Dirichlet series
| L(s) = 1 | − 5·2-s + 3-s + 4·4-s + 3·5-s − 5·6-s + 2·7-s + 22·8-s − 16·9-s − 15·10-s + 21·11-s + 4·12-s + 11·13-s − 10·14-s + 3·15-s − 50·16-s + 15·17-s + 80·18-s + 7·19-s + 12·20-s + 2·21-s − 105·22-s + 26·23-s + 22·24-s − 40·25-s − 55·26-s − 10·27-s + 8·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 0.577·3-s + 2·4-s + 1.34·5-s − 2.04·6-s + 0.755·7-s + 7.77·8-s − 5.33·9-s − 4.74·10-s + 6.33·11-s + 1.15·12-s + 3.05·13-s − 2.67·14-s + 0.774·15-s − 12.5·16-s + 3.63·17-s + 18.8·18-s + 1.60·19-s + 2.68·20-s + 0.436·21-s − 22.3·22-s + 5.42·23-s + 4.49·24-s − 8·25-s − 10.7·26-s − 1.92·27-s + 1.51·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(43^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.42259\times 10^{23}\) |
| Root analytic conductor: | \(3.84243\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 43^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(84.85733651\) |
| \(L(\frac12)\) | \(\approx\) | \(84.85733651\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 5 T + 21 T^{2} + 63 T^{3} + 171 T^{4} + 395 T^{5} + 861 T^{6} + 1697 T^{7} + 101 p^{5} T^{8} + 2865 p T^{9} + 9969 T^{10} + 8231 p T^{11} + 27043 T^{12} + 1337 p^{5} T^{13} + 67961 T^{14} + 104635 T^{15} + 80965 p T^{16} + 121325 p T^{17} + 181939 p T^{18} + 526409 T^{19} + 758669 T^{20} + 526409 p T^{21} + 181939 p^{3} T^{22} + 121325 p^{4} T^{23} + 80965 p^{5} T^{24} + 104635 p^{5} T^{25} + 67961 p^{6} T^{26} + 1337 p^{12} T^{27} + 27043 p^{8} T^{28} + 8231 p^{10} T^{29} + 9969 p^{10} T^{30} + 2865 p^{12} T^{31} + 101 p^{17} T^{32} + 1697 p^{13} T^{33} + 861 p^{14} T^{34} + 395 p^{15} T^{35} + 171 p^{16} T^{36} + 63 p^{17} T^{37} + 21 p^{18} T^{38} + 5 p^{19} T^{39} + p^{20} T^{40} \) |
| 3 | \( 1 - T + 17 T^{2} - 23 T^{3} + 167 T^{4} - 232 T^{5} + 1222 T^{6} - 1630 T^{7} + 2389 p T^{8} - 3068 p T^{9} + 35773 T^{10} - 4862 p^{2} T^{11} + 17462 p^{2} T^{12} - 182474 T^{13} + 206216 p T^{14} - 76081 p^{2} T^{15} + 2226544 T^{16} - 784292 p T^{17} + 7416388 T^{18} - 7539203 T^{19} + 23008870 T^{20} - 7539203 p T^{21} + 7416388 p^{2} T^{22} - 784292 p^{4} T^{23} + 2226544 p^{4} T^{24} - 76081 p^{7} T^{25} + 206216 p^{7} T^{26} - 182474 p^{7} T^{27} + 17462 p^{10} T^{28} - 4862 p^{11} T^{29} + 35773 p^{10} T^{30} - 3068 p^{12} T^{31} + 2389 p^{13} T^{32} - 1630 p^{13} T^{33} + 1222 p^{14} T^{34} - 232 p^{15} T^{35} + 167 p^{16} T^{36} - 23 p^{17} T^{37} + 17 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) | |
| 5 | \( 1 - 3 T + 49 T^{2} - 144 T^{3} + 1263 T^{4} - 3607 T^{5} + 22534 T^{6} - 61984 T^{7} + 309298 T^{8} - 162742 p T^{9} + 689623 p T^{10} - 345426 p^{2} T^{11} + 32236817 T^{12} - 76617151 T^{13} + 257923006 T^{14} - 580352696 T^{15} + 357900278 p T^{16} - 3803096299 T^{17} + 10857165918 T^{18} - 21729721248 T^{19} + 57872565353 T^{20} - 21729721248 p T^{21} + 10857165918 p^{2} T^{22} - 3803096299 p^{3} T^{23} + 357900278 p^{5} T^{24} - 580352696 p^{5} T^{25} + 257923006 p^{6} T^{26} - 76617151 p^{7} T^{27} + 32236817 p^{8} T^{28} - 345426 p^{11} T^{29} + 689623 p^{11} T^{30} - 162742 p^{12} T^{31} + 309298 p^{12} T^{32} - 61984 p^{13} T^{33} + 22534 p^{14} T^{34} - 3607 p^{15} T^{35} + 1263 p^{16} T^{36} - 144 p^{17} T^{37} + 49 