Dirichlet series
| L(s) = 1 | + 2·2-s + 20·3-s − 3·4-s − 5-s + 40·6-s + 9·7-s − 8·8-s + 210·9-s − 2·10-s − 60·12-s + 21·13-s + 18·14-s − 20·15-s + 4·16-s − 4·17-s + 420·18-s + 7·19-s + 3·20-s + 180·21-s + 20·23-s − 160·24-s − 26·25-s + 42·26-s + 1.54e3·27-s − 27·28-s + 20·29-s − 40·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 11.5·3-s − 3/2·4-s − 0.447·5-s + 16.3·6-s + 3.40·7-s − 2.82·8-s + 70·9-s − 0.632·10-s − 17.3·12-s + 5.82·13-s + 4.81·14-s − 5.16·15-s + 16-s − 0.970·17-s + 98.9·18-s + 1.60·19-s + 0.670·20-s + 39.2·21-s + 4.17·23-s − 32.6·24-s − 5.19·25-s + 8.23·26-s + 296.·27-s − 5.10·28-s + 3.71·29-s − 7.30·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 23^{20} \cdot 29^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 23^{20} \cdot 29^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 23^{20} \cdot 29^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17620\times 10^{24}\) |
| Root analytic conductor: | \(3.99725\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 23^{20} \cdot 29^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(462615.7040\) |
| \(L(\frac12)\) | \(\approx\) | \(462615.7040\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{20} \) |
| 23 | \( ( 1 - T )^{20} \) | |
| 29 | \( ( 1 - T )^{20} \) | |
| good | 2 | \( 1 - p T + 7 T^{2} - 3 p^{2} T^{3} + 25 T^{4} - 19 p T^{5} + 67 T^{6} - 83 T^{7} + 17 p^{3} T^{8} - 37 p^{2} T^{9} + 221 T^{10} - 117 p T^{11} + 321 T^{12} - 167 p T^{13} + 509 T^{14} - 535 T^{15} + 813 T^{16} - 415 p T^{17} + 261 p^{2} T^{18} - 139 p^{3} T^{19} + 187 p^{3} T^{20} - 139 p^{4} T^{21} + 261 p^{4} T^{22} - 415 p^{4} T^{23} + 813 p^{4} T^{24} - 535 p^{5} T^{25} + 509 p^{6} T^{26} - 167 p^{8} T^{27} + 321 p^{8} T^{28} - 117 p^{10} T^{29} + 221 p^{10} T^{30} - 37 p^{13} T^{31} + 17 p^{15} T^{32} - 83 p^{13} T^{33} + 67 p^{14} T^{34} - 19 p^{16} T^{35} + 25 p^{16} T^{36} - 3 p^{19} T^{37} + 7 p^{18} T^{38} - p^{20} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + T + 27 T^{2} + 39 T^{3} + 82 p T^{4} + 717 T^{5} + 4736 T^{6} + 8813 T^{7} + 9122 p T^{8} + 84699 T^{9} + 378699 T^{10} + 688957 T^{11} + 2777258 T^{12} + 4928589 T^{13} + 18374081 T^{14} + 6329339 p T^{15} + 111227573 T^{16} + 185409648 T^{17} + 124258781 p T^{18} + 1001839322 T^{19} + 3220723704 T^{20} + 1001839322 p T^{21} + 124258781 p^{3} T^{22} + 185409648 p^{3} T^{23} + 111227573 p^{4} T^{24} + 6329339 p^{6} T^{25} + 18374081 p^{6} T^{26} + 4928589 p^{7} T^{27} + 2777258 p^{8} T^{28} + 688957 p^{9} T^{29} + 378699 p^{10} T^{30} + 84699 p^{11} T^{31} + 9122 p^{13} T^{32} + 8813 p^{13} T^{33} + 4736 p^{14} T^{34} + 717 p^{15} T^{35} + 82 p^{17} T^{36} + 39 p^{17} T^{37} + 27 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 9 T + 93 T^{2} - 583 T^{3} + 3711 T^{4} - 18395 T^{5} + 89974 T^{6} - 373916 T^{7} + 217529 p T^{8} - 5492013 T^{9} + 19423811 T^{10} - 62344298 T^{11} + 197250576 T^{12} - 576616234 T^{13} + 239600135 p T^{14} - 654611823 p T^{15} + 12620567920 T^{16} - 33184332359 T^{17} + 12742209327 p T^{18} - 231925425628 T^{19} + 622248430242 T^{20} - 231925425628 p T^{21} + 12742209327 p^{3} T^{22} - 