Dirichlet series
| L(s) = 1 | − 8·2-s + 24·4-s − 3·5-s + 20·7-s − 24·8-s + 24·10-s − 26·11-s − 4·13-s − 160·14-s − 38·16-s − 4·17-s + 19-s − 72·20-s + 208·22-s − 31·23-s − 32·25-s + 32·26-s + 480·28-s − 16·29-s + 6·31-s + 139·32-s + 32·34-s − 60·35-s + 2·37-s − 8·38-s + 72·40-s − 25·41-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 12·4-s − 1.34·5-s + 7.55·7-s − 8.48·8-s + 7.58·10-s − 7.83·11-s − 1.10·13-s − 42.7·14-s − 9.5·16-s − 0.970·17-s + 0.229·19-s − 16.0·20-s + 44.3·22-s − 6.46·23-s − 6.39·25-s + 6.27·26-s + 90.7·28-s − 2.97·29-s + 1.07·31-s + 24.5·32-s + 5.48·34-s − 10.1·35-s + 0.328·37-s − 1.29·38-s + 11.3·40-s − 3.90·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 7^{20} \cdot 127^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 7^{20} \cdot 127^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 7^{20} \cdot 127^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.28359\times 10^{36}\) |
| Root analytic conductor: | \(7.99301\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 3^{40} \cdot 7^{20} \cdot 127^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T )^{20} \) | |
| 127 | \( ( 1 + T )^{20} \) | |
| good | 2 | \( 1 + p^{3} T + 5 p^{3} T^{2} + 19 p^{3} T^{3} + 243 p T^{4} + 1365 T^{5} + 3477 T^{6} + 8181 T^{7} + 9013 p T^{8} + 18761 p T^{9} + 74311 T^{10} + 8795 p^{4} T^{11} + 255933 T^{12} + 448587 T^{13} + 760073 T^{14} + 623893 p T^{15} + 1988749 T^{16} + 96301 p^{5} T^{17} + 4648149 T^{18} + 6829961 T^{19} + 9782715 T^{20} + 6829961 p T^{21} + 4648149 p^{2} T^{22} + 96301 p^{8} T^{23} + 1988749 p^{4} T^{24} + 623893 p^{6} T^{25} + 760073 p^{6} T^{26} + 448587 p^{7} T^{27} + 255933 p^{8} T^{28} + 8795 p^{13} T^{29} + 74311 p^{10} T^{30} + 18761 p^{12} T^{31} + 9013 p^{13} T^{32} + 8181 p^{13} T^{33} + 3477 p^{14} T^{34} + 1365 p^{15} T^{35} + 243 p^{17} T^{36} + 19 p^{20} T^{37} + 5 p^{21} T^{38} + p^{22} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + 3 T + 41 T^{2} + 102 T^{3} + 852 T^{4} + 1789 T^{5} + 11896 T^{6} + 847 p^{2} T^{7} + 25219 p T^{8} + 190114 T^{9} + 218257 p T^{10} + 1389004 T^{11} + 8094983 T^{12} + 8669778 T^{13} + 53198371 T^{14} + 48017284 T^{15} + 316996324 T^{16} + 49079464 p T^{17} + 1744322639 T^{18} + 1214931511 T^{19} + 8987882722 T^{20} + 1214931511 p T^{21} + 1744322639 p^{2} T^{22} + 49079464 p^{4} T^{23} + 316996324 p^{4} T^{24} + 48017284 p^{5} T^{25} + 53198371 p^{6} T^{26} + 8669778 p^{7} T^{27} + 8094983 p^{8} T^{28} + 1389004 p^{9} T^{29} + 218257 p^{11} T^{30} + 190114 p^{11} T^{31} + 25219 p^{13} T^{32} + 847 p^{15} T^{33} + 11896 p^{14} T^{34} + 1789 p^{15} T^{35} + 852 p^{16} T^{36} + 102 p^{17} T^{37} + 41 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 26 T + 41 p T^{2} + 5745 T^{3} + 5490 p T^{4} + 539974 T^{5} + 4256816 T^{6} + 30041049 T^{7} + 193010981 T^{8} + 1139446445 T^{9} + 6242249652 T^{10} + 2903222965 p T^{11} + 153602642308 T^{12} + 697892207262 T^{13} + 3010361369953 T^{14} + 1124955887056 p T^{15} + 48663255845401 T^{16} + 183604003037714 T^{17} + 666451122560284 T^{18} + 2331322386267096 T^{19} + 7870077151801638 T^{20} + 2331322386267096 p T^{21} + 666451122560284 