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 2 T + 62 T^{2} - 65 T^{3} + 1889 T^{4} - 519 T^{5} + 39331 T^{6} + 15569 T^{7} + 91388 p T^{8} + 616487 T^{9} + 8694547 T^{10} + 12252311 T^{11} + 14613658 p T^{12} + 176127090 T^{13} + 1061310639 T^{14} + 288773085 p T^{15} + 9802789013 T^{16} + 19335318101 T^{17} + 80959704741 T^{18} + 157473424267 T^{19} + 598780926318 T^{20} + 157473424267 p T^{21} + 80959704741 p^{2} T^{22} + 19335318101 p^{3} T^{23} + 9802789013 p^{4} T^{24} + 288773085 p^{6} T^{25} + 1061310639 p^{6} T^{26} + 176127090 p^{7} T^{27} + 14613658 p^{9} T^{28} + 12252311 p^{9} T^{29} + 8694547 p^{10} T^{30} + 616487 p^{11} T^{31} + 91388 p^{13} T^{32} + 15569 p^{13} T^{33} + 39331 p^{14} T^{34} - 519 p^{15} T^{35} + 1889 p^{16} T^{36} - 65 p^{17} T^{37} + 62 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 21 T + 332 T^{2} - 3830 T^{3} + 37841 T^{4} - 320137 T^{5} + 2436666 T^{6} - 16742723 T^{7} + 106159161 T^{8} - 623189861 T^{9} + 311540898 p T^{10} - 17699462467 T^{11} + 7859993283 p T^{12} - 400240623516 T^{13} + 1763631303304 T^{14} - 673392366347 p T^{15} + 29739414733198 T^{16} - 114225849306908 T^{17} + 420481686442108 T^{18} - 1483819660190874 T^{19} + 5024676407893132 T^{20} - 1483819660190874 p T^{21} + 420481686442108 p^{2} T^{22} - 114225849306908 p^{3} T^{23} + 29739414733198 p^{4} T^{24} - 673392366347 p^{6} T^{25} + 1763631303304 p^{6} T^{26} - 400240623516 p^{7} T^{27} + 7859993283 p^{9} T^{28} - 17699462467 p^{9} T^{29} + 311540898 p^{11} T^{30} - 623189861 p^{11} T^{31} + 106159161 p^{12} T^{32} - 16742723 p^{13} T^{33} + 2436666 p^{14} T^{34} - 320137 p^{15} T^{35} + 37841 p^{16} T^{36} - 3830 p^{17} T^{37} + 332 p^{18} T^{38} - 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 - 11 T + 196 T^{2} - 1713 T^{3} + 18160 T^{4} - 133662 T^{5} + 1079881 T^{6} - 6932011 T^{7} + 46713697 T^{8} - 267489078 T^{9} + 1571773670 T^{10} - 8150101646 T^{11} + 42825515208 T^{12} - 203146993039 T^{13} + 969462848845 T^{14} - 4235770049611 T^{15} + 18539666968204 T^{16} - 74930665858929 T^{17} + 302660024363703 T^{18} - 1134071638747758 T^{19} + 4242290418159309 T^{20} - 1134071638747758 p T^{21} + 302660024363703 p^{2} T^{22} - 74930665858929 p^{3} T^{23} + 18539666968204 p^{4} T^{24} - 4235770049611 p^{5} T^{25} + 969462848845 p^{6} T^{26} - 203146993039 p^{7} T^{27} + 42825515208 p^{8} T^{28} - 8150101646 p^{9} T^{29} + 1571773670 p^{10} T^{30} - 267489078 p^{11} T^{31} + 46713697 p^{12} T^{32} - 6932011 p^{13} T^{33} + 1079881 p^{14} T^{34} - 133662 p^{15} T^{35} + 18160 p^{16} T^{36} - 1713 p^{17} T^{37} + 196 p^{18} T^{38} - 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 15 T + 313 T^{2} - 3573 T^{3} + 44424 T^{4} - 414231 T^{5} + 3917027 T^{6} - 31128563 T^{7} + 244617241 T^{8} - 1703384965 T^{9} + 11619991396 T^{10} - 72257120664 T^{11} + 438711655391 T^{12} - 2468350927685 T^{13} + 13546454355779 T^{14} - 69584613261749 T^{15} + 348603683348992 T^{16} - 1644463538759774 T^{17} + 7566533178075647 T^{18} - 32888168110340244 T^{19} + 139446945683457267 T^{20} - 32888168110340244 p T^{21} + 7566533178075647 p^{2} T^{22} - 1644463538759774 p^{3} T^{23} + 348603683348992 p^{4} T^{24} - 69584613261749 p^{5} T^{25} + 13546454355779 p^{6} T^{26} - 