33184332359 p^{3} T^{23} + 12620567920 p^{4} T^{24} - 654611823 p^{6} T^{25} + 239600135 p^{7} T^{26} - 576616234 p^{7} T^{27} + 197250576 p^{8} T^{28} - 62344298 p^{9} T^{29} + 19423811 p^{10} T^{30} - 5492013 p^{11} T^{31} + 217529 p^{13} T^{32} - 373916 p^{13} T^{33} + 89974 p^{14} T^{34} - 18395 p^{15} T^{35} + 3711 p^{16} T^{36} - 583 p^{17} T^{37} + 93 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 73 T^{2} + 47 T^{3} + 3033 T^{4} + 3811 T^{5} + 91636 T^{6} + 14896 p T^{7} + 2231051 T^{8} + 450336 p T^{9} + 46062463 T^{10} + 116768915 T^{11} + 829973932 T^{12} + 2266227913 T^{13} + 13265281121 T^{14} + 37356687648 T^{15} + 189734815108 T^{16} + 533198439656 T^{17} + 2439922967707 T^{18} + 6663484842834 T^{19} + 28267289746614 T^{20} + 6663484842834 p T^{21} + 2439922967707 p^{2} T^{22} + 533198439656 p^{3} T^{23} + 189734815108 p^{4} T^{24} + 37356687648 p^{5} T^{25} + 13265281121 p^{6} T^{26} + 2266227913 p^{7} T^{27} + 829973932 p^{8} T^{28} + 116768915 p^{9} T^{29} + 46062463 p^{10} T^{30} + 450336 p^{12} T^{31} + 2231051 p^{12} T^{32} + 14896 p^{14} T^{33} + 91636 p^{14} T^{34} + 3811 p^{15} T^{35} + 3033 p^{16} T^{36} + 47 p^{17} T^{37} + 73 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 21 T + 301 T^{2} - 3163 T^{3} + 27637 T^{4} - 206057 T^{5} + 1364242 T^{6} - 8139280 T^{7} + 44625688 T^{8} - 227253048 T^{9} + 1088899906 T^{10} - 4956051849 T^{11} + 21651182355 T^{12} - 91555662621 T^{13} + 377574054643 T^{14} - 1525983394811 T^{15} + 6058490078215 T^{16} - 23625172277730 T^{17} + 90322393264348 T^{18} - 337747420812184 T^{19} + 1232943029423520 T^{20} - 337747420812184 p T^{21} + 90322393264348 p^{2} T^{22} - 23625172277730 p^{3} T^{23} + 6058490078215 p^{4} T^{24} - 1525983394811 p^{5} T^{25} + 377574054643 p^{6} T^{26} - 91555662621 p^{7} T^{27} + 21651182355 p^{8} T^{28} - 4956051849 p^{9} T^{29} + 1088899906 p^{10} T^{30} - 227253048 p^{11} T^{31} + 44625688 p^{12} T^{32} - 8139280 p^{13} T^{33} + 1364242 p^{14} T^{34} - 206057 p^{15} T^{35} + 27637 p^{16} T^{36} - 3163 p^{17} T^{37} + 301 p^{18} T^{38} - 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 4 T + 168 T^{2} + 619 T^{3} + 13433 T^{4} + 47287 T^{5} + 688249 T^{6} + 2398160 T^{7} + 25709641 T^{8} + 91244692 T^{9} + 757956498 T^{10} + 2784205789 T^{11} + 18734675684 T^{12} + 71205881473 T^{13} + 24026777780 p T^{14} + 1580820886684 T^{15} + 8161638451030 T^{16} + 31416124339492 T^{17} + 152493040628985 T^{18} + 33715686015640 p T^{19} + 2675816091260806 T^{20} + 33715686015640 p^{2} T^{21} + 152493040628985 p^{2} T^{22} + 31416124339492 p^{3} T^{23} + 8161638451030 p^{4} T^{24} + 1580820886684 p^{5} T^{25} + 24026777780 p^{7} T^{26} + 71205881473 p^{7} T^{27} + 18734675684 p^{8} T^{28} + 2784205789 p^{9} T^{29} + 757956498 p^{10} T^{30} + 91244692 p^{11} T^{31} + 25709641 p^{12} T^{32} + 2398160 p^{13} T^{33} + 688249 p^{14} T^{34} + 47287 p^{15} T^{35} + 13433 p^{16} T^{36} + 619 p^{17} T^{37} + 168 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 7 T + 206 T^{2} - 1441 T^{3} + 21594 T^{4} - 148311 T^{5} + 1532868 T^{6} - 10173127 T^{7} + 82557992 