Dirichlet series
| L(s) = 1 | + 20·2-s − 5·3-s + 210·4-s − 6·5-s − 100·6-s − 12·7-s + 1.54e3·8-s − 18·9-s − 120·10-s − 3·11-s − 1.05e3·12-s − 13·13-s − 240·14-s + 30·15-s + 8.85e3·16-s − 14·17-s − 360·18-s − 21·19-s − 1.26e3·20-s + 60·21-s − 60·22-s + 20·23-s − 7.70e3·24-s − 39·25-s − 260·26-s + 135·27-s − 2.52e3·28-s + ⋯ |
| L(s) = 1 | + 14.1·2-s − 2.88·3-s + 105·4-s − 2.68·5-s − 40.8·6-s − 4.53·7-s + 544.·8-s − 6·9-s − 37.9·10-s − 0.904·11-s − 303.·12-s − 3.60·13-s − 64.1·14-s + 7.74·15-s + 2.21e3·16-s − 3.39·17-s − 84.8·18-s − 4.81·19-s − 281.·20-s + 13.0·21-s − 12.7·22-s + 4.17·23-s − 1.57e3·24-s − 7.79·25-s − 50.9·26-s + 25.9·27-s − 476.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 23^{20} \cdot 131^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 23^{20} \cdot 131^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 23^{20} \cdot 131^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.42715\times 10^{33}\) |
| Root analytic conductor: | \(6.93670\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 2^{20} \cdot 23^{20} \cdot 131^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
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 | 2 | \( ( 1 - T )^{20} \) |
| 23 | \( ( 1 - T )^{20} \) | |
| 131 | \( ( 1 + T )^{20} \) | |
| good | 3 | \( 1 + 5 T + 43 T^{2} + 170 T^{3} + 290 p T^{4} + 971 p T^{5} + 11363 T^{6} + 33431 T^{7} + 36278 p T^{8} + 287590 T^{9} + 817133 T^{10} + 1966679 T^{11} + 5003300 T^{12} + 3690019 p T^{13} + 25617412 T^{14} + 52423138 T^{15} + 111471658 T^{16} + 70590170 p T^{17} + 416452564 T^{18} + 245265761 p T^{19} + 447779366 p T^{20} + 245265761 p^{2} T^{21} + 416452564 p^{2} T^{22} + 70590170 p^{4} T^{23} + 111471658 p^{4} T^{24} + 52423138 p^{5} T^{25} + 25617412 p^{6} T^{26} + 3690019 p^{8} T^{27} + 5003300 p^{8} T^{28} + 1966679 p^{9} T^{29} + 817133 p^{10} T^{30} + 287590 p^{11} T^{31} + 36278 p^{13} T^{32} + 33431 p^{13} T^{33} + 11363 p^{14} T^{34} + 971 p^{16} T^{35} + 290 p^{17} T^{36} + 170 p^{17} T^{37} + 43 p^{18} T^{38} + 5 p^{19} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + 6 T + 3 p^{2} T^{2} + 381 T^{3} + 538 p T^{4} + 2374 p T^{5} + 61882 T^{6} + 48368 p T^{7} + 1031036 T^{8} + 3622669 T^{9} + 13300861 T^{10} + 42508071 T^{11} + 138513654 T^{12} + 406195814 T^{13} + 1197041252 T^{14} + 3241483406 T^{15} + 1749012777 p T^{16} + 21956418274 T^{17} + 54653761616 T^{18} + 127482517206 T^{19} + 294154111133 T^{20} + 127482517206 p T^{21} + 54653761616 p^{2} T^{22} + 21956418274 p^{3} T^{23} + 1749012777 p^{5} T^{24} + 3241483406 p^{5} T^{25} + 1197041252 p^{6} T^{26} + 406195814 p^{7} T^{27} + 138513654 p^{8} T^{28} + 42508071 p^{9} T^{29} + 13300861 p^{10} T^{30} + 3622669 p^{11} T^{31} + 1031036 p^{12} T^{32} + 48368 p^{14} T^{33} + 61882 p^{14} T^{34} + 2374 p^{16} T^{35} + 538 p^{17} T^{36} + 381 p^{17} T^{37} + 3 p^{20} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 + 12 T + 3 p^{2} T^{2} + 1170 T^{3} + 8891 T^{4} + 55000 T^{5} + 323177 T^{6} + 1668589 T^{7} + 1173453 p T^{8} + 5260603 p T^{9} + 22592361 p T^{10} + 630789928 T^{11} + 2420077368 T^{12} + 8725743805 T^{13} + 30356087150 T^{14} + 99978175894 T^{15} + 318410565688 T^{16} + 