2468350927685 p^{7} T^{27} + 438711655391 p^{8} T^{28} - 72257120664 p^{9} T^{29} + 11619991396 p^{10} T^{30} - 1703384965 p^{11} T^{31} + 244617241 p^{12} T^{32} - 31128563 p^{13} T^{33} + 3917027 p^{14} T^{34} - 414231 p^{15} T^{35} + 44424 p^{16} T^{36} - 3573 p^{17} T^{37} + 313 p^{18} T^{38} - 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 7 T + 231 T^{2} - 1374 T^{3} + 25024 T^{4} - 130485 T^{5} + 1715592 T^{6} - 8012436 T^{7} + 84195440 T^{8} - 357387230 T^{9} + 3159874579 T^{10} - 12319711684 T^{11} + 94598235142 T^{12} - 342087763955 T^{13} + 2339303113735 T^{14} - 7951572377870 T^{15} + 49783554013471 T^{16} - 162524198764971 T^{17} + 966128590282455 T^{18} - 3115875295016196 T^{19} + 18283539240533924 T^{20} - 3115875295016196 p T^{21} + 966128590282455 p^{2} T^{22} - 162524198764971 p^{3} T^{23} + 49783554013471 p^{4} T^{24} - 7951572377870 p^{5} T^{25} + 2339303113735 p^{6} T^{26} - 342087763955 p^{7} T^{27} + 94598235142 p^{8} T^{28} - 12319711684 p^{9} T^{29} + 3159874579 p^{10} T^{30} - 357387230 p^{11} T^{31} + 84195440 p^{12} T^{32} - 8012436 p^{13} T^{33} + 1715592 p^{14} T^{34} - 130485 p^{15} T^{35} + 25024 p^{16} T^{36} - 1374 p^{17} T^{37} + 231 p^{18} T^{38} - 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 26 T + 593 T^{2} - 9245 T^{3} + 129577 T^{4} - 1511139 T^{5} + 16221238 T^{6} - 154957328 T^{7} + 1384484378 T^{8} - 11370338208 T^{9} + 88282943166 T^{10} - 641718697525 T^{11} + 4442215536019 T^{12} - 29114088050089 T^{13} + 182617431976691 T^{14} - 1092252819275905 T^{15} + 6272762653585341 T^{16} - 34502594610117177 T^{17} + 182584167899486144 T^{18} - 927641876546830932 T^{19} + 4538624829212785064 T^{20} - 927641876546830932 p T^{21} + 182584167899486144 p^{2} T^{22} - 34502594610117177 p^{3} T^{23} + 6272762653585341 p^{4} T^{24} - 1092252819275905 p^{5} T^{25} + 182617431976691 p^{6} T^{26} - 29114088050089 p^{7} T^{27} + 4442215536019 p^{8} T^{28} - 641718697525 p^{9} T^{29} + 88282943166 p^{10} T^{30} - 11370338208 p^{11} T^{31} + 1384484378 p^{12} T^{32} - 154957328 p^{13} T^{33} + 16221238 p^{14} T^{34} - 1511139 p^{15} T^{35} + 129577 p^{16} T^{36} - 9245 p^{17} T^{37} + 593 p^{18} T^{38} - 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 12 T + 389 T^{2} - 3720 T^{3} + 2377 p T^{4} - 541228 T^{5} + 7485382 T^{6} - 48633620 T^{7} + 560017432 T^{8} - 2961118542 T^{9} + 30502328697 T^{10} - 123368346582 T^{11} + 1233796038955 T^{12} - 3114560706046 T^{13} + 36613152683904 T^{14} - 7331957410626 T^{15} + 751283965485840 T^{16} + 3447445803430502 T^{17} + 9273961941957196 T^{18} + 180596486019813978 T^{19} + 99640287316737551 T^{20} + 180596486019813978 p T^{21} + 9273961941957196 p^{2} T^{22} + 3447445803430502 p^{3} T^{23} + 751283965485840 p^{4} T^{24} - 7331957410626 p^{5} T^{25} + 36613152683904 p^{6} T^{26} - 3114560706046 p^{7} T^{27} + 1233796038955 p^{8} T^{28} - 123368346582 p^{9} T^{29} + 30502328697 p^{10} T^{30} - 2961118542 p^{11} T^{31} + 560017432 p^{12} T^{32} - 48633620 p^{13} T^{33} + 7485382 p^{14} T^{34} - 541228 p^{15} T^{35} + 2377 p^{17} T^{36} - 3720 p^{17} T^{37} + 389 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 26 T + 699 T^{2} - 11988 T^{3} + 197159 T^{4} - 2606034 