p^{2} T^{22} + 183604003037714 p^{3} T^{23} + 48663255845401 p^{4} T^{24} + 1124955887056 p^{6} T^{25} + 3010361369953 p^{6} T^{26} + 697892207262 p^{7} T^{27} + 153602642308 p^{8} T^{28} + 2903222965 p^{10} T^{29} + 6242249652 p^{10} T^{30} + 1139446445 p^{11} T^{31} + 193010981 p^{12} T^{32} + 30041049 p^{13} T^{33} + 4256816 p^{14} T^{34} + 539974 p^{15} T^{35} + 5490 p^{17} T^{36} + 5745 p^{17} T^{37} + 41 p^{19} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 4 T + 147 T^{2} + 612 T^{3} + 10965 T^{4} + 3610 p T^{5} + 42414 p T^{6} + 2393971 T^{7} + 20938477 T^{8} + 90968199 T^{9} + 637211719 T^{10} + 2733708791 T^{11} + 16093496584 T^{12} + 5180177443 p T^{13} + 344740815492 T^{14} + 1391161764554 T^{15} + 6350216567005 T^{16} + 24452183743506 T^{17} + 101441444392152 T^{18} + 28374692424098 p T^{19} + 1411805054266120 T^{20} + 28374692424098 p^{2} T^{21} + 101441444392152 p^{2} T^{22} + 24452183743506 p^{3} T^{23} + 6350216567005 p^{4} T^{24} + 1391161764554 p^{5} T^{25} + 344740815492 p^{6} T^{26} + 5180177443 p^{8} T^{27} + 16093496584 p^{8} T^{28} + 2733708791 p^{9} T^{29} + 637211719 p^{10} T^{30} + 90968199 p^{11} T^{31} + 20938477 p^{12} T^{32} + 2393971 p^{13} T^{33} + 42414 p^{15} T^{34} + 3610 p^{16} T^{35} + 10965 p^{16} T^{36} + 612 p^{17} T^{37} + 147 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 4 T + 183 T^{2} + 657 T^{3} + 16646 T^{4} + 53973 T^{5} + 1003885 T^{6} + 173882 p T^{7} + 45271848 T^{8} + 121833860 T^{9} + 1633898241 T^{10} + 4051003521 T^{11} + 49301704657 T^{12} + 113655742290 T^{13} + 1280577022539 T^{14} + 2768680397715 T^{15} + 29171755167306 T^{16} + 59503142298315 T^{17} + 589114026053452 T^{18} + 1136153545521719 T^{19} + 10601403457616084 T^{20} + 1136153545521719 p T^{21} + 589114026053452 p^{2} T^{22} + 59503142298315 p^{3} T^{23} + 29171755167306 p^{4} T^{24} + 2768680397715 p^{5} T^{25} + 1280577022539 p^{6} T^{26} + 113655742290 p^{7} T^{27} + 49301704657 p^{8} T^{28} + 4051003521 p^{9} T^{29} + 1633898241 p^{10} T^{30} + 121833860 p^{11} T^{31} + 45271848 p^{12} T^{32} + 173882 p^{14} T^{33} + 1003885 p^{14} T^{34} + 53973 p^{15} T^{35} + 16646 p^{16} T^{36} + 657 p^{17} T^{37} + 183 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - T + 168 T^{2} - 22 T^{3} + 14607 T^{4} + 7772 T^{5} + 881922 T^{6} + 903038 T^{7} + 41475084 T^{8} + 56620098 T^{9} + 1610059343 T^{10} + 2542393063 T^{11} + 53307795046 T^{12} + 90285835997 T^{13} + 1535187711513 T^{14} + 2663902287560 T^{15} + 38939941596646 T^{16} + 67089130901567 T^{17} + 877089383861008 T^{18} + 1463016008143316 T^{19} + 17627732755396580 T^{20} + 1463016008143316 p T^{21} + 877089383861008 p^{2} T^{22} + 67089130901567 p^{3} T^{23} + 38939941596646 p^{4} T^{24} + 2663902287560 p^{5} T^{25} + 1535187711513 p^{6} T^{26} + 90285835997 p^{7} T^{27} + 53307795046 p^{8} T^{28} + 2542393063 p^{9} T^{29} + 1610059343 p^{10} T^{30} + 56620098 p^{11} T^{31} + 41475084 p^{12} T^{32} + 903038 p^{13} T^{33} + 881922 p^{14} T^{34} + 7772 p^{15} T^{35} + 14607 p^{16} T^{36} - 22 p^{17} T^{37} + 168 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 31 T + 692 T^{2} + 11153 T^{3} + 