964332838117 T^{17} + 404008856248 p T^{18} + 7906019233338 T^{19} + 21415945449380 T^{20} + 7906019233338 p T^{21} + 404008856248 p^{3} T^{22} + 964332838117 p^{3} T^{23} + 318410565688 p^{4} T^{24} + 99978175894 p^{5} T^{25} + 30356087150 p^{6} T^{26} + 8725743805 p^{7} T^{27} + 2420077368 p^{8} T^{28} + 630789928 p^{9} T^{29} + 22592361 p^{11} T^{30} + 5260603 p^{12} T^{31} + 1173453 p^{13} T^{32} + 1668589 p^{13} T^{33} + 323177 p^{14} T^{34} + 55000 p^{15} T^{35} + 8891 p^{16} T^{36} + 1170 p^{17} T^{37} + 3 p^{20} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 3 T + 131 T^{2} + 321 T^{3} + 8135 T^{4} + 15880 T^{5} + 321429 T^{6} + 481417 T^{7} + 9179096 T^{8} + 10029665 T^{9} + 204842879 T^{10} + 154673252 T^{11} + 3783896214 T^{12} + 1927511248 T^{13} + 60456516650 T^{14} + 22207375194 T^{15} + 78173929475 p T^{16} + 265628161228 T^{17} + 11027077922255 T^{18} + 3195229405816 T^{19} + 127695328671322 T^{20} + 3195229405816 p T^{21} + 11027077922255 p^{2} T^{22} + 265628161228 p^{3} T^{23} + 78173929475 p^{5} T^{24} + 22207375194 p^{5} T^{25} + 60456516650 p^{6} T^{26} + 1927511248 p^{7} T^{27} + 3783896214 p^{8} T^{28} + 154673252 p^{9} T^{29} + 204842879 p^{10} T^{30} + 10029665 p^{11} T^{31} + 9179096 p^{12} T^{32} + 481417 p^{13} T^{33} + 321429 p^{14} T^{34} + 15880 p^{15} T^{35} + 8135 p^{16} T^{36} + 321 p^{17} T^{37} + 131 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + p T + 250 T^{2} + 2515 T^{3} + 28423 T^{4} + 235369 T^{5} + 2004624 T^{6} + 14191771 T^{7} + 99780516 T^{8} + 619217464 T^{9} + 3759589966 T^{10} + 20807541823 T^{11} + 111940867944 T^{12} + 559237823792 T^{13} + 208349012198 p T^{14} + 12317082402164 T^{15} + 54249221358905 T^{16} + 225804772898792 T^{17} + 910064764244018 T^{18} + 3478200391442993 T^{19} + 12871278826699238 T^{20} + 3478200391442993 p T^{21} + 910064764244018 p^{2} T^{22} + 225804772898792 p^{3} T^{23} + 54249221358905 p^{4} T^{24} + 12317082402164 p^{5} T^{25} + 208349012198 p^{7} T^{26} + 559237823792 p^{7} T^{27} + 111940867944 p^{8} T^{28} + 20807541823 p^{9} T^{29} + 3759589966 p^{10} T^{30} + 619217464 p^{11} T^{31} + 99780516 p^{12} T^{32} + 14191771 p^{13} T^{33} + 2004624 p^{14} T^{34} + 235369 p^{15} T^{35} + 28423 p^{16} T^{36} + 2515 p^{17} T^{37} + 250 p^{18} T^{38} + p^{20} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 14 T + 257 T^{2} + 2549 T^{3} + 27893 T^{4} + 218211 T^{5} + 1801951 T^{6} + 11737184 T^{7} + 79817016 T^{8} + 446660195 T^{9} + 2619243701 T^{10} + 12853016346 T^{11} + 67082057368 T^{12} + 293396794586 T^{13} + 82532650420 p T^{14} + 5571417931583 T^{15} + 25265396829208 T^{16} + 93765328911505 T^{17} + 420595892488941 T^{18} + 1525578603027681 T^{19} + 7009077428022231 T^{20} + 1525578603027681 p T^{21} + 420595892488941 p^{2} T^{22} + 93765328911505 p^{3} T^{23} + 25265396829208 p^{4} T^{24} + 5571417931583 p^{5} T^{25} + 82532650420 p^{7} T^{26} + 293396794586 p^{7} T^{27} + 67082057368 p^{8} T^{28} + 12853016346 p^{9} T^{29} + 2619243701 p^{10} T^{30} + 446660195 p^{11} T^{31} + 79817016 p^{12} T^{32} + 11737184 p^{13} T^{33} + 1801951 p^{14} T^{34} + 218211 p^{15} T^{35} + 27893 p^{16} T^{36} + 2549 p^{17} T^{37} + 257 