T^{5} + 32787948 T^{6} - 358975262 T^{7} + 3752975389 T^{8} - 35462352562 T^{9} + 321545092610 T^{10} - 2692085875236 T^{11} + 21727037022202 T^{12} - 164069990196176 T^{13} + 1198764838137908 T^{14} - 8264662661616974 T^{15} + 55280859892476778 T^{16} - 350744272756180416 T^{17} + 2162834271239151139 T^{18} - 12687257023224161930 T^{19} + 72392713215803297198 T^{20} - 12687257023224161930 p T^{21} + 2162834271239151139 p^{2} T^{22} - 350744272756180416 p^{3} T^{23} + 55280859892476778 p^{4} T^{24} - 8264662661616974 p^{5} T^{25} + 1198764838137908 p^{6} T^{26} - 164069990196176 p^{7} T^{27} + 21727037022202 p^{8} T^{28} - 2692085875236 p^{9} T^{29} + 321545092610 p^{10} T^{30} - 35462352562 p^{11} T^{31} + 3752975389 p^{12} T^{32} - 358975262 p^{13} T^{33} + 32787948 p^{14} T^{34} - 2606034 p^{15} T^{35} + 197159 p^{16} T^{36} - 11988 p^{17} T^{37} + 699 p^{18} T^{38} - 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 19 T + 674 T^{2} - 10602 T^{3} + 214944 T^{4} - 2880765 T^{5} + 43380980 T^{6} - 506355490 T^{7} + 6241509967 T^{8} - 64506815954 T^{9} + 682665477543 T^{10} - 6324666556184 T^{11} + 59031781806262 T^{12} - 494695786144745 T^{13} + 4139373674392274 T^{14} - 31576171482499992 T^{15} + 239318805423602520 T^{16} - 1668642371011655963 T^{17} + 11528995954659392273 T^{18} - 73635723896403709878 T^{19} + \)\(46\!\cdots\!96\)\( T^{20} - 73635723896403709878 p T^{21} + 11528995954659392273 p^{2} T^{22} - 1668642371011655963 p^{3} T^{23} + 239318805423602520 p^{4} T^{24} - 31576171482499992 p^{5} T^{25} + 4139373674392274 p^{6} T^{26} - 494695786144745 p^{7} T^{27} + 59031781806262 p^{8} T^{28} - 6324666556184 p^{9} T^{29} + 682665477543 p^{10} T^{30} - 64506815954 p^{11} T^{31} + 6241509967 p^{12} T^{32} - 506355490 p^{13} T^{33} + 43380980 p^{14} T^{34} - 2880765 p^{15} T^{35} + 214944 p^{16} T^{36} - 10602 p^{17} T^{37} + 674 p^{18} T^{38} - 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 27 T + 869 T^{2} - 16530 T^{3} + 319061 T^{4} - 116085 p T^{5} + 69410492 T^{6} - 858796296 T^{7} + 10290120942 T^{8} - 109274591678 T^{9} + 1122948542213 T^{10} - 10479929878806 T^{11} + 94920152010059 T^{12} - 793170034277493 T^{13} + 6465307666310432 T^{14} - 49195805087779524 T^{15} + 367588572508585940 T^{16} - 2588560044156107325 T^{17} + 18023329911574085292 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(78\!\cdots\!59\)\( T^{20} - \)\(11\!\cdots\!60\)\( p T^{21} + 18023329911574085292 p^{2} T^{22} - 2588560044156107325 p^{3} T^{23} + 367588572508585940 p^{4} T^{24} - 49195805087779524 p^{5} T^{25} + 6465307666310432 p^{6} T^{26} - 793170034277493 p^{7} T^{27} + 94920152010059 p^{8} T^{28} - 10479929878806 p^{9} T^{29} + 1122948542213 p^{10} T^{30} - 109274591678 p^{11} T^{31} + 10290120942 p^{12} T^{32} - 858796296 p^{13} T^{33} + 69410492 p^{14} T^{34} - 116085 p^{16} T^{35} + 319061 p^{16} T^{36} - 16530 p^{17} T^{37} + 869 p^{18} T^{38} - 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 45 T + 1511 T^{2} - 36937 T^{3} + 770808 T^{4} - 13736882 T^{5} + 219798824 T^{6} - 67393142 p T^{7} + 42058010785 T^{8} - 516011995467 T^{9} + 5923697054548 T^{10} - 63764307510743 T^{11} + 648591155045327 T^{12} - 132836554451959 p T^{13} + 57164463799000625 T^{14} - 498296134307400984 T^{15} + 4149644163892787994 T^{16} - 33024657601896825176 T^{17} + \)\(25\!