T^{8} - 522709431 T^{9} + 3579868022 T^{10} - 21420969929 T^{11} + 129424965062 T^{12} - 727382090943 T^{13} + 3987290830474 T^{14} - 20976856836709 T^{15} + 106125178431079 T^{16} - 522115505502926 T^{17} + 2460886478873694 T^{18} - 11327402839776084 T^{19} + 49943143618540832 T^{20} - 11327402839776084 p T^{21} + 2460886478873694 p^{2} T^{22} - 522115505502926 p^{3} T^{23} + 106125178431079 p^{4} T^{24} - 20976856836709 p^{5} T^{25} + 3987290830474 p^{6} T^{26} - 727382090943 p^{7} T^{27} + 129424965062 p^{8} T^{28} - 21420969929 p^{9} T^{29} + 3579868022 p^{10} T^{30} - 522709431 p^{11} T^{31} + 82557992 p^{12} T^{32} - 10173127 p^{13} T^{33} + 1532868 p^{14} T^{34} - 148311 p^{15} T^{35} + 21594 p^{16} T^{36} - 1441 p^{17} T^{37} + 206 p^{18} T^{38} - 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 28 T + 734 T^{2} - 12443 T^{3} + 195429 T^{4} - 2441390 T^{5} + 28616380 T^{6} - 286109548 T^{7} + 2720474403 T^{8} - 22805231081 T^{9} + 184428733252 T^{10} - 1340489657632 T^{11} + 9562630355712 T^{12} - 62167212657734 T^{13} + 405178142932552 T^{14} - 2433533660272167 T^{15} + 14981131940984620 T^{16} - 85549761961121342 T^{17} + 508708968702325082 T^{18} - 2807453646036113993 T^{19} + 16234616358335702678 T^{20} - 2807453646036113993 p T^{21} + 508708968702325082 p^{2} T^{22} - 85549761961121342 p^{3} T^{23} + 14981131940984620 p^{4} T^{24} - 2433533660272167 p^{5} T^{25} + 405178142932552 p^{6} T^{26} - 62167212657734 p^{7} T^{27} + 9562630355712 p^{8} T^{28} - 1340489657632 p^{9} T^{29} + 184428733252 p^{10} T^{30} - 22805231081 p^{11} T^{31} + 2720474403 p^{12} T^{32} - 286109548 p^{13} T^{33} + 28616380 p^{14} T^{34} - 2441390 p^{15} T^{35} + 195429 p^{16} T^{36} - 12443 p^{17} T^{37} + 734 p^{18} T^{38} - 28 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 14 T + 384 T^{2} - 3803 T^{3} + 61172 T^{4} - 464658 T^{5} + 5850756 T^{6} - 36539181 T^{7} + 419482784 T^{8} - 2343294339 T^{9} + 26315153804 T^{10} - 138788107216 T^{11} + 1498002688124 T^{12} - 7419001222863 T^{13} + 2041895451192 p T^{14} - 349617353886428 T^{15} + 3424235107091679 T^{16} - 15141415688005494 T^{17} + 144053314275522840 T^{18} - 614955419688354484 T^{19} + 5591421800765102880 T^{20} - 614955419688354484 p T^{21} + 144053314275522840 p^{2} T^{22} - 15141415688005494 p^{3} T^{23} + 3424235107091679 p^{4} T^{24} - 349617353886428 p^{5} T^{25} + 2041895451192 p^{7} T^{26} - 7419001222863 p^{7} T^{27} + 1498002688124 p^{8} T^{28} - 138788107216 p^{9} T^{29} + 26315153804 p^{10} T^{30} - 2343294339 p^{11} T^{31} + 419482784 p^{12} T^{32} - 36539181 p^{13} T^{33} + 5850756 p^{14} T^{34} - 464658 p^{15} T^{35} + 61172 p^{16} T^{36} - 3803 p^{17} T^{37} + 384 p^{18} T^{38} - 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 7 T + 479 T^{2} - 3033 T^{3} + 111924 T^{4} - 649297 T^{5} + 17050658 T^{6} - 92016777 T^{7} + 1912340620 T^{8} - 9765477291 T^{9} + 4126877733 p T^{10} - 831514315257 T^{11} + 12361443058494 T^{12} - 59254741384169 T^{13} + 770247956920639 T^{14} - 3624728678993295 T^{15} + 41904106814273651 T^{16} - 193018006334304006 T^{17} + 2022878003909591075 