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 21 T + 451 T^{2} + 6362 T^{3} + 84400 T^{4} + 923179 T^{5} + 9436044 T^{6} + 85628621 T^{7} + 729772014 T^{8} + 300361828 p T^{9} + 42168273342 T^{10} + 290911980160 T^{11} + 1906043983671 T^{12} + 11778286141925 T^{13} + 69391734738037 T^{14} + 387962218052004 T^{15} + 2073638466666164 T^{16} + 10557286158037905 T^{17} + 51473695046135409 T^{18} + 239535359041619767 T^{19} + 1068395447119897552 T^{20} + 239535359041619767 p T^{21} + 51473695046135409 p^{2} T^{22} + 10557286158037905 p^{3} T^{23} + 2073638466666164 p^{4} T^{24} + 387962218052004 p^{5} T^{25} + 69391734738037 p^{6} T^{26} + 11778286141925 p^{7} T^{27} + 1906043983671 p^{8} T^{28} + 290911980160 p^{9} T^{29} + 42168273342 p^{10} T^{30} + 300361828 p^{12} T^{31} + 729772014 p^{12} T^{32} + 85628621 p^{13} T^{33} + 9436044 p^{14} T^{34} + 923179 p^{15} T^{35} + 84400 p^{16} T^{36} + 6362 p^{17} T^{37} + 451 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 27 T + 671 T^{2} + 11017 T^{3} + 165315 T^{4} + 2010743 T^{5} + 22663351 T^{6} + 221757410 T^{7} + 2035049503 T^{8} + 16732130981 T^{9} + 130229187383 T^{10} + 923100618802 T^{11} + 6243404043424 T^{12} + 38808061014968 T^{13} + 232367815425918 T^{14} + 1288269677620003 T^{15} + 7002351817865045 T^{16} + 35777926622125189 T^{17} + 185989094839062606 T^{18} + 939301419801050704 T^{19} + 5068588391553922034 T^{20} + 939301419801050704 p T^{21} + 185989094839062606 p^{2} T^{22} + 35777926622125189 p^{3} T^{23} + 7002351817865045 p^{4} T^{24} + 1288269677620003 p^{5} T^{25} + 232367815425918 p^{6} T^{26} + 38808061014968 p^{7} T^{27} + 6243404043424 p^{8} T^{28} + 923100618802 p^{9} T^{29} + 130229187383 p^{10} T^{30} + 16732130981 p^{11} T^{31} + 2035049503 p^{12} T^{32} + 221757410 p^{13} T^{33} + 22663351 p^{14} T^{34} + 2010743 p^{15} T^{35} + 165315 p^{16} T^{36} + 11017 p^{17} T^{37} + 671 p^{18} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 + 27 T + 23 p T^{2} + 12137 T^{3} + 195684 T^{4} + 2532878 T^{5} + 31207842 T^{6} + 333285818 T^{7} + 3414953124 T^{8} + 31478051657 T^{9} + 280310736443 T^{10} + 2293712746101 T^{11} + 18232824796335 T^{12} + 134951431179116 T^{13} + 974527353169652 T^{14} + 6609003430534677 T^{15} + 43870230254309681 T^{16} + 274973038486893379 T^{17} + 1690597180418362117 T^{18} + 9844637256758697796 T^{19} + 56293976094785761312 T^{20} + 9844637256758697796 p T^{21} + 1690597180418362117 p^{2} T^{22} + 274973038486893379 p^{3} T^{23} + 43870230254309681 p^{4} T^{24} + 6609003430534677 p^{5} T^{25} + 974527353169652 p^{6} T^{26} + 134951431179116 p^{7} T^{27} + 18232824796335 p^{8} T^{28} + 2293712746101 p^{9} T^{29} + 280310736443 p^{10} T^{30} + 31478051657 p^{11} T^{31} + 3414953124 p^{12} T^{32} + 333285818 p^{13} T^{33} + 31207842 p^{14} T^{34} + 2532878 p^{15} T^{35} + 195684 p^{16} T^{36} + 12137 p^{17} T^{37} + 23 p^{19} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + 19 T + 647 T^{2} + 9963 T^{3} + 195436 T^{4} + 2548820 T^{5} + 37291257 T^{6} + 423559055 T^{7} + 5089644511 T^{8} + 51308513336 T^{9} + 14356914660 p T^{10} + 4816778803176 T^{11} + 1193315912481 p T^{12} + 363561122402284 T^{13} + 3000111142403417 T^{14} + 22582633158474953 T^{15} + 169535397164083678 T^{16} + 1171649398494154643 T^{17} + 8054932557801109683 T^{18} + 51227599492139006671 T^{19} + \)\(32\!