\cdots\!32\)\( T^{18} - \)\(18\!\cdots\!69\)\( T^{19} + \)\(12\!\cdots\!66\)\( T^{20} - \)\(18\!\cdots\!69\)\( p T^{21} + \)\(25\!\cdots\!32\)\( p^{2} T^{22} - 33024657601896825176 p^{3} T^{23} + 4149644163892787994 p^{4} T^{24} - 498296134307400984 p^{5} T^{25} + 57164463799000625 p^{6} T^{26} - 132836554451959 p^{8} T^{27} + 648591155045327 p^{8} T^{28} - 63764307510743 p^{9} T^{29} + 5923697054548 p^{10} T^{30} - 516011995467 p^{11} T^{31} + 42058010785 p^{12} T^{32} - 67393142 p^{14} T^{33} + 219798824 p^{14} T^{34} - 13736882 p^{15} T^{35} + 770808 p^{16} T^{36} - 36937 p^{17} T^{37} + 1511 p^{18} T^{38} - 45 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 - 3 T + 613 T^{2} - 1912 T^{3} + 186977 T^{4} - 607031 T^{5} + 37799602 T^{6} - 127535552 T^{7} + 5694013788 T^{8} - 19875117064 T^{9} + 681406942949 T^{10} - 2441370740336 T^{11} + 67442970127123 T^{12} - 245242274964199 T^{13} + 5672683623486178 T^{14} - 20630924101460484 T^{15} + 413096049704419240 T^{16} - 1476192887327507123 T^{17} + 26367839384243376638 T^{18} - 90699112484413664918 T^{19} + \)\(14\!\cdots\!27\)\( T^{20} - 90699112484413664918 p T^{21} + 26367839384243376638 p^{2} T^{22} - 1476192887327507123 p^{3} T^{23} + 413096049704419240 p^{4} T^{24} - 20630924101460484 p^{5} T^{25} + 5672683623486178 p^{6} T^{26} - 245242274964199 p^{7} T^{27} + 67442970127123 p^{8} T^{28} - 2441370740336 p^{9} T^{29} + 681406942949 p^{10} T^{30} - 19875117064 p^{11} T^{31} + 5694013788 p^{12} T^{32} - 127535552 p^{13} T^{33} + 37799602 p^{14} T^{34} - 607031 p^{15} T^{35} + 186977 p^{16} T^{36} - 1912 p^{17} T^{37} + 613 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 66 T + 2815 T^{2} - 88361 T^{3} + 2275828 T^{4} - 49883249 T^{5} + 962488724 T^{6} - 16630945847 T^{7} + 261502441352 T^{8} - 3778845268886 T^{9} + 50649777859055 T^{10} - 633662617235908 T^{11} + 7443048277471240 T^{12} - 82425509095651651 T^{13} + 863910544933685903 T^{14} - 8593161165219341990 T^{15} + 81320010966712563121 T^{16} - \)\(73\!\cdots\!79\)\( T^{17} + \)\(63\!\cdots\!87\)\( T^{18} - \)\(51\!\cdots\!89\)\( T^{19} + \)\(40\!\cdots\!16\)\( T^{20} - \)\(51\!\cdots\!89\)\( p T^{21} + \)\(63\!\cdots\!87\)\( p^{2} T^{22} - \)\(73\!\cdots\!79\)\( p^{3} T^{23} + 81320010966712563121 p^{4} T^{24} - 8593161165219341990 p^{5} T^{25} + 863910544933685903 p^{6} T^{26} - 82425509095651651 p^{7} T^{27} + 7443048277471240 p^{8} T^{28} - 633662617235908 p^{9} T^{29} + 50649777859055 p^{10} T^{30} - 3778845268886 p^{11} T^{31} + 261502441352 p^{12} T^{32} - 16630945847 p^{13} T^{33} + 962488724 p^{14} T^{34} - 49883249 p^{15} T^{35} + 2275828 p^{16} T^{36} - 88361 p^{17} T^{37} + 2815 p^{18} T^{38} - 66 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 30 T + 922 T^{2} + 17978 T^{3} + 345594 T^{4} + 5305743 T^{5} + 80142022 T^{6} + 1050374877 T^{7} + 13559425579 T^{8} + 158226315366 T^{9} + 1819771079659 T^{10} + 19368804087758 T^{11} + 203166699504050 T^{12} + 2001054980174280 T^{13} + 19412282312992375 T^{14} + 178471358955453852 T^{15} + 1615021614573728110 T^{16} + 13931247955336092241 T^{17} + \)\(11\!\cdots\!07\)\( T^{18} + \)\(95\!