Dirichlet series
| L(s) = 1 | − 4·3-s − 10·4-s + 2·5-s − 2·7-s + 8·9-s + 40·12-s − 8·15-s + 55·16-s + 20·17-s − 6·19-s − 20·20-s + 8·21-s − 3·25-s − 8·27-s + 20·28-s − 18·29-s + 12·31-s − 4·35-s − 80·36-s + 16·45-s − 22·47-s − 220·48-s + 2·49-s − 80·51-s − 4·53-s + 24·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 5·4-s + 0.894·5-s − 0.755·7-s + 8/3·9-s + 11.5·12-s − 2.06·15-s + 55/4·16-s + 4.85·17-s − 1.37·19-s − 4.47·20-s + 1.74·21-s − 3/5·25-s − 1.53·27-s + 3.77·28-s − 3.34·29-s + 2.15·31-s − 0.676·35-s − 13.3·36-s + 2.38·45-s − 3.20·47-s − 31.7·48-s + 2/7·49-s − 11.2·51-s − 0.549·53-s + 3.17·57-s + 1.30·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 37^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 37^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 5^{20} \cdot 37^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.56789\times 10^{9}\) |
| Root analytic conductor: | \(1.71885\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 5^{20} \cdot 37^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03821321021\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03821321021\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{10} \) |
| 5 | \( 1 - 2 T + 7 T^{2} - 8 T^{3} + 24 T^{4} - 22 T^{5} + 36 p T^{6} - 434 T^{7} + 831 T^{8} - 158 p T^{9} + 2298 T^{10} - 158 p^{2} T^{11} + 831 p^{2} T^{12} - 434 p^{3} T^{13} + 36 p^{5} T^{14} - 22 p^{5} T^{15} + 24 p^{6} T^{16} - 8 p^{7} T^{17} + 7 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 20 T^{2} - 96 T^{3} - 595 T^{4} - 7200 T^{5} - 23936 T^{6} - 54240 T^{7} - 208534 T^{8} + 15712608 T^{9} + 8354392 T^{10} + 15712608 p T^{11} - 208534 p^{2} T^{12} - 54240 p^{3} T^{13} - 23936 p^{4} T^{14} - 7200 p^{5} T^{15} - 595 p^{6} T^{16} - 96 p^{7} T^{17} + 20 p^{8} T^{18} + p^{10} T^{20} \) | |
| good | 3 | \( 1 + 4 T + 8 T^{2} + 8 T^{3} - 11 T^{4} - 62 T^{5} - 128 T^{6} - 116 T^{7} + 146 T^{8} + 274 p T^{9} + 1706 T^{10} + 692 p T^{11} + 1202 T^{12} - 1856 T^{13} - 7214 T^{14} - 15374 T^{15} - 21203 T^{16} - 9940 T^{17} + 44396 T^{18} + 177670 T^{19} + 369970 T^{20} + 177670 p T^{21} + 44396 p^{2} T^{22} - 9940 p^{3} T^{23} - 21203 p^{4} T^{24} - 15374 p^{5} T^{25} - 7214 p^{6} T^{26} - 1856 p^{7} T^{27} + 1202 p^{8} T^{28} + 692 p^{10} T^{29} + 1706 p^{10} T^{30} + 274 p^{12} T^{31} + 146 p^{12} T^{32} - 116 p^{13} T^{33} - 128 p^{14} T^{34} - 62 p^{15} T^{35} - 11 p^{16} T^{36} + 8 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) |
| 7 | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} - 13 T^{4} + 92 T^{5} + 44 p T^{6} + 788 T^{7} + 2433 T^{8} + 2942 T^{9} + 414 p^{2} T^{10} + 18402 T^{11} - 18980 T^{12} - 106 p T^{13} + 389594 T^{14} - 32602 T^{15} + 7367070 T^{16} + 5570858 T^{17} - 484366 p^{2} T^{18} + 3205786 p T^{19} - 347596702 T^{20} + 3205786 p^{2} T^{21} - 484366 p^{4} T^{22} + 5570858 p^{3} T^{23} + 7367070 p^{4} T^{24} - 32602 p^{5} T^{25} + 389594 p^{6} T^{26} - 106 p^{8} T^{27} - 18980 p^{8} T^{28} + 18402 p^{9} T^{29} + 414 p^{12} T^{30} + 2942 p^{11} T^{31} + 2433 p^{12} T^{32} + 788 p^{13} T^{33} + 44 p^{15} T^{34} + 92 p^{15} T^{35} - 13 p^{16} T^{36} + 2 p^{18} T^{37} + 2 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 108 T^{2} + 6246 T^{4} - 250586 