6576 p T^{4} + 1734815 T^{5} + 17720485 T^{6} + 161884250 T^{7} + 1358181700 T^{8} + 10498603348 T^{9} + 76056972053 T^{10} + 517541354301 T^{11} + 3348917400973 T^{12} + 20629077972721 T^{13} + 122097347658439 T^{14} + 694208163015300 T^{15} + 3818664081897215 T^{16} + 20293567559402971 T^{17} + 104766417002832779 T^{18} + 524141981778219074 T^{19} + 2552226400117829118 T^{20} + 524141981778219074 p T^{21} + 104766417002832779 p^{2} T^{22} + 20293567559402971 p^{3} T^{23} + 3818664081897215 p^{4} T^{24} + 694208163015300 p^{5} T^{25} + 122097347658439 p^{6} T^{26} + 20629077972721 p^{7} T^{27} + 3348917400973 p^{8} T^{28} + 517541354301 p^{9} T^{29} + 76056972053 p^{10} T^{30} + 10498603348 p^{11} T^{31} + 1358181700 p^{12} T^{32} + 161884250 p^{13} T^{33} + 17720485 p^{14} T^{34} + 1734815 p^{15} T^{35} + 6576 p^{17} T^{36} + 11153 p^{17} T^{37} + 692 p^{18} T^{38} + 31 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 16 T + 356 T^{2} + 4090 T^{3} + 54774 T^{4} + 503499 T^{5} + 5185947 T^{6} + 40664010 T^{7} + 355565661 T^{8} + 2486474985 T^{9} + 19496647316 T^{10} + 125387959304 T^{11} + 909855586995 T^{12} + 5484428987080 T^{13} + 37452135569274 T^{14} + 213835811773071 T^{15} + 1385703445491510 T^{16} + 7532573989645114 T^{17} + 46469947093723675 T^{18} + 240781898786988183 T^{19} + 1414696368543280646 T^{20} + 240781898786988183 p T^{21} + 46469947093723675 p^{2} T^{22} + 7532573989645114 p^{3} T^{23} + 1385703445491510 p^{4} T^{24} + 213835811773071 p^{5} T^{25} + 37452135569274 p^{6} T^{26} + 5484428987080 p^{7} T^{27} + 909855586995 p^{8} T^{28} + 125387959304 p^{9} T^{29} + 19496647316 p^{10} T^{30} + 2486474985 p^{11} T^{31} + 355565661 p^{12} T^{32} + 40664010 p^{13} T^{33} + 5185947 p^{14} T^{34} + 503499 p^{15} T^{35} + 54774 p^{16} T^{36} + 4090 p^{17} T^{37} + 356 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 6 T + 318 T^{2} - 1518 T^{3} + 50004 T^{4} - 189505 T^{5} + 5236681 T^{6} - 15524147 T^{7} + 13349965 p T^{8} - 934471947 T^{9} + 26425681433 T^{10} - 43790363774 T^{11} + 1421493925351 T^{12} - 1653825118500 T^{13} + 66176781521440 T^{14} - 51939856393273 T^{15} + 2713695257514807 T^{16} - 1433424696071581 T^{17} + 99105886692289330 T^{18} - 39032877693138405 T^{19} + 3242764231950792144 T^{20} - 39032877693138405 p T^{21} + 99105886692289330 p^{2} T^{22} - 1433424696071581 p^{3} T^{23} + 2713695257514807 p^{4} T^{24} - 51939856393273 p^{5} T^{25} + 66176781521440 p^{6} T^{26} - 1653825118500 p^{7} T^{27} + 1421493925351 p^{8} T^{28} - 43790363774 p^{9} T^{29} + 26425681433 p^{10} T^{30} - 934471947 p^{11} T^{31} + 13349965 p^{13} T^{32} - 15524147 p^{13} T^{33} + 5236681 p^{14} T^{34} - 189505 p^{15} T^{35} + 50004 p^{16} T^{36} - 1518 p^{17} T^{37} + 318 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 2 T + 436 T^{2} - 636 T^{3} + 94484 T^{4} - 96806 T^{5} + 13573717 T^{6} - 9079400 T^{7} + 1452889835 T^{8} - 545246494 T^{9} + 123414777016 T^{10} - 16711275309 T^{11} + 8652333784042 T^{12} + 463394612292 T^{13} + 514000404908958 T^{14} + 98194267690708 T^{15} + 26349138021430845 T^{16} + 7256885305064918 T^{17} + 1179935948825639157 