564440865 p^{14} T^{34} - 27369850 p^{15} T^{35} + 1319282 p^{16} T^{36} - 48477 p^{17} T^{37} + 1811 p^{18} T^{38} - 40 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 10 T + 938 T^{2} - 8915 T^{3} + 437497 T^{4} - 4047595 T^{5} + 135198370 T^{6} - 1234894620 T^{7} + 31094638166 T^{8} - 282741954734 T^{9} + 5672941324764 T^{10} - 51540855872926 T^{11} + 855297468978862 T^{12} - 7757780277247158 T^{13} + 109698914421900540 T^{14} - 987728555306949406 T^{15} + 12229665929165323865 T^{16} - \)\(10\!\cdots\!02\)\( T^{17} + \)\(12\!\cdots\!04\)\( T^{18} - \)\(10\!\cdots\!60\)\( T^{19} + \)\(10\!\cdots\!94\)\( T^{20} - \)\(10\!\cdots\!60\)\( p T^{21} + \)\(12\!\cdots\!04\)\( p^{2} T^{22} - \)\(10\!\cdots\!02\)\( p^{3} T^{23} + 12229665929165323865 p^{4} T^{24} - 987728555306949406 p^{5} T^{25} + 109698914421900540 p^{6} T^{26} - 7757780277247158 p^{7} T^{27} + 855297468978862 p^{8} T^{28} - 51540855872926 p^{9} T^{29} + 5672941324764 p^{10} T^{30} - 282741954734 p^{11} T^{31} + 31094638166 p^{12} T^{32} - 1234894620 p^{13} T^{33} + 135198370 p^{14} T^{34} - 4047595 p^{15} T^{35} + 437497 p^{16} T^{36} - 8915 p^{17} T^{37} + 938 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 17 T + 846 T^{2} - 13042 T^{3} + 368030 T^{4} - 5072331 T^{5} + 107208364 T^{6} - 1327865531 T^{7} + 23321848051 T^{8} - 261915573704 T^{9} + 4032316846037 T^{10} - 41403697198507 T^{11} + 578136461074171 T^{12} - 5471945286248959 T^{13} + 71086330842501118 T^{14} - 626350947584957666 T^{15} + 7727728719959675218 T^{16} - 64202582847234225602 T^{17} + \)\(76\!\cdots\!23\)\( T^{18} - \)\(60\!\cdots\!61\)\( T^{19} + \)\(70\!\cdots\!34\)\( T^{20} - \)\(60\!\cdots\!61\)\( p T^{21} + \)\(76\!\cdots\!23\)\( p^{2} T^{22} - 64202582847234225602 p^{3} T^{23} + 7727728719959675218 p^{4} T^{24} - 626350947584957666 p^{5} T^{25} + 71086330842501118 p^{6} T^{26} - 5471945286248959 p^{7} T^{27} + 578136461074171 p^{8} T^{28} - 41403697198507 p^{9} T^{29} + 4032316846037 p^{10} T^{30} - 261915573704 p^{11} T^{31} + 23321848051 p^{12} T^{32} - 1327865531 p^{13} T^{33} + 107208364 p^{14} T^{34} - 5072331 p^{15} T^{35} + 368030 p^{16} T^{36} - 13042 p^{17} T^{37} + 846 p^{18} T^{38} - 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 2 T + 909 T^{2} + 2372 T^{3} + 414784 T^{4} + 1241355 T^{5} + 128264569 T^{6} + 402347181 T^{7} + 30420162456 T^{8} + 94138802251 T^{9} + 5901608783321 T^{10} + 17396802161192 T^{11} + 972566457145183 T^{12} + 27701833079543 p T^{13} + 139522059303651822 T^{14} + 359592587249436223 T^{15} + 17715158909625088629 T^{16} + 42681247133183014345 T^{17} + \)\(20\!\cdots\!75\)\( T^{18} + \)\(45\!\cdots\!08\)\( T^{19} + \)\(20\!\cdots\!22\)\( T^{20} + \)\(45\!\cdots\!08\)\( p T^{21} + \)\(20\!\cdots\!75\)\( p^{2} T^{22} + 42681247133183014345 p^{3} T^{23} + 17715158909625088629 p^{4} T^{24} + 359592587249436223 p^{5} T^{25} + 139522059303651822 p^{6} T^{26} + 27701833079543 p^{8} T^{27} + 972566457145183 p^{8} T^{28} + 17396802161192 p^{9} T^{29} + 5901608783321 p^{10} T^{30} + 94138802251 p^{11} T^{31} + 30420162456 p^{12} T^{32} + 402347181 p^{13} T^{33} + 128264569 p^{14} T^{34} + 1241355 p^{15} T^{35} + 414784 p^{16} T^{36} + 2372 p^{17} T^{37} + 909 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−1.73029958987328197491649812994, −1.70914131474761651252852242654, −1.63852898850977037978643357840, −1.38194892723015730939017018321, −1.33833354440766041077971234667, −1.24031538323172862260975826496, −1.22611323990696017491648358568, −1.16566686397888874461010076703, −1.08853994531427669139110634633, −1.04448704838976984587362053117, −1.02885049409031313938020464006, −1.00055850149985688035798482302, −0.968985558252578111790719246672, −0.967662436879940794729309554136, −0.865040811756209858316726757438, −0.801945806107209832441291280493, −0.71969028736183347835621649904, −0.70071711878840978766869378702, −0.68213659704272313313424854261, −0.59384383696360518239275427839, −0.55951541197316299594621925251, −0.54106577049150719717688651410, −0.39534080498663190999548091850, −0.31974390989821635180494324565, −0.17775921658594464433021962390, 0.17775921658594464433021962390, 0.31974390989821635180494324565, 0.39534080498663190999548091850, 0.54106577049150719717688651410, 0.55951541197316299594621925251, 0.59384383696360518239275427839, 0.68213659704272313313424854261, 0.70071711878840978766869378702, 0.71969028736183347835621649904, 0.801945806107209832441291280493, 0.865040811756209858316726757438, 0.967662436879940794729309554136, 0.968985558252578111790719246672, 1.00055850149985688035798482302, 1.02885049409031313938020464006, 1.04448704838976984587362053117, 1.08853994531427669139110634633, 1.16566686397888874461010076703, 1.22611323990696017491648358568, 1.24031538323172862260975826496, 1.33833354440766041077971234667, 1.38194892723015730939017018321, 1.63852898850977037978643357840, 1.70914131474761651252852242654, 1.73029958987328197491649812994
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.