T^{18} - 9007468644099125876 T^{19} + 87485259650824791324 T^{20} - 9007468644099125876 p T^{21} + 2022878003909591075 p^{2} T^{22} - 193018006334304006 p^{3} T^{23} + 41904106814273651 p^{4} T^{24} - 3624728678993295 p^{5} T^{25} + 770247956920639 p^{6} T^{26} - 59254741384169 p^{7} T^{27} + 12361443058494 p^{8} T^{28} - 831514315257 p^{9} T^{29} + 4126877733 p^{11} T^{30} - 9765477291 p^{11} T^{31} + 1912340620 p^{12} T^{32} - 92016777 p^{13} T^{33} + 17050658 p^{14} T^{34} - 649297 p^{15} T^{35} + 111924 p^{16} T^{36} - 3033 p^{17} T^{37} + 479 p^{18} T^{38} - 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 3 T + 347 T^{2} - 881 T^{3} + 62672 T^{4} - 150103 T^{5} + 7836966 T^{6} - 19114103 T^{7} + 760747202 T^{8} - 1976582771 T^{9} + 61037238543 T^{10} - 171148563521 T^{11} + 4209454598974 T^{12} - 12607701901679 T^{13} + 256102399277881 T^{14} - 800311005423689 T^{15} + 13977998993261533 T^{16} - 44276426802708506 T^{17} + 691244233050498015 T^{18} - 2154155385698601420 T^{19} + 31126330649301424100 T^{20} - 2154155385698601420 p T^{21} + 691244233050498015 p^{2} T^{22} - 44276426802708506 p^{3} T^{23} + 13977998993261533 p^{4} T^{24} - 800311005423689 p^{5} T^{25} + 256102399277881 p^{6} T^{26} - 12607701901679 p^{7} T^{27} + 4209454598974 p^{8} T^{28} - 171148563521 p^{9} T^{29} + 61037238543 p^{10} T^{30} - 1976582771 p^{11} T^{31} + 760747202 p^{12} T^{32} - 19114103 p^{13} T^{33} + 7836966 p^{14} T^{34} - 150103 p^{15} T^{35} + 62672 p^{16} T^{36} - 881 p^{17} T^{37} + 347 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 3 T + 220 T^{2} - 455 T^{3} + 29369 T^{4} - 32791 T^{5} + 3053842 T^{6} - 1350802 T^{7} + 263467845 T^{8} + 35923295 T^{9} + 19795789858 T^{10} + 11956563230 T^{11} + 1339059446476 T^{12} + 1185220997750 T^{13} + 82693897224342 T^{14} + 86528288754835 T^{15} + 4705956226316154 T^{16} + 5264505627410837 T^{17} + 247969141553411418 T^{18} + 275049900373265592 T^{19} + 12108087115108416246 T^{20} + 275049900373265592 p T^{21} + 247969141553411418 p^{2} T^{22} + 5264505627410837 p^{3} T^{23} + 4705956226316154 p^{4} T^{24} + 86528288754835 p^{5} T^{25} + 82693897224342 p^{6} T^{26} + 1185220997750 p^{7} T^{27} + 1339059446476 p^{8} T^{28} + 11956563230 p^{9} T^{29} + 19795789858 p^{10} T^{30} + 35923295 p^{11} T^{31} + 263467845 p^{12} T^{32} - 1350802 p^{13} T^{33} + 3053842 p^{14} T^{34} - 32791 p^{15} T^{35} + 29369 p^{16} T^{36} - 455 p^{17} T^{37} + 220 p^{18} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 19 T + 771 T^{2} + 13045 T^{3} + 297433 T^{4} + 4446370 T^{5} + 75343953 T^{6} + 1000209050 T^{7} + 13969343581 T^{8} + 166226634293 T^{9} + 2010454187510 T^{10} + 21643768295159 T^{11} + 232886942885068 T^{12} + 2285759140151649 T^{13} + 22232806260698970 T^{14} + 200086387459444507 T^{15} + 1776686645323641746 T^{16} + 14718052278784708623 T^{17} + \)\(12\!\cdots\!64\)\( T^{18} + \)\(91\!\cdots\!37\)\( T^{19} + \)\(68\!\cdots\!62\)\( T^{20} + \)\(91\!\cdots\!37\)\( p T^{21} + \)\(12\!