T^{18} + 361679909566421909 T^{19} + 46488733595387500290 T^{20} + 361679909566421909 p T^{21} + 1179935948825639157 p^{2} T^{22} + 7256885305064918 p^{3} T^{23} + 26349138021430845 p^{4} T^{24} + 98194267690708 p^{5} T^{25} + 514000404908958 p^{6} T^{26} + 463394612292 p^{7} T^{27} + 8652333784042 p^{8} T^{28} - 16711275309 p^{9} T^{29} + 123414777016 p^{10} T^{30} - 545246494 p^{11} T^{31} + 1452889835 p^{12} T^{32} - 9079400 p^{13} T^{33} + 13573717 p^{14} T^{34} - 96806 p^{15} T^{35} + 94484 p^{16} T^{36} - 636 p^{17} T^{37} + 436 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 25 T + 696 T^{2} + 12098 T^{3} + 210607 T^{4} + 2933554 T^{5} + 39929227 T^{6} + 474261550 T^{7} + 5464594445 T^{8} + 57229450190 T^{9} + 580264984256 T^{10} + 133346110921 p T^{11} + 49861714090633 T^{12} + 428109710343371 T^{13} + 3559450256532479 T^{14} + 28082973862518910 T^{15} + 214665825107524323 T^{16} + 1564541885634587045 T^{17} + 11052155258273894274 T^{18} + 1820288728900917376 p T^{19} + \)\(48\!\cdots\!02\)\( T^{20} + 1820288728900917376 p^{2} T^{21} + 11052155258273894274 p^{2} T^{22} + 1564541885634587045 p^{3} T^{23} + 214665825107524323 p^{4} T^{24} + 28082973862518910 p^{5} T^{25} + 3559450256532479 p^{6} T^{26} + 428109710343371 p^{7} T^{27} + 49861714090633 p^{8} T^{28} + 133346110921 p^{10} T^{29} + 580264984256 p^{10} T^{30} + 57229450190 p^{11} T^{31} + 5464594445 p^{12} T^{32} + 474261550 p^{13} T^{33} + 39929227 p^{14} T^{34} + 2933554 p^{15} T^{35} + 210607 p^{16} T^{36} + 12098 p^{17} T^{37} + 696 p^{18} T^{38} + 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 13 T + 471 T^{2} - 6283 T^{3} + 120422 T^{4} - 1515822 T^{5} + 21422518 T^{6} - 245586530 T^{7} + 2891592693 T^{8} - 30027599711 T^{9} + 310306876222 T^{10} - 2936062274715 T^{11} + 27307558269017 T^{12} - 237138824626569 T^{13} + 2013092147719312 T^{14} - 16135263794711645 T^{15} + 126104976690593190 T^{16} - 936396434766212447 T^{17} + 6773410299108949813 T^{18} - 46695524115296682981 T^{19} + \)\(31\!\cdots\!58\)\( T^{20} - 46695524115296682981 p T^{21} + 6773410299108949813 p^{2} T^{22} - 936396434766212447 p^{3} T^{23} + 126104976690593190 p^{4} T^{24} - 16135263794711645 p^{5} T^{25} + 2013092147719312 p^{6} T^{26} - 237138824626569 p^{7} T^{27} + 27307558269017 p^{8} T^{28} - 2936062274715 p^{9} T^{29} + 310306876222 p^{10} T^{30} - 30027599711 p^{11} T^{31} + 2891592693 p^{12} T^{32} - 245586530 p^{13} T^{33} + 21422518 p^{14} T^{34} - 1515822 p^{15} T^{35} + 120422 p^{16} T^{36} - 6283 p^{17} T^{37} + 471 p^{18} T^{38} - 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 + 19 T + 774 T^{2} + 11872 T^{3} + 273276 T^{4} + 3551724 T^{5} + 60026983 T^{6} + 681902290 T^{7} + 9358794360 T^{8} + 94919407161 T^{9} + 1115094426107 T^{10} + 10251596671240 T^{11} + 106457690519673 T^{12} + 897170926294968 T^{13} + 8415041365773569 T^{14} + 65562559430606980 T^{15} + 563889426336727830 T^{16} + 4086881137017435528 T^{17} + 692936328535401897 p T^{18} + \)\(22\!\cdots\!48\)\( T^{19} + \)\(16\!\cdots\!12\)\( T^{20} + \)\(22\!