\cdots\!05\)\( T^{19} + \)\(76\!\cdots\!03\)\( T^{20} + \)\(95\!\cdots\!05\)\( p T^{21} + \)\(11\!\cdots\!07\)\( p^{2} T^{22} + 13931247955336092241 p^{3} T^{23} + 1615021614573728110 p^{4} T^{24} + 178471358955453852 p^{5} T^{25} + 19412282312992375 p^{6} T^{26} + 2001054980174280 p^{7} T^{27} + 203166699504050 p^{8} T^{28} + 19368804087758 p^{9} T^{29} + 1819771079659 p^{10} T^{30} + 158226315366 p^{11} T^{31} + 13559425579 p^{12} T^{32} + 1050374877 p^{13} T^{33} + 80142022 p^{14} T^{34} + 5305743 p^{15} T^{35} + 345594 p^{16} T^{36} + 17978 p^{17} T^{37} + 922 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 6 T + 713 T^{2} + 4865 T^{3} + 256067 T^{4} + 1917283 T^{5} + 61955432 T^{6} + 492179942 T^{7} + 11362573071 T^{8} + 93015796636 T^{9} + 1680301496756 T^{10} + 13848234931211 T^{11} + 207778921318378 T^{12} + 1693635064497520 T^{13} + 21986354136885960 T^{14} + 174788972094032572 T^{15} + 2021682240803889472 T^{16} + 15486466214090967294 T^{17} + \)\(16\!\cdots\!83\)\( T^{18} + \)\(11\!\cdots\!21\)\( T^{19} + \)\(11\!\cdots\!62\)\( T^{20} + \)\(11\!\cdots\!21\)\( p T^{21} + \)\(16\!\cdots\!83\)\( p^{2} T^{22} + 15486466214090967294 p^{3} T^{23} + 2021682240803889472 p^{4} T^{24} + 174788972094032572 p^{5} T^{25} + 21986354136885960 p^{6} T^{26} + 1693635064497520 p^{7} T^{27} + 207778921318378 p^{8} T^{28} + 13848234931211 p^{9} T^{29} + 1680301496756 p^{10} T^{30} + 93015796636 p^{11} T^{31} + 11362573071 p^{12} T^{32} + 492179942 p^{13} T^{33} + 61955432 p^{14} T^{34} + 1917283 p^{15} T^{35} + 256067 p^{16} T^{36} + 4865 p^{17} T^{37} + 713 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 31 T + 1322 T^{2} - 31320 T^{3} + 798364 T^{4} - 15452678 T^{5} + 299944301 T^{6} - 4942514823 T^{7} + 79706547295 T^{8} - 1148868791647 T^{9} + 16066350850621 T^{10} - 206368868523456 T^{11} + 2564945996247701 T^{12} - 29747128248307392 T^{13} + 333708285975141834 T^{14} - 3526768418326305726 T^{15} + 36069118289707342370 T^{16} - \)\(34\!\cdots\!13\)\( T^{17} + \)\(32\!\cdots\!14\)\( T^{18} - \)\(29\!\cdots\!04\)\( T^{19} + \)\(25\!\cdots\!70\)\( T^{20} - \)\(29\!\cdots\!04\)\( p T^{21} + \)\(32\!\cdots\!14\)\( p^{2} T^{22} - \)\(34\!\cdots\!13\)\( p^{3} T^{23} + 36069118289707342370 p^{4} T^{24} - 3526768418326305726 p^{5} T^{25} + 333708285975141834 p^{6} T^{26} - 29747128248307392 p^{7} T^{27} + 2564945996247701 p^{8} T^{28} - 206368868523456 p^{9} T^{29} + 16066350850621 p^{10} T^{30} - 1148868791647 p^{11} T^{31} + 79706547295 p^{12} T^{32} - 4942514823 p^{13} T^{33} + 299944301 p^{14} T^{34} - 15452678 p^{15} T^{35} + 798364 p^{16} T^{36} - 31320 p^{17} T^{37} + 1322 p^{18} T^{38} - 31 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 26 T + 1291 T^{2} - 27722 T^{3} + 783640 T^{4} - 14380525 T^{5} + 300791939 T^{6} - 4827381656 T^{7} + 82453306014 T^{8} - 1176453930867 T^{9} + 17238057078813 T^{10} - 221303711720328 T^{11} + 2861428005828764 T^{12} - 33345380649687651 T^{13} + 387137229185140814 T^{14} - 4120923032974543647 T^{15} + 43437310660293055700 T^{16} - \)\(42\!\cdots\!70\)\( T^{17} + \)\(40\!\cdots\!62\)\( T^{18} - \)\(36\!\cdots\!52\)\( T^{19} + \)\(32\!\cdots\!57\)\( T^{20} - \)\(36\!\cdots\!52\)\( p T^{21} + \)\(40\!\cdots\!62\)\( p^{2} T^{22} - \)\(42\!