\cdots\!38\)\( T^{20} + 51227599492139006671 p T^{21} + 8054932557801109683 p^{2} T^{22} + 1171649398494154643 p^{3} T^{23} + 169535397164083678 p^{4} T^{24} + 22582633158474953 p^{5} T^{25} + 3000111142403417 p^{6} T^{26} + 363561122402284 p^{7} T^{27} + 1193315912481 p^{9} T^{28} + 4816778803176 p^{9} T^{29} + 14356914660 p^{11} T^{30} + 51308513336 p^{11} T^{31} + 5089644511 p^{12} T^{32} + 423559055 p^{13} T^{33} + 37291257 p^{14} T^{34} + 2548820 p^{15} T^{35} + 195436 p^{16} T^{36} + 9963 p^{17} T^{37} + 647 p^{18} T^{38} + 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 17 T + 614 T^{2} + 8450 T^{3} + 172614 T^{4} + 2008685 T^{5} + 30118031 T^{6} + 305182851 T^{7} + 3712005492 T^{8} + 33479163477 T^{9} + 348309302875 T^{10} + 2846798532185 T^{11} + 26203538538417 T^{12} + 197127228039002 T^{13} + 1645086735274430 T^{14} + 11547517225333148 T^{15} + 89050943966169610 T^{16} + 589691058124247137 T^{17} + 4264569325585298427 T^{18} + 26826917650906222371 T^{19} + \)\(18\!\cdots\!49\)\( T^{20} + 26826917650906222371 p T^{21} + 4264569325585298427 p^{2} T^{22} + 589691058124247137 p^{3} T^{23} + 89050943966169610 p^{4} T^{24} + 11547517225333148 p^{5} T^{25} + 1645086735274430 p^{6} T^{26} + 197127228039002 p^{7} T^{27} + 26203538538417 p^{8} T^{28} + 2846798532185 p^{9} T^{29} + 348309302875 p^{10} T^{30} + 33479163477 p^{11} T^{31} + 3712005492 p^{12} T^{32} + 305182851 p^{13} T^{33} + 30118031 p^{14} T^{34} + 2008685 p^{15} T^{35} + 172614 p^{16} T^{36} + 8450 p^{17} T^{37} + 614 p^{18} T^{38} + 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 27 T + 848 T^{2} + 16241 T^{3} + 309305 T^{4} + 4686811 T^{5} + 68377421 T^{6} + 869072333 T^{7} + 10577445269 T^{8} + 116877285139 T^{9} + 1238610560284 T^{10} + 12181830342590 T^{11} + 115255509326132 T^{12} + 1025169834050054 T^{13} + 8797100750992930 T^{14} + 71536734877722021 T^{15} + 562459634866818242 T^{16} + 4211267863165516099 T^{17} + 30531924168395012207 T^{18} + \)\(21\!\cdots\!41\)\( T^{19} + \)\(14\!\cdots\!06\)\( T^{20} + \)\(21\!\cdots\!41\)\( p T^{21} + 30531924168395012207 p^{2} T^{22} + 4211267863165516099 p^{3} T^{23} + 562459634866818242 p^{4} T^{24} + 71536734877722021 p^{5} T^{25} + 8797100750992930 p^{6} T^{26} + 1025169834050054 p^{7} T^{27} + 115255509326132 p^{8} T^{28} + 12181830342590 p^{9} T^{29} + 1238610560284 p^{10} T^{30} + 116877285139 p^{11} T^{31} + 10577445269 p^{12} T^{32} + 869072333 p^{13} T^{33} + 68377421 p^{14} T^{34} + 4686811 p^{15} T^{35} + 309305 p^{16} T^{36} + 16241 p^{17} T^{37} + 848 p^{18} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 + 28 T + 993 T^{2} + 20685 T^{3} + 439032 T^{4} + 7360562 T^{5} + 118848621 T^{6} + 1678339164 T^{7} + 22476337393 T^{8} + 275221030231 T^{9} + 3189017382204 T^{10} + 34529640553953 T^{11} + 354602683045808 T^{12} + 3441770986824135 T^{13} + 31792828545292042 T^{14} + 279268232138503491 T^{15} + 2342138326084974085 T^{16} + 18740110031998932636 T^{17} + \)\(14\!\cdots\!31\)\( T^{18} + \)\(10\!\cdots\!93\)\( T^{19} + \)\(73\!