\cdots\!48\)\( p T^{21} + 692936328535401897 p^{3} T^{22} + 4086881137017435528 p^{3} T^{23} + 563889426336727830 p^{4} T^{24} + 65562559430606980 p^{5} T^{25} + 8415041365773569 p^{6} T^{26} + 897170926294968 p^{7} T^{27} + 106457690519673 p^{8} T^{28} + 10251596671240 p^{9} T^{29} + 1115094426107 p^{10} T^{30} + 94919407161 p^{11} T^{31} + 9358794360 p^{12} T^{32} + 681902290 p^{13} T^{33} + 60026983 p^{14} T^{34} + 3551724 p^{15} T^{35} + 273276 p^{16} T^{36} + 11872 p^{17} T^{37} + 774 p^{18} T^{38} + 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 24 T + 752 T^{2} + 12714 T^{3} + 4559 p T^{4} + 3275489 T^{5} + 47980334 T^{6} + 556894589 T^{7} + 6886482945 T^{8} + 71132417785 T^{9} + 778084195202 T^{10} + 7330484721748 T^{11} + 72925359990441 T^{12} + 637044558919314 T^{13} + 5866379765742918 T^{14} + 48032253020185635 T^{15} + 413966785474461514 T^{16} + 3196625429282140999 T^{17} + 25947703842724984442 T^{18} + \)\(18\!\cdots\!27\)\( T^{19} + \)\(14\!\cdots\!16\)\( T^{20} + \)\(18\!\cdots\!27\)\( p T^{21} + 25947703842724984442 p^{2} T^{22} + 3196625429282140999 p^{3} T^{23} + 413966785474461514 p^{4} T^{24} + 48032253020185635 p^{5} T^{25} + 5866379765742918 p^{6} T^{26} + 637044558919314 p^{7} T^{27} + 72925359990441 p^{8} T^{28} + 7330484721748 p^{9} T^{29} + 778084195202 p^{10} T^{30} + 71132417785 p^{11} T^{31} + 6886482945 p^{12} T^{32} + 556894589 p^{13} T^{33} + 47980334 p^{14} T^{34} + 3275489 p^{15} T^{35} + 4559 p^{17} T^{36} + 12714 p^{17} T^{37} + 752 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 23 T + 835 T^{2} + 14786 T^{3} + 314559 T^{4} + 4632805 T^{5} + 74385661 T^{6} + 955001616 T^{7} + 216078899 p T^{8} + 147128015000 T^{9} + 1716324613351 T^{10} + 18163752339784 T^{11} + 3230096029237 p T^{12} + 1872219326295796 T^{13} + 17979789322374275 T^{14} + 165115335732762556 T^{15} + 1467551308043410298 T^{16} + 12644429046237961845 T^{17} + \)\(10\!\cdots\!46\)\( T^{18} + \)\(84\!\cdots\!39\)\( T^{19} + \)\(65\!\cdots\!12\)\( T^{20} + \)\(84\!\cdots\!39\)\( p T^{21} + \)\(10\!\cdots\!46\)\( p^{2} T^{22} + 12644429046237961845 p^{3} T^{23} + 1467551308043410298 p^{4} T^{24} + 165115335732762556 p^{5} T^{25} + 17979789322374275 p^{6} T^{26} + 1872219326295796 p^{7} T^{27} + 3230096029237 p^{9} T^{28} + 18163752339784 p^{9} T^{29} + 1716324613351 p^{10} T^{30} + 147128015000 p^{11} T^{31} + 216078899 p^{13} T^{32} + 955001616 p^{13} T^{33} + 74385661 p^{14} T^{34} + 4632805 p^{15} T^{35} + 314559 p^{16} T^{36} + 14786 p^{17} T^{37} + 835 p^{18} T^{38} + 23 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 27 T + 875 T^{2} + 17033 T^{3} + 331994 T^{4} + 5116610 T^{5} + 76175542 T^{6} + 16107742 p T^{7} + 12203408370 T^{8} + 136820170557 T^{9} + 1484538070309 T^{10} + 14880081183760 T^{11} + 145384403908907 T^{12} + 1332681712720025 T^{13} + 12007469649060554 T^{14} + 102707680608618470 T^{15} + 871687045460596794 T^{16} + 7092973046188373144 T^{17} + 57801758671098466844 T^{18} + \)\(45\!\cdots\!44\)\( T^{19} + \)\(36\!\cdots\!60\)\( T^{20} + \)\(45\!