T^{6} + 702435 p T^{8} - 193075702 T^{10} + 4034961265 T^{12} - 71942366094 T^{14} + 1108584758210 T^{16} - 14883472017910 T^{18} + 174879069451146 T^{20} - 14883472017910 p^{2} T^{22} + 1108584758210 p^{4} T^{24} - 71942366094 p^{6} T^{26} + 4034961265 p^{8} T^{28} - 193075702 p^{10} T^{30} + 702435 p^{13} T^{32} - 250586 p^{14} T^{34} + 6246 p^{16} T^{36} - 108 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 94 T^{2} + 4345 T^{4} - 128712 T^{6} + 2702694 T^{8} - 42913290 T^{10} + 568477842 T^{12} - 7650805254 T^{14} + 122229091545 T^{16} - 2037966245818 T^{18} + 29378039998186 T^{20} - 2037966245818 p^{2} T^{22} + 122229091545 p^{4} T^{24} - 7650805254 p^{6} T^{26} + 568477842 p^{8} T^{28} - 42913290 p^{10} T^{30} + 2702694 p^{12} T^{32} - 128712 p^{14} T^{34} + 4345 p^{16} T^{36} - 94 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( ( 1 - 10 T + 113 T^{2} - 634 T^{3} + 3863 T^{4} - 12680 T^{5} + 47188 T^{6} - 6064 T^{7} - 261400 T^{8} + 4034524 T^{9} - 14319242 T^{10} + 4034524 p T^{11} - 261400 p^{2} T^{12} - 6064 p^{3} T^{13} + 47188 p^{4} T^{14} - 12680 p^{5} T^{15} + 3863 p^{6} T^{16} - 634 p^{7} T^{17} + 113 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 + 6 T + 18 T^{2} - 14 T^{3} - 910 T^{4} - 3666 T^{5} - 5518 T^{6} + 18026 T^{7} + 366381 T^{8} + 1417104 T^{9} + 3353800 T^{10} + 7862448 T^{11} - 54140360 T^{12} - 414417120 T^{13} - 1581037080 T^{14} - 6408910240 T^{15} - 3629532958 T^{16} + 70133937612 T^{17} + 274674638172 T^{18} + 978584443684 T^{19} + 3359099901228 T^{20} + 978584443684 p T^{21} + 274674638172 p^{2} T^{22} + 70133937612 p^{3} T^{23} - 3629532958 p^{4} T^{24} - 6408910240 p^{5} T^{25} - 1581037080 p^{6} T^{26} - 414417120 p^{7} T^{27} - 54140360 p^{8} T^{28} + 7862448 p^{9} T^{29} + 3353800 p^{10} T^{30} + 1417104 p^{11} T^{31} + 366381 p^{12} T^{32} + 18026 p^{13} T^{33} - 5518 p^{14} T^{34} - 3666 p^{15} T^{35} - 910 p^{16} T^{36} - 14 p^{17} T^{37} + 18 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 226 T^{2} + 24977 T^{4} - 1802240 T^{6} + 95917470 T^{8} - 4048066714 T^{10} + 142781997386 T^{12} - 4389829381470 T^{14} + 121586937404065 T^{16} - 3102410101445766 T^{18} + 73784461812787402 T^{20} - 3102410101445766 p^{2} T^{22} + 121586937404065 p^{4} T^{24} - 4389829381470 p^{6} T^{26} + 142781997386 p^{8} T^{28} - 4048066714 p^{10} T^{30} + 95917470 p^{12} T^{32} - 1802240 p^{14} T^{34} + 24977 p^{16} T^{36} - 226 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( 1 + 18 T + 162 T^{2} + 1362 T^{3} + 12774 T^{4} + 97956 T^{5} + 621342 T^{6} + 4096542 T^{7} + 26767981 T^{8} + 147030468 T^{9} + 750291966 T^{10} + 4041512496 T^{11} + 19116836065 T^{12} + 68802600192 T^{13} + 223103658582 T^{14} + 457072047264 T^{15} - 4079313694802 T^{16} - 51380228770812 T^{17} - 348451102363572 T^{18} - 2347856733467382 T^{19} - 14428340269140998 T^{20} - 2347856733467382 p T^{21} - 348451102363572 p^{2} T^{22} - 51380228770812 p^{3} T^{23} - 4079313694802 p^{4} T^{24} + 457072047264 p^{5} T^{25} + 223103658582 p^{6} T^{26} + 68802600192 p^{7} T^{27} + 19116836065 p^{8} T^{28} + 4041512496 p^{9} T^{29} + 750291966 p^{10} T^{30} + 147030468 p^{11} T^{31} + 26767981 p^{12} T^{32} + 4096542 p^{13} T^{33} + 621342 