\cdots\!70\)\( p^{3} T^{23} + 43437310660293055700 p^{4} T^{24} - 4120923032974543647 p^{5} T^{25} + 387137229185140814 p^{6} T^{26} - 33345380649687651 p^{7} T^{27} + 2861428005828764 p^{8} T^{28} - 221303711720328 p^{9} T^{29} + 17238057078813 p^{10} T^{30} - 1176453930867 p^{11} T^{31} + 82453306014 p^{12} T^{32} - 4827381656 p^{13} T^{33} + 300791939 p^{14} T^{34} - 14380525 p^{15} T^{35} + 783640 p^{16} T^{36} - 27722 p^{17} T^{37} + 1291 p^{18} T^{38} - 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 39 T + 1449 T^{2} - 35558 T^{3} + 812562 T^{4} - 15192159 T^{5} + 267563164 T^{6} - 4155506201 T^{7} + 61579924350 T^{8} - 834905764293 T^{9} + 10910568449802 T^{10} - 133193600438580 T^{11} + 1578428678113741 T^{12} - 17695120582970783 T^{13} + 193525885871052352 T^{14} - 2018288570921152211 T^{15} + 20606670235364380219 T^{16} - \)\(20\!\cdots\!02\)\( T^{17} + \)\(19\!\cdots\!13\)\( T^{18} - \)\(17\!\cdots\!42\)\( T^{19} + \)\(16\!\cdots\!26\)\( T^{20} - \)\(17\!\cdots\!42\)\( p T^{21} + \)\(19\!\cdots\!13\)\( p^{2} T^{22} - \)\(20\!\cdots\!02\)\( p^{3} T^{23} + 20606670235364380219 p^{4} T^{24} - 2018288570921152211 p^{5} T^{25} + 193525885871052352 p^{6} T^{26} - 17695120582970783 p^{7} T^{27} + 1578428678113741 p^{8} T^{28} - 133193600438580 p^{9} T^{29} + 10910568449802 p^{10} T^{30} - 834905764293 p^{11} T^{31} + 61579924350 p^{12} T^{32} - 4155506201 p^{13} T^{33} + 267563164 p^{14} T^{34} - 15192159 p^{15} T^{35} + 812562 p^{16} T^{36} - 35558 p^{17} T^{37} + 1449 p^{18} T^{38} - 39 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 13 T + 940 T^{2} - 11084 T^{3} + 440421 T^{4} - 4812065 T^{5} + 137986658 T^{6} - 1414251517 T^{7} + 32565902809 T^{8} - 315087302047 T^{9} + 6164011043080 T^{10} - 56448298149495 T^{11} + 970896449158067 T^{12} - 8418999649219894 T^{13} + 1568967282549500 p T^{14} - 1068214850843374843 T^{15} + 15093540496126075108 T^{16} - \)\(11\!\cdots\!18\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} - \)\(11\!\cdots\!02\)\( T^{19} + \)\(13\!\cdots\!36\)\( T^{20} - \)\(11\!\cdots\!02\)\( p T^{21} + \)\(15\!\cdots\!46\)\( p^{2} T^{22} - \)\(11\!\cdots\!18\)\( p^{3} T^{23} + 15093540496126075108 p^{4} T^{24} - 1068214850843374843 p^{5} T^{25} + 1568967282549500 p^{7} T^{26} - 8418999649219894 p^{7} T^{27} + 970896449158067 p^{8} T^{28} - 56448298149495 p^{9} T^{29} + 6164011043080 p^{10} T^{30} - 315087302047 p^{11} T^{31} + 32565902809 p^{12} T^{32} - 1414251517 p^{13} T^{33} + 137986658 p^{14} T^{34} - 4812065 p^{15} T^{35} + 440421 p^{16} T^{36} - 11084 p^{17} T^{37} + 940 p^{18} T^{38} - 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 4 T + 1028 T^{2} - 4088 T^{3} + 523306 T^{4} - 2066896 T^{5} + 176115864 T^{6} - 687204558 T^{7} + 44130361600 T^{8} - 168664694084 T^{9} + 8788135497517 T^{10} - 32564477960810 T^{11} + 1448948480228385 T^{12} - 5153430855025836 T^{13} + 203304254976165983 T^{14} - 688359262903258760 T^{15} + 24738386446626412881 T^{16} - 79304421418613845292 T^{17} + \)\(26\!\cdots\!32\)\( T^{18} - \)\(79\!\cdots\!92\)\( T^{19} + \)\(24\!\cdots\!25\)\( T^{20} - \)\(79\!\cdots\!92\)\( p T^{21} + \)\(26\!