\cdots\!44\)\( p T^{21} + 57801758671098466844 p^{2} T^{22} + 7092973046188373144 p^{3} T^{23} + 871687045460596794 p^{4} T^{24} + 102707680608618470 p^{5} T^{25} + 12007469649060554 p^{6} T^{26} + 1332681712720025 p^{7} T^{27} + 145384403908907 p^{8} T^{28} + 14880081183760 p^{9} T^{29} + 1484538070309 p^{10} T^{30} + 136820170557 p^{11} T^{31} + 12203408370 p^{12} T^{32} + 16107742 p^{14} T^{33} + 76175542 p^{14} T^{34} + 5116610 p^{15} T^{35} + 331994 p^{16} T^{36} + 17033 p^{17} T^{37} + 875 p^{18} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 9 T + 591 T^{2} - 4752 T^{3} + 170755 T^{4} - 1206847 T^{5} + 479120 p T^{6} - 197318136 T^{7} + 4462893877 T^{8} - 23737930247 T^{9} + 500089616732 T^{10} - 2319275613331 T^{11} + 48393933613827 T^{12} - 200872169616621 T^{13} + 4252058182022998 T^{14} - 16383764790112677 T^{15} + 347117079633002046 T^{16} - 1274967067532347537 T^{17} + 26328418982796754431 T^{18} - 92887292534777448481 T^{19} + \)\(18\!\cdots\!00\)\( T^{20} - 92887292534777448481 p T^{21} + 26328418982796754431 p^{2} T^{22} - 1274967067532347537 p^{3} T^{23} + 347117079633002046 p^{4} T^{24} - 16383764790112677 p^{5} T^{25} + 4252058182022998 p^{6} T^{26} - 200872169616621 p^{7} T^{27} + 48393933613827 p^{8} T^{28} - 2319275613331 p^{9} T^{29} + 500089616732 p^{10} T^{30} - 23737930247 p^{11} T^{31} + 4462893877 p^{12} T^{32} - 197318136 p^{13} T^{33} + 479120 p^{15} T^{34} - 1206847 p^{15} T^{35} + 170755 p^{16} T^{36} - 4752 p^{17} T^{37} + 591 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 63 T + 2596 T^{2} + 78560 T^{3} + 1968706 T^{4} + 42263598 T^{5} + 807471444 T^{6} + 13947310273 T^{7} + 221613295558 T^{8} + 3266810090497 T^{9} + 45096447088326 T^{10} + 585913588994251 T^{11} + 7204919382169096 T^{12} + 84116808873980238 T^{13} + 935676611913596316 T^{14} + 9935597569031351192 T^{15} + \)\(10\!\cdots\!58\)\( T^{16} + \)\(98\!\cdots\!37\)\( T^{17} + \)\(91\!\cdots\!02\)\( T^{18} + \)\(82\!\cdots\!29\)\( T^{19} + \)\(70\!\cdots\!14\)\( T^{20} + \)\(82\!\cdots\!29\)\( p T^{21} + \)\(91\!\cdots\!02\)\( p^{2} T^{22} + \)\(98\!\cdots\!37\)\( p^{3} T^{23} + \)\(10\!\cdots\!58\)\( p^{4} T^{24} + 9935597569031351192 p^{5} T^{25} + 935676611913596316 p^{6} T^{26} + 84116808873980238 p^{7} T^{27} + 7204919382169096 p^{8} T^{28} + 585913588994251 p^{9} T^{29} + 45096447088326 p^{10} T^{30} + 3266810090497 p^{11} T^{31} + 221613295558 p^{12} T^{32} + 13947310273 p^{13} T^{33} + 807471444 p^{14} T^{34} + 42263598 p^{15} T^{35} + 1968706 p^{16} T^{36} + 78560 p^{17} T^{37} + 2596 p^{18} T^{38} + 63 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 21 T + 916 T^{2} + 15097 T^{3} + 380738 T^{4} + 5228211 T^{5} + 98720492 T^{6} + 1169044476 T^{7} + 18261023340 T^{8} + 190285062363 T^{9} + 2593445804090 T^{10} + 24049645680336 T^{11} + 296353479113844 T^{12} + 2462421964424588 T^{13} + 28282525841124194 T^{14} + 212323680749559685 T^{15} + 2347288002490873928 T^{16} + 16218471867962234439 T^{17} + \)\(17\!\cdots\!16\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{19} + \)\(13\!\cdots\!66\)\( T^{20} + \)\(11\!\cdots\!68\)\( p T^{21} + \)\(17\!