p^{14} T^{34} + 97956 p^{15} T^{35} + 12774 p^{16} T^{36} + 1362 p^{17} T^{37} + 162 p^{18} T^{38} + 18 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 12 T + 72 T^{2} - 356 T^{3} + 842 T^{4} - 2678 T^{5} + 34880 T^{6} - 352488 T^{7} + 3252705 T^{8} - 18452408 T^{9} + 86936746 T^{10} - 446102644 T^{11} + 3173287089 T^{12} - 23534687474 T^{13} + 137636473708 T^{14} - 746291521086 T^{15} + 3204450303850 T^{16} - 15379961302370 T^{17} + 100682544666346 T^{18} - 721087077048368 T^{19} + 4807016938508210 T^{20} - 721087077048368 p T^{21} + 100682544666346 p^{2} T^{22} - 15379961302370 p^{3} T^{23} + 3204450303850 p^{4} T^{24} - 746291521086 p^{5} T^{25} + 137636473708 p^{6} T^{26} - 23534687474 p^{7} T^{27} + 3173287089 p^{8} T^{28} - 446102644 p^{9} T^{29} + 86936746 p^{10} T^{30} - 18452408 p^{11} T^{31} + 3252705 p^{12} T^{32} - 352488 p^{13} T^{33} + 34880 p^{14} T^{34} - 2678 p^{15} T^{35} + 842 p^{16} T^{36} - 356 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 352 T^{2} + 61574 T^{4} - 7003054 T^{6} + 568260057 T^{8} - 33681655358 T^{10} + 1387007520881 T^{12} - 28500433650714 T^{14} - 953873260275422 T^{16} + 124908350856883494 T^{18} - 6591784544503261414 T^{20} + 124908350856883494 p^{2} T^{22} - 953873260275422 p^{4} T^{24} - 28500433650714 p^{6} T^{26} + 1387007520881 p^{8} T^{28} - 33681655358 p^{10} T^{30} + 568260057 p^{12} T^{32} - 7003054 p^{14} T^{34} + 61574 p^{16} T^{36} - 352 p^{18} T^{38} + p^{20} T^{40} \) | |
| 43 | \( 1 - 354 T^{2} + 63039 T^{4} - 7623538 T^{6} + 710615601 T^{8} - 54726872648 T^{10} + 3635929325836 T^{12} - 214460361184392 T^{14} + 11441580221572478 T^{16} - 558086522199979532 T^{18} + 25015574442066574890 T^{20} - 558086522199979532 p^{2} T^{22} + 11441580221572478 p^{4} T^{24} - 214460361184392 p^{6} T^{26} + 3635929325836 p^{8} T^{28} - 54726872648 p^{10} T^{30} + 710615601 p^{12} T^{32} - 7623538 p^{14} T^{34} + 63039 p^{16} T^{36} - 354 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 + 22 T + 242 T^{2} + 2850 T^{3} + 37242 T^{4} + 358438 T^{5} + 2934322 T^{6} + 27887618 T^{7} + 246271693 T^{8} + 1680777376 T^{9} + 11817943864 T^{10} + 94471485344 T^{11} + 568828685560 T^{12} + 2523781077136 T^{13} + 15660162847288 T^{14} + 83189365412336 T^{15} - 121832291681678 T^{16} - 3359933133422756 T^{17} - 18402746388641876 T^{18} - 236767748206273516 T^{19} - 2459396131898505956 T^{20} - 236767748206273516 p T^{21} - 18402746388641876 p^{2} T^{22} - 3359933133422756 p^{3} T^{23} - 121832291681678 p^{4} T^{24} + 83189365412336 p^{5} T^{25} + 15660162847288 p^{6} T^{26} + 2523781077136 p^{7} T^{27} + 568828685560 p^{8} T^{28} + 94471485344 p^{9} T^{29} + 11817943864 p^{10} T^{30} + 1680777376 p^{11} T^{31} + 246271693 p^{12} T^{32} + 27887618 p^{13} T^{33} + 2934322 p^{14} T^{34} + 358438 p^{15} T^{35} + 37242 p^{16} T^{36} + 2850 p^{17} T^{37} + 242 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 4 T + 8 T^{2} + 244 T^{3} + 8219 T^{4} + 26424 T^{5} + 69712 T^{6} + 1733832 T^{7} + 47124225 T^{8} + 131098908 T^{9} + 346607800 T^{10} + 8240525052 T^{11} + 188985530268 T^{12} + 445206552516 T^{13} + 1112980846312 T^{14} + 26854745251908 T^{15} + 630349805473710 T^{16} + 1368206353814820 T^{17} + 2972097812137288 