\cdots\!32\)\( p^{2} T^{22} - 79304421418613845292 p^{3} T^{23} + 24738386446626412881 p^{4} T^{24} - 688359262903258760 p^{5} T^{25} + 203304254976165983 p^{6} T^{26} - 5153430855025836 p^{7} T^{27} + 1448948480228385 p^{8} T^{28} - 32564477960810 p^{9} T^{29} + 8788135497517 p^{10} T^{30} - 168664694084 p^{11} T^{31} + 44130361600 p^{12} T^{32} - 687204558 p^{13} T^{33} + 176115864 p^{14} T^{34} - 2066896 p^{15} T^{35} + 523306 p^{16} T^{36} - 4088 p^{17} T^{37} + 1028 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 2 T + 1189 T^{2} + 3583 T^{3} + 696112 T^{4} + 2780155 T^{5} + 268164947 T^{6} + 1317002885 T^{7} + 76580507893 T^{8} + 438837705949 T^{9} + 17300229878303 T^{10} + 110945920301791 T^{11} + 3218716322383712 T^{12} + 22271189365451886 T^{13} + 506456339637067704 T^{14} + 3652951957138677156 T^{15} + 68610447555198868641 T^{16} + \)\(49\!\cdots\!58\)\( T^{17} + \)\(80\!\cdots\!66\)\( T^{18} + \)\(57\!\cdots\!15\)\( T^{19} + \)\(83\!\cdots\!29\)\( T^{20} + \)\(57\!\cdots\!15\)\( p T^{21} + \)\(80\!\cdots\!66\)\( p^{2} T^{22} + \)\(49\!\cdots\!58\)\( p^{3} T^{23} + 68610447555198868641 p^{4} T^{24} + 3652951957138677156 p^{5} T^{25} + 506456339637067704 p^{6} T^{26} + 22271189365451886 p^{7} T^{27} + 3218716322383712 p^{8} T^{28} + 110945920301791 p^{9} T^{29} + 17300229878303 p^{10} T^{30} + 438837705949 p^{11} T^{31} + 76580507893 p^{12} T^{32} + 1317002885 p^{13} T^{33} + 268164947 p^{14} T^{34} + 2780155 p^{15} T^{35} + 696112 p^{16} T^{36} + 3583 p^{17} T^{37} + 1189 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−1.84955437748766934474076940840, −1.81899723996019667042294949756, −1.77802480389169138121023281421, −1.61647241201005553306279457376, −1.57326109122745137607234605100, −1.56371472572697813657947335912, −1.56235303906004159397483735034, −1.39908032929780423650383653962, −1.31598956612876023637526308771, −1.30606760638705983726352778001, −1.25944455751197198942832090575, −1.04947310905042239760553806900, −0.914756205192393253912993678609, −0.813651541516450602353860340115, −0.804356276545355559601070410399, −0.801977571939337646201486381293, −0.796729432199634871335599822437, −0.78827954107093305543494942803, −0.74337306341142718011422245714, −0.71935314128020327375392575197, −0.60833321310490738539897442199, −0.58316495248009369764445753991, −0.49317088632185569576052197214, −0.43270524972498343399365080946, −0.22422670711372285098967437004, 0.22422670711372285098967437004, 0.43270524972498343399365080946, 0.49317088632185569576052197214, 0.58316495248009369764445753991, 0.60833321310490738539897442199, 0.71935314128020327375392575197, 0.74337306341142718011422245714, 0.78827954107093305543494942803, 0.796729432199634871335599822437, 0.801977571939337646201486381293, 0.804356276545355559601070410399, 0.813651541516450602353860340115, 0.914756205192393253912993678609, 1.04947310905042239760553806900, 1.25944455751197198942832090575, 1.30606760638705983726352778001, 1.31598956612876023637526308771, 1.39908032929780423650383653962, 1.56235303906004159397483735034, 1.56371472572697813657947335912, 1.57326109122745137607234605100, 1.61647241201005553306279457376, 1.77802480389169138121023281421, 1.81899723996019667042294949756, 1.84955437748766934474076940840
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.