\cdots\!26\)\( T^{20} + \)\(10\!\cdots\!93\)\( p T^{21} + \)\(14\!\cdots\!31\)\( p^{2} T^{22} + 18740110031998932636 p^{3} T^{23} + 2342138326084974085 p^{4} T^{24} + 279268232138503491 p^{5} T^{25} + 31792828545292042 p^{6} T^{26} + 3441770986824135 p^{7} T^{27} + 354602683045808 p^{8} T^{28} + 34529640553953 p^{9} T^{29} + 3189017382204 p^{10} T^{30} + 275221030231 p^{11} T^{31} + 22476337393 p^{12} T^{32} + 1678339164 p^{13} T^{33} + 118848621 p^{14} T^{34} + 7360562 p^{15} T^{35} + 439032 p^{16} T^{36} + 20685 p^{17} T^{37} + 993 p^{18} T^{38} + 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 47 T + 1681 T^{2} + 43968 T^{3} + 981807 T^{4} + 18761686 T^{5} + 320983201 T^{6} + 4945844776 T^{7} + 69979455888 T^{8} + 913949652973 T^{9} + 11132689880723 T^{10} + 126995499372888 T^{11} + 1365445272517663 T^{12} + 13882487655437830 T^{13} + 134049218942923013 T^{14} + 1232441923026567905 T^{15} + 10822039562950719883 T^{16} + 90925914919540226034 T^{17} + \)\(73\!\cdots\!17\)\( T^{18} + \)\(56\!\cdots\!89\)\( T^{19} + \)\(79\!\cdots\!62\)\( p T^{20} + \)\(56\!\cdots\!89\)\( p T^{21} + \)\(73\!\cdots\!17\)\( p^{2} T^{22} + 90925914919540226034 p^{3} T^{23} + 10822039562950719883 p^{4} T^{24} + 1232441923026567905 p^{5} T^{25} + 134049218942923013 p^{6} T^{26} + 13882487655437830 p^{7} T^{27} + 1365445272517663 p^{8} T^{28} + 126995499372888 p^{9} T^{29} + 11132689880723 p^{10} T^{30} + 913949652973 p^{11} T^{31} + 69979455888 p^{12} T^{32} + 4945844776 p^{13} T^{33} + 320983201 p^{14} T^{34} + 18761686 p^{15} T^{35} + 981807 p^{16} T^{36} + 43968 p^{17} T^{37} + 1681 p^{18} T^{38} + 47 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 16 T + 827 T^{2} + 11877 T^{3} + 335576 T^{4} + 4383449 T^{5} + 88976111 T^{6} + 1067077138 T^{7} + 17310676371 T^{8} + 191886219097 T^{9} + 2629904793335 T^{10} + 27070380562048 T^{11} + 324088376542843 T^{12} + 3106822957430109 T^{13} + 33208019826739010 T^{14} + 296892323633288999 T^{15} + 2876225056598832830 T^{16} + 23979334991434569270 T^{17} + \)\(21\!\cdots\!94\)\( T^{18} + \)\(16\!\cdots\!05\)\( T^{19} + \)\(13\!\cdots\!58\)\( T^{20} + \)\(16\!\cdots\!05\)\( p T^{21} + \)\(21\!\cdots\!94\)\( p^{2} T^{22} + 23979334991434569270 p^{3} T^{23} + 2876225056598832830 p^{4} T^{24} + 296892323633288999 p^{5} T^{25} + 33208019826739010 p^{6} T^{26} + 3106822957430109 p^{7} T^{27} + 324088376542843 p^{8} T^{28} + 27070380562048 p^{9} T^{29} + 2629904793335 p^{10} T^{30} + 191886219097 p^{11} T^{31} + 17310676371 p^{12} T^{32} + 1067077138 p^{13} T^{33} + 88976111 p^{14} T^{34} + 4383449 p^{15} T^{35} + 335576 p^{16} T^{36} + 11877 p^{17} T^{37} + 827 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 9 T + 796 T^{2} + 5942 T^{3} + 302091 T^{4} + 1870209 T^{5} + 73537618 T^{6} + 375202370 T^{7} + 13029424208 T^{8} + 54247970616 T^{9} + 1806973241077 T^{10} + 6080157655422 T^{11} + 205721234015499 T^{12} + 557440679325247 T^{13} + 19860791343813298 T^{14} + 43771868722018452 T^{15} + 1661098684851801070 T^{16} + 3068191863509903970 T^{17} + 1998699732567873333 p T^{18} + \)\(19\!\cdots\!64\)\( T^{19} + \)\(79\!\cdots\!13\)\( T^{20} + \)\(19\!