T^{18} + 75729690856918020 T^{19} + 1861950102177542994 T^{20} + 75729690856918020 p T^{21} + 2972097812137288 p^{2} T^{22} + 1368206353814820 p^{3} T^{23} + 630349805473710 p^{4} T^{24} + 26854745251908 p^{5} T^{25} + 1112980846312 p^{6} T^{26} + 445206552516 p^{7} T^{27} + 188985530268 p^{8} T^{28} + 8240525052 p^{9} T^{29} + 346607800 p^{10} T^{30} + 131098908 p^{11} T^{31} + 47124225 p^{12} T^{32} + 1733832 p^{13} T^{33} + 69712 p^{14} T^{34} + 26424 p^{15} T^{35} + 8219 p^{16} T^{36} + 244 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 10 T + 50 T^{2} - 750 T^{3} - 4518 T^{4} + 113630 T^{5} - 629150 T^{6} + 9767050 T^{7} - 30375683 T^{8} - 587055952 T^{9} + 3677444920 T^{10} - 61181697200 T^{11} + 444477457528 T^{12} + 1537149132032 T^{13} - 13214791082120 T^{14} + 253761679799488 T^{15} - 2948023922769038 T^{16} + 1739855647067372 T^{17} + 20743752877614892 T^{18} - 637954094659884188 T^{19} + 12621622708654139356 T^{20} - 637954094659884188 p T^{21} + 20743752877614892 p^{2} T^{22} + 1739855647067372 p^{3} T^{23} - 2948023922769038 p^{4} T^{24} + 253761679799488 p^{5} T^{25} - 13214791082120 p^{6} T^{26} + 1537149132032 p^{7} T^{27} + 444477457528 p^{8} T^{28} - 61181697200 p^{9} T^{29} + 3677444920 p^{10} T^{30} - 587055952 p^{11} T^{31} - 30375683 p^{12} T^{32} + 9767050 p^{13} T^{33} - 629150 p^{14} T^{34} + 113630 p^{15} T^{35} - 4518 p^{16} T^{36} - 750 p^{17} T^{37} + 50 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 10 T + 50 T^{2} - 1102 T^{3} + 5102 T^{4} + 59504 T^{5} - 242938 T^{6} + 5929502 T^{7} - 118784235 T^{8} + 389703476 T^{9} - 1482301530 T^{10} + 26651068668 T^{11} + 367516403617 T^{12} - 4425202085464 T^{13} + 15980633545190 T^{14} - 312428322858268 T^{15} + 1705812391183350 T^{16} + 9703738660203308 T^{17} - 35710800258586660 T^{18} + 716846940620847082 T^{19} - 13985029345046462326 T^{20} + 716846940620847082 p T^{21} - 35710800258586660 p^{2} T^{22} + 9703738660203308 p^{3} T^{23} + 1705812391183350 p^{4} T^{24} - 312428322858268 p^{5} T^{25} + 15980633545190 p^{6} T^{26} - 4425202085464 p^{7} T^{27} + 367516403617 p^{8} T^{28} + 26651068668 p^{9} T^{29} - 1482301530 p^{10} T^{30} + 389703476 p^{11} T^{31} - 118784235 p^{12} T^{32} + 5929502 p^{13} T^{33} - 242938 p^{14} T^{34} + 59504 p^{15} T^{35} + 5102 p^{16} T^{36} - 1102 p^{17} T^{37} + 50 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 8 T + 32 T^{2} - 488 T^{3} - 6847 T^{4} + 54710 T^{5} + 775856 T^{6} + 9762612 T^{7} + 17977950 T^{8} - 391274494 T^{9} - 1660766926 T^{10} + 2237557496 T^{11} + 476538792886 T^{12} + 3188344635284 T^{13} + 10093685098882 T^{14} - 3393588714954 T^{15} - 603365752730255 T^{16} + 4100495597791448 T^{17} + 55837510583568844 T^{18} + 535879391991961986 T^{19} + 5676916329019635570 T^{20} + 535879391991961986 p T^{21} + 55837510583568844 p^{2} T^{22} + 4100495597791448 p^{3} T^{23} - 603365752730255 p^{4} T^{24} - 3393588714954 p^{5} T^{25} + 10093685098882 p^{6} T^{26} + 3188344635284 p^{7} T^{27} + 476538792886 p^{8} T^{28} + 2237557496 p^{9} T^{29} - 1660766926 p^{10} T^{30} - 391274494 p^{11} T^{31} + 17977950 p^{12} T^{32} + 9762612 p^{13} T^{33} + 775856 p^{14} T^{34} + 54710 p^{15} T^{35} - 6847 p^{16} T^{36} - 488 p^{17} T^{37} + 