\cdots\!16\)\( p^{2} T^{22} + 16218471867962234439 p^{3} T^{23} + 2347288002490873928 p^{4} T^{24} + 212323680749559685 p^{5} T^{25} + 28282525841124194 p^{6} T^{26} + 2462421964424588 p^{7} T^{27} + 296353479113844 p^{8} T^{28} + 24049645680336 p^{9} T^{29} + 2593445804090 p^{10} T^{30} + 190285062363 p^{11} T^{31} + 18261023340 p^{12} T^{32} + 1169044476 p^{13} T^{33} + 98720492 p^{14} T^{34} + 5228211 p^{15} T^{35} + 380738 p^{16} T^{36} + 15097 p^{17} T^{37} + 916 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 18 T + 1184 T^{2} - 19119 T^{3} + 677618 T^{4} - 9886627 T^{5} + 249669964 T^{6} - 3315237533 T^{7} + 66604748933 T^{8} - 810231025074 T^{9} + 13722285935097 T^{10} - 153803296119898 T^{11} + 2274301424731523 T^{12} - 23596328615581358 T^{13} + 311767076012223811 T^{14} - 3004511485146333437 T^{15} + 456206555425044077 p T^{16} - \)\(32\!\cdots\!52\)\( T^{17} + \)\(35\!\cdots\!28\)\( T^{18} - \)\(29\!\cdots\!26\)\( T^{19} + \)\(30\!\cdots\!92\)\( T^{20} - \)\(29\!\cdots\!26\)\( p T^{21} + \)\(35\!\cdots\!28\)\( p^{2} T^{22} - \)\(32\!\cdots\!52\)\( p^{3} T^{23} + 456206555425044077 p^{5} T^{24} - 3004511485146333437 p^{5} T^{25} + 311767076012223811 p^{6} T^{26} - 23596328615581358 p^{7} T^{27} + 2274301424731523 p^{8} T^{28} - 153803296119898 p^{9} T^{29} + 13722285935097 p^{10} T^{30} - 810231025074 p^{11} T^{31} + 66604748933 p^{12} T^{32} - 3315237533 p^{13} T^{33} + 249669964 p^{14} T^{34} - 9886627 p^{15} T^{35} + 677618 p^{16} T^{36} - 19119 p^{17} T^{37} + 1184 p^{18} T^{38} - 18 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - T + 887 T^{2} - 696 T^{3} + 387443 T^{4} - 295714 T^{5} + 111591936 T^{6} - 99249840 T^{7} + 23963641195 T^{8} - 26427369351 T^{9} + 4117436531106 T^{10} - 5497886185864 T^{11} + 593163051161905 T^{12} - 904000829396146 T^{13} + 73957076079070724 T^{14} - 120559785802295129 T^{15} + 8138620605331576032 T^{16} - 13425709742136785765 T^{17} + \)\(79\!\cdots\!35\)\( T^{18} - \)\(12\!\cdots\!00\)\( T^{19} + \)\(70\!\cdots\!44\)\( T^{20} - \)\(12\!\cdots\!00\)\( p T^{21} + \)\(79\!\cdots\!35\)\( p^{2} T^{22} - 13425709742136785765 p^{3} T^{23} + 8138620605331576032 p^{4} T^{24} - 120559785802295129 p^{5} T^{25} + 73957076079070724 p^{6} T^{26} - 904000829396146 p^{7} T^{27} + 593163051161905 p^{8} T^{28} - 5497886185864 p^{9} T^{29} + 4117436531106 p^{10} T^{30} - 26427369351 p^{11} T^{31} + 23963641195 p^{12} T^{32} - 99249840 p^{13} T^{33} + 111591936 p^{14} T^{34} - 295714 p^{15} T^{35} + 387443 p^{16} T^{36} - 696 p^{17} T^{37} + 887 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 16 T + 1054 T^{2} - 12313 T^{3} + 495452 T^{4} - 4181350 T^{5} + 143352715 T^{6} - 815178841 T^{7} + 29862289153 T^{8} - 95871706672 T^{9} + 4986555809317 T^{10} - 5188938821125 T^{11} + 718166918489319 T^{12} + 484338156975993 T^{13} + 92377826244411923 T^{14} + 171552993935159192 T^{15} + 10685149175823232350 T^{16} + 27269038279907589269 T^{17} + \)\(11\!\cdots\!99\)\( T^{18} + \)\(31\!\cdots\!79\)\( T^{19} + \)\(10\!\cdots\!30\)\( T^{20} + \)\(31\!\cdots\!79\)\( p T^{21} + \)\(11\!