\cdots\!64\)\( p T^{21} + 1998699732567873333 p^{3} T^{22} + 3068191863509903970 p^{3} T^{23} + 1661098684851801070 p^{4} T^{24} + 43771868722018452 p^{5} T^{25} + 19860791343813298 p^{6} T^{26} + 557440679325247 p^{7} T^{27} + 205721234015499 p^{8} T^{28} + 6080157655422 p^{9} T^{29} + 1806973241077 p^{10} T^{30} + 54247970616 p^{11} T^{31} + 13029424208 p^{12} T^{32} + 375202370 p^{13} T^{33} + 73537618 p^{14} T^{34} + 1870209 p^{15} T^{35} + 302091 p^{16} T^{36} + 5942 p^{17} T^{37} + 796 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 8 T + 799 T^{2} + 5806 T^{3} + 301830 T^{4} + 1937254 T^{5} + 71371796 T^{6} + 387292940 T^{7} + 175755114 p T^{8} + 49738934215 T^{9} + 1428010560154 T^{10} + 3805498703355 T^{11} + 130231227462876 T^{12} + 55241662470200 T^{13} + 8941893613771951 T^{14} - 29357008106891146 T^{15} + 454372620941213478 T^{16} - 4793715363180557428 T^{17} + 17319841290126240121 T^{18} - \)\(45\!\cdots\!06\)\( T^{19} + \)\(75\!\cdots\!62\)\( T^{20} - \)\(45\!\cdots\!06\)\( p T^{21} + 17319841290126240121 p^{2} T^{22} - 4793715363180557428 p^{3} T^{23} + 454372620941213478 p^{4} T^{24} - 29357008106891146 p^{5} T^{25} + 8941893613771951 p^{6} T^{26} + 55241662470200 p^{7} T^{27} + 130231227462876 p^{8} T^{28} + 3805498703355 p^{9} T^{29} + 1428010560154 p^{10} T^{30} + 49738934215 p^{11} T^{31} + 175755114 p^{13} T^{32} + 387292940 p^{13} T^{33} + 71371796 p^{14} T^{34} + 1937254 p^{15} T^{35} + 301830 p^{16} T^{36} + 5806 p^{17} T^{37} + 799 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 30 T + 992 T^{2} + 20245 T^{3} + 423314 T^{4} + 6961967 T^{5} + 114875051 T^{6} + 1624601233 T^{7} + 22857241197 T^{8} + 287457100846 T^{9} + 3585730364979 T^{10} + 40953613841521 T^{11} + 463695638168566 T^{12} + 4879948332713678 T^{13} + 50953110655371853 T^{14} + 499437946117170912 T^{15} + 4863790579276901412 T^{16} + 44755863471035471874 T^{17} + \)\(40\!\cdots\!20\)\( T^{18} + \)\(35\!\cdots\!28\)\( T^{19} + \)\(30\!\cdots\!68\)\( T^{20} + \)\(35\!\cdots\!28\)\( p T^{21} + \)\(40\!\cdots\!20\)\( p^{2} T^{22} + 44755863471035471874 p^{3} T^{23} + 4863790579276901412 p^{4} T^{24} + 499437946117170912 p^{5} T^{25} + 50953110655371853 p^{6} T^{26} + 4879948332713678 p^{7} T^{27} + 463695638168566 p^{8} T^{28} + 40953613841521 p^{9} T^{29} + 3585730364979 p^{10} T^{30} + 287457100846 p^{11} T^{31} + 22857241197 p^{12} T^{32} + 1624601233 p^{13} T^{33} + 114875051 p^{14} T^{34} + 6961967 p^{15} T^{35} + 423314 p^{16} T^{36} + 20245 p^{17} T^{37} + 992 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 26 T + 1197 T^{2} + 24592 T^{3} + 657890 T^{4} + 11341711 T^{5} + 227311894 T^{6} + 3409169339 T^{7} + 56312125437 T^{8} + 752189598694 T^{9} + 10746088856380 T^{10} + 129888755624308 T^{11} + 1650660275351722 T^{12} + 18250686856715718 T^{13} + 209964989934409142 T^{14} + 2139004944961456839 T^{15} + 22530176407805990018 T^{16} + \)\(21\!\cdots\!22\)\( T^{17} + \)\(20\!\cdots\!55\)\( T^{18} + \)\(18\!\cdots\!31\)\( T^{19} + \)\(16\!\cdots\!00\)\( T^{20} + \)\(18\!\cdots\!31\)\( p T^{21} + \)\(20\!\cdots\!55\)\( p^{2} T^{22} + \)\(21\!