32 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 8 T + 358 T^{2} - 3528 T^{3} + 71837 T^{4} - 714480 T^{5} + 10091912 T^{6} - 93443184 T^{7} + 1059740690 T^{8} - 8807048448 T^{9} + 85572573860 T^{10} - 8807048448 p T^{11} + 1059740690 p^{2} T^{12} - 93443184 p^{3} T^{13} + 10091912 p^{4} T^{14} - 714480 p^{5} T^{15} + 71837 p^{6} T^{16} - 3528 p^{7} T^{17} + 358 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 - 6 T + 18 T^{2} + 562 T^{3} - 14119 T^{4} + 108894 T^{5} - 241300 T^{6} - 456484 T^{7} + 76026450 T^{8} - 797248254 T^{9} + 5680506334 T^{10} - 41325048144 T^{11} - 34807496942 T^{12} + 3834627960888 T^{13} - 33769021112502 T^{14} + 340280052432398 T^{15} - 2392553004779731 T^{16} + 2111098936325898 T^{17} + 58818750761960634 T^{18} - 1557695701948058312 T^{19} + 21271415429064564522 T^{20} - 1557695701948058312 p T^{21} + 58818750761960634 p^{2} T^{22} + 2111098936325898 p^{3} T^{23} - 2392553004779731 p^{4} T^{24} + 340280052432398 p^{5} T^{25} - 33769021112502 p^{6} T^{26} + 3834627960888 p^{7} T^{27} - 34807496942 p^{8} T^{28} - 41325048144 p^{9} T^{29} + 5680506334 p^{10} T^{30} - 797248254 p^{11} T^{31} + 76026450 p^{12} T^{32} - 456484 p^{13} T^{33} - 241300 p^{14} T^{34} + 108894 p^{15} T^{35} - 14119 p^{16} T^{36} + 562 p^{17} T^{37} + 18 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 12 T + 72 T^{2} - 1112 T^{3} - 8443 T^{4} + 115194 T^{5} + 2608496 T^{6} + 9820532 T^{7} - 128718798 T^{8} - 797212938 T^{9} + 17439127162 T^{10} + 239390403444 T^{11} + 208510209682 T^{12} - 13355234894376 T^{13} - 2089384921230 T^{14} + 1850585672029274 T^{15} + 14611420367070221 T^{16} - 36570001364063700 T^{17} - 735236365087925700 T^{18} + 5114564831154485014 T^{19} + \)\(12\!\cdots\!54\)\( T^{20} + 5114564831154485014 p T^{21} - 735236365087925700 p^{2} T^{22} - 36570001364063700 p^{3} T^{23} + 14611420367070221 p^{4} T^{24} + 1850585672029274 p^{5} T^{25} - 2089384921230 p^{6} T^{26} - 13355234894376 p^{7} T^{27} + 208510209682 p^{8} T^{28} + 239390403444 p^{9} T^{29} + 17439127162 p^{10} T^{30} - 797212938 p^{11} T^{31} - 128718798 p^{12} T^{32} + 9820532 p^{13} T^{33} + 2608496 p^{14} T^{34} + 115194 p^{15} T^{35} - 8443 p^{16} T^{36} - 1112 p^{17} T^{37} + 72 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 6 T + 18 T^{2} - 114 T^{3} - 19414 T^{4} + 142722 T^{5} - 500382 T^{6} + 8638950 T^{7} + 82229277 T^{8} - 1506484128 T^{9} + 7044305976 T^{10} - 151513542432 T^{11} + 833183379000 T^{12} + 9000931297968 T^{13} - 51518837635848 T^{14} + 1146665167608528 T^{15} - 11610303227411214 T^{16} - 20289230745331644 T^{17} + 147936291046807788 T^{18} - 3537898317581946036 T^{19} + 83574848493417194940 T^{20} - 3537898317581946036 p T^{21} + 147936291046807788 p^{2} T^{22} - 20289230745331644 p^{3} T^{23} - 11610303227411214 p^{4} T^{24} + 1146665167608528 p^{5} T^{25} - 51518837635848 p^{6} T^{26} + 9000931297968 p^{7} T^{27} + 833183379000 p^{8} T^{28} - 151513542432 p^{9} T^{29} + 7044305976 p^{10} T^{30} - 1506484128 p^{11} T^{31} + 82229277 p^{12} T^{32} + 8638950 p^{13} T^{33} - 500382 p^{14} T^{34} + 142722 p^{15} T^{35} - 19414 p^{16} T^{36} - 114 p^{17} T^{37} + 18 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 44 T + 968 T^{2} - 14340 T^{3} + 147114 T^{4} - 994100 T^{5} + 4151848 T^{6} - 26693884 T^{7} + 841021741 T^{8} - 17319514784 T^{9} + 214062340096 T^{10} - 1644971664736 T^{11} + 5031538965688 T^{12} + 35630284360000 T^{13} - 259118724244928 T^{14} - 6303883871551168 T^{15} + 153220556960450482 T^{16} - 1590841799833299800 T^{17} + 7749373125946350928 T^{18} + 23707026920525031032 T^{19} - \)\(66\!