\cdots\!99\)\( p^{2} T^{22} + 27269038279907589269 p^{3} T^{23} + 10685149175823232350 p^{4} T^{24} + 171552993935159192 p^{5} T^{25} + 92377826244411923 p^{6} T^{26} + 484338156975993 p^{7} T^{27} + 718166918489319 p^{8} T^{28} - 5188938821125 p^{9} T^{29} + 4986555809317 p^{10} T^{30} - 95871706672 p^{11} T^{31} + 29862289153 p^{12} T^{32} - 815178841 p^{13} T^{33} + 143352715 p^{14} T^{34} - 4181350 p^{15} T^{35} + 495452 p^{16} T^{36} - 12313 p^{17} T^{37} + 1054 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 32 T + 1491 T^{2} + 35175 T^{3} + 973478 T^{4} + 18675375 T^{5} + 393370752 T^{6} + 6490900374 T^{7} + 114399025400 T^{8} + 1681555576436 T^{9} + 26020622897892 T^{10} + 348211972917488 T^{11} + 4860356046062111 T^{12} + 59995895481335366 T^{13} + 767559931026210100 T^{14} + 8808788952687065688 T^{15} + \)\(10\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!30\)\( T^{17} + \)\(12\!\cdots\!97\)\( T^{18} + \)\(12\!\cdots\!32\)\( T^{19} + \)\(12\!\cdots\!86\)\( T^{20} + \)\(12\!\cdots\!32\)\( p T^{21} + \)\(12\!\cdots\!97\)\( p^{2} T^{22} + \)\(11\!\cdots\!30\)\( p^{3} T^{23} + \)\(10\!\cdots\!79\)\( p^{4} T^{24} + 8808788952687065688 p^{5} T^{25} + 767559931026210100 p^{6} T^{26} + 59995895481335366 p^{7} T^{27} + 4860356046062111 p^{8} T^{28} + 348211972917488 p^{9} T^{29} + 26020622897892 p^{10} T^{30} + 1681555576436 p^{11} T^{31} + 114399025400 p^{12} T^{32} + 6490900374 p^{13} T^{33} + 393370752 p^{14} T^{34} + 18675375 p^{15} T^{35} + 973478 p^{16} T^{36} + 35175 p^{17} T^{37} + 1491 p^{18} T^{38} + 32 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−1.96674659692090982513028736781, −1.90349956311661912168333728208, −1.86111524910402065999084853111, −1.85986417963948528184789806517, −1.75165894168644963281739209349, −1.73189301996649854994406094661, −1.71362146251955903212444086090, −1.62503100978663418218297203270, −1.43668394921816853478810816779, −1.41119168918214746406955273841, −1.40205457009655107181472856018, −1.34902644505515969270518934333, −1.34801238512757822375336792981, −1.29448644546565863804629226845, −1.28296683208745391952473449322, −1.27308794877082216581891051395, −1.16389039536521753080588530230, −1.11661045602762026394249167634, −1.07578138569337477850749547308, −1.00390754946349407716120429494, −0.956473081048047111614106253759, −0.955338175428465973200979205611, −0.915789602895728398221764767934, −0.896268805103735423933126610693, −0.880094081354726456716004195305, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.880094081354726456716004195305, 0.896268805103735423933126610693, 0.915789602895728398221764767934, 0.955338175428465973200979205611, 0.956473081048047111614106253759, 1.00390754946349407716120429494, 1.07578138569337477850749547308, 1.11661045602762026394249167634, 1.16389039536521753080588530230, 1.27308794877082216581891051395, 1.28296683208745391952473449322, 1.29448644546565863804629226845, 1.34801238512757822375336792981, 1.34902644505515969270518934333, 1.40205457009655107181472856018, 1.41119168918214746406955273841, 1.43668394921816853478810816779, 1.62503100978663418218297203270, 1.71362146251955903212444086090, 1.73189301996649854994406094661, 1.75165894168644963281739209349, 1.85986417963948528184789806517, 1.86111524910402065999084853111, 1.90349956311661912168333728208, 1.96674659692090982513028736781
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.