\cdots\!22\)\( p^{3} T^{23} + 22530176407805990018 p^{4} T^{24} + 2139004944961456839 p^{5} T^{25} + 209964989934409142 p^{6} T^{26} + 18250686856715718 p^{7} T^{27} + 1650660275351722 p^{8} T^{28} + 129888755624308 p^{9} T^{29} + 10746088856380 p^{10} T^{30} + 752189598694 p^{11} T^{31} + 56312125437 p^{12} T^{32} + 3409169339 p^{13} T^{33} + 227311894 p^{14} T^{34} + 11341711 p^{15} T^{35} + 657890 p^{16} T^{36} + 24592 p^{17} T^{37} + 1197 p^{18} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 35 T + 1358 T^{2} + 32138 T^{3} + 763329 T^{4} + 14201535 T^{5} + 259877581 T^{6} + 4078954210 T^{7} + 62660095384 T^{8} + 863796466425 T^{9} + 11643015327162 T^{10} + 144609897419381 T^{11} + 1756305094950960 T^{12} + 19992624445757313 T^{13} + 222610909075945230 T^{14} + 2349868391539924272 T^{15} + 24271098688322182345 T^{16} + \)\(23\!\cdots\!68\)\( T^{17} + \)\(23\!\cdots\!49\)\( T^{18} + \)\(21\!\cdots\!15\)\( T^{19} + \)\(19\!\cdots\!54\)\( T^{20} + \)\(21\!\cdots\!15\)\( p T^{21} + \)\(23\!\cdots\!49\)\( p^{2} T^{22} + \)\(23\!\cdots\!68\)\( p^{3} T^{23} + 24271098688322182345 p^{4} T^{24} + 2349868391539924272 p^{5} T^{25} + 222610909075945230 p^{6} T^{26} + 19992624445757313 p^{7} T^{27} + 1756305094950960 p^{8} T^{28} + 144609897419381 p^{9} T^{29} + 11643015327162 p^{10} T^{30} + 863796466425 p^{11} T^{31} + 62660095384 p^{12} T^{32} + 4078954210 p^{13} T^{33} + 259877581 p^{14} T^{34} + 14201535 p^{15} T^{35} + 763329 p^{16} T^{36} + 32138 p^{17} T^{37} + 1358 p^{18} T^{38} + 35 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 2 T + 1054 T^{2} - 2189 T^{3} + 549076 T^{4} - 1221114 T^{5} + 188559139 T^{6} - 454561506 T^{7} + 47980488365 T^{8} - 125031361088 T^{9} + 9632903576114 T^{10} - 26783048587634 T^{11} + 1585644441482620 T^{12} - 4612318693332748 T^{13} + 219438930094848336 T^{14} - 651863119546389719 T^{15} + 25963241327252864126 T^{16} - 76632059974815401333 T^{17} + \)\(26\!\cdots\!16\)\( T^{18} - \)\(75\!\cdots\!51\)\( T^{19} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(75\!\cdots\!51\)\( p T^{21} + \)\(26\!\cdots\!16\)\( p^{2} T^{22} - 76632059974815401333 p^{3} T^{23} + 25963241327252864126 p^{4} T^{24} - 651863119546389719 p^{5} T^{25} + 219438930094848336 p^{6} T^{26} - 4612318693332748 p^{7} T^{27} + 1585644441482620 p^{8} T^{28} - 26783048587634 p^{9} T^{29} + 9632903576114 p^{10} T^{30} - 125031361088 p^{11} T^{31} + 47980488365 p^{12} T^{32} - 454561506 p^{13} T^{33} + 188559139 p^{14} T^{34} - 1221114 p^{15} T^{35} + 549076 p^{16} T^{36} - 2189 p^{17} T^{37} + 1054 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 25 T + 1122 T^{2} + 25128 T^{3} + 647461 T^{4} + 12646106 T^{5} + 249354041 T^{6} + 4264554695 T^{7} + 71079322173 T^{8} + 1081231776971 T^{9} + 15902633104810 T^{10} + 218513406337704 T^{11} + 2899837771736837 T^{12} + 36411084279396553 T^{13} + 441808943107334142 T^{14} + 5109424918721731119 T^{15} + 57157923736166926274 T^{16} + \)\(61\!\cdots\!00\)\( T^{17} + \)\(63\!\cdots\!73\)\( T^{18} + \)\(63\!\cdots\!03\)\( T^{19} + \)\(60\!\cdots\!40\)\( T^{20} + \)\(63\!\cdots\!03\)\( p T^{21} + \)\(63\!\cdots\!73\)\( p^{2} T^{22} + \)\(61\!