\cdots\!36\)\( T^{20} + 23707026920525031032 p T^{21} + 7749373125946350928 p^{2} T^{22} - 1590841799833299800 p^{3} T^{23} + 153220556960450482 p^{4} T^{24} - 6303883871551168 p^{5} T^{25} - 259118724244928 p^{6} T^{26} + 35630284360000 p^{7} T^{27} + 5031538965688 p^{8} T^{28} - 1644971664736 p^{9} T^{29} + 214062340096 p^{10} T^{30} - 17319514784 p^{11} T^{31} + 841021741 p^{12} T^{32} - 26693884 p^{13} T^{33} + 4151848 p^{14} T^{34} - 994100 p^{15} T^{35} + 147114 p^{16} T^{36} - 14340 p^{17} T^{37} + 968 p^{18} T^{38} - 44 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( ( 1 + 36 T + 977 T^{2} + 19470 T^{3} + 346835 T^{4} + 5300586 T^{5} + 74835292 T^{6} + 947404182 T^{7} + 11259882044 T^{8} + 122542439502 T^{9} + 1259269182022 T^{10} + 122542439502 p T^{11} + 11259882044 p^{2} T^{12} + 947404182 p^{3} T^{13} + 74835292 p^{4} T^{14} + 5300586 p^{5} T^{15} + 346835 p^{6} T^{16} + 19470 p^{7} T^{17} + 977 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
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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
−2.85345777455880806090813447733, −2.74293543952200716531778777994, −2.61117089670679367428637336462, −2.39200391937552066787820599003, −2.36019406771387180579590747974, −2.29900703654871576691779603968, −2.22920124009962490139407743789, −2.18654795156572766397752280003, −2.08560810095328927114473618733, −2.02262626103259848706792541555, −1.78620947290343957983582991398, −1.76772786516786713595633535844, −1.73592808571019692886988627564, −1.66411611642409934407879915869, −1.41149547565734070857009715608, −1.38698989164421709151728534743, −1.29340164760864806089322263104, −0.962423269816020330762629472929, −0.951422231544588587023417947894, −0.912822125550698145406151575044, −0.859022198824899713090228786327, −0.70432887695903331489632602672, −0.65970699845296022466153006354, −0.17634696682800565398427978327, −0.099964061267268606280460614837, 0.099964061267268606280460614837, 0.17634696682800565398427978327, 0.65970699845296022466153006354, 0.70432887695903331489632602672, 0.859022198824899713090228786327, 0.912822125550698145406151575044, 0.951422231544588587023417947894, 0.962423269816020330762629472929, 1.29340164760864806089322263104, 1.38698989164421709151728534743, 1.41149547565734070857009715608, 1.66411611642409934407879915869, 1.73592808571019692886988627564, 1.76772786516786713595633535844, 1.78620947290343957983582991398, 2.02262626103259848706792541555, 2.08560810095328927114473618733, 2.18654795156572766397752280003, 2.22920124009962490139407743789, 2.29900703654871576691779603968, 2.36019406771387180579590747974, 2.39200391937552066787820599003, 2.61117089670679367428637336462, 2.74293543952200716531778777994, 2.85345777455880806090813447733
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.