\cdots\!00\)\( p^{3} T^{23} + 57157923736166926274 p^{4} T^{24} + 5109424918721731119 p^{5} T^{25} + 441808943107334142 p^{6} T^{26} + 36411084279396553 p^{7} T^{27} + 2899837771736837 p^{8} T^{28} + 218513406337704 p^{9} T^{29} + 15902633104810 p^{10} T^{30} + 1081231776971 p^{11} T^{31} + 71079322173 p^{12} T^{32} + 4264554695 p^{13} T^{33} + 249354041 p^{14} T^{34} + 12646106 p^{15} T^{35} + 647461 p^{16} T^{36} + 25128 p^{17} T^{37} + 1122 p^{18} T^{38} + 25 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 2 T + 1027 T^{2} - 1383 T^{3} + 524261 T^{4} - 464294 T^{5} + 177455904 T^{6} - 103803620 T^{7} + 44747258506 T^{8} - 18322151133 T^{9} + 8955705191589 T^{10} - 2941979482999 T^{11} + 1482863476002801 T^{12} - 467189237697526 T^{13} + 209629148472931131 T^{14} - 71617841081070944 T^{15} + 25982714719775057886 T^{16} - 9836112063168080376 T^{17} + \)\(28\!\cdots\!04\)\( T^{18} - \)\(11\!\cdots\!71\)\( T^{19} + \)\(29\!\cdots\!04\)\( T^{20} - \)\(11\!\cdots\!71\)\( p T^{21} + \)\(28\!\cdots\!04\)\( p^{2} T^{22} - 9836112063168080376 p^{3} T^{23} + 25982714719775057886 p^{4} T^{24} - 71617841081070944 p^{5} T^{25} + 209629148472931131 p^{6} T^{26} - 467189237697526 p^{7} T^{27} + 1482863476002801 p^{8} T^{28} - 2941979482999 p^{9} T^{29} + 8955705191589 p^{10} T^{30} - 18322151133 p^{11} T^{31} + 44747258506 p^{12} T^{32} - 103803620 p^{13} T^{33} + 177455904 p^{14} T^{34} - 464294 p^{15} T^{35} + 524261 p^{16} T^{36} - 1383 p^{17} T^{37} + 1027 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.29267812084981612686586638458, −2.27416270668718082596446320680, −2.22645580234863752672402295896, −2.20315526801503746597463371930, −2.13040197054249708208052790449, −2.02161837619030121053290745476, −1.89724085104450806554613771574, −1.82655860477872897635925746470, −1.73568512348176238208564960499, −1.73389774518231934379088664642, −1.73036561899018693135586952220, −1.70141901208101478297020941682, −1.62620463660748164773409917285, −1.54578541585865273620317791666, −1.52997470004133870697115714364, −1.49143280698500783100739751693, −1.45922782604309696756919832189, −1.45535210193165130837851126370, −1.36950521045049169998370783175, −1.32397380663959964333884397890, −1.30636654842061360204049909134, −1.23390296826707447294112262249, −1.15583835859046667761759441228, −1.14395246589879247585361032360, −0.965989545190015659742593461455, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.965989545190015659742593461455, 1.14395246589879247585361032360, 1.15583835859046667761759441228, 1.23390296826707447294112262249, 1.30636654842061360204049909134, 1.32397380663959964333884397890, 1.36950521045049169998370783175, 1.45535210193165130837851126370, 1.45922782604309696756919832189, 1.49143280698500783100739751693, 1.52997470004133870697115714364, 1.54578541585865273620317791666, 1.62620463660748164773409917285, 1.70141901208101478297020941682, 1.73036561899018693135586952220, 1.73389774518231934379088664642, 1.73568512348176238208564960499, 1.82655860477872897635925746470, 1.89724085104450806554613771574, 2.02161837619030121053290745476, 2.13040197054249708208052790449, 2.20315526801503746597463371930, 2.22645580234863752672402295896, 2.27416270668718082596446320680, 2.29267812084981612686586638458
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.