\cdots\!64\)\( p^{2} T^{22} + 14718052278784708623 p^{3} T^{23} + 1776686645323641746 p^{4} T^{24} + 200086387459444507 p^{5} T^{25} + 22232806260698970 p^{6} T^{26} + 2285759140151649 p^{7} T^{27} + 232886942885068 p^{8} T^{28} + 21643768295159 p^{9} T^{29} + 2010454187510 p^{10} T^{30} + 166226634293 p^{11} T^{31} + 13969343581 p^{12} T^{32} + 1000209050 p^{13} T^{33} + 75343953 p^{14} T^{34} + 4446370 p^{15} T^{35} + 297433 p^{16} T^{36} + 13045 p^{17} T^{37} + 771 p^{18} T^{38} + 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 20 T + 850 T^{2} - 14944 T^{3} + 359194 T^{4} - 5554132 T^{5} + 99417320 T^{6} - 1364452652 T^{7} + 20136013888 T^{8} - 248042865464 T^{9} + 3169580665114 T^{10} - 35396074222236 T^{11} + 402433552754022 T^{12} - 4106531088799116 T^{13} + 42228011673478494 T^{14} - 396014313052177864 T^{15} + 3721339647187461071 T^{16} - 32195426419238706616 T^{17} + \)\(27\!\cdots\!14\)\( T^{18} - \)\(22\!\cdots\!08\)\( T^{19} + \)\(17\!\cdots\!32\)\( T^{20} - \)\(22\!\cdots\!08\)\( p T^{21} + \)\(27\!\cdots\!14\)\( p^{2} T^{22} - 32195426419238706616 p^{3} T^{23} + 3721339647187461071 p^{4} T^{24} - 396014313052177864 p^{5} T^{25} + 42228011673478494 p^{6} T^{26} - 4106531088799116 p^{7} T^{27} + 402433552754022 p^{8} T^{28} - 35396074222236 p^{9} T^{29} + 3169580665114 p^{10} T^{30} - 248042865464 p^{11} T^{31} + 20136013888 p^{12} T^{32} - 1364452652 p^{13} T^{33} + 99417320 p^{14} T^{34} - 5554132 p^{15} T^{35} + 359194 p^{16} T^{36} - 14944 p^{17} T^{37} + 850 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 15 T + 632 T^{2} - 9178 T^{3} + 214368 T^{4} - 2896282 T^{5} + 50100198 T^{6} - 623402438 T^{7} + 8906308926 T^{8} - 102103383754 T^{9} + 1270478817982 T^{10} - 13469241787695 T^{11} + 150321742884866 T^{12} - 1479978782393335 T^{13} + 15075703455738050 T^{14} - 138321830366923864 T^{15} + 1300058924253812145 T^{16} - 11142344236564741201 T^{17} + 97284222403134078882 T^{18} - \)\(77\!\cdots\!14\)\( T^{19} + \)\(63\!\cdots\!00\)\( T^{20} - \)\(77\!\cdots\!14\)\( p T^{21} + 97284222403134078882 p^{2} T^{22} - 11142344236564741201 p^{3} T^{23} + 1300058924253812145 p^{4} T^{24} - 138321830366923864 p^{5} T^{25} + 15075703455738050 p^{6} T^{26} - 1479978782393335 p^{7} T^{27} + 150321742884866 p^{8} T^{28} - 13469241787695 p^{9} T^{29} + 1270478817982 p^{10} T^{30} - 102103383754 p^{11} T^{31} + 8906308926 p^{12} T^{32} - 623402438 p^{13} T^{33} + 50100198 p^{14} T^{34} - 2896282 p^{15} T^{35} + 214368 p^{16} T^{36} - 9178 p^{17} T^{37} + 632 p^{18} T^{38} - 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 20 T + 814 T^{2} - 12903 T^{3} + 305961 T^{4} - 4047653 T^{5} + 72254961 T^{6} - 824255236 T^{7} + 12233558357 T^{8} - 122983674302 T^{9} + 1601693140998 T^{10} - 14406388234225 T^{11} + 170654313648350 T^{12} - 1389771538600015 T^{13} + 15404833617631740 T^{14} - 115030417424373978 T^{15} + 1223404814530070450 T^{16} - 8525247538987629756 T^{17} + 88790021430906710023 T^{18} - \)\(59\!\cdots\!00\)\( T^{19} + \)\(60\!\cdots\!06\)\( T^{20} - \)\(59\!\cdots\!00\)\( p T^{21} + 88790021430906710023 p^{2} T^{22} - 8525247538987629756 p^{3} T^{23} + 1223404814530070450 p^{4} T^{24} - 115030417424373978 p^{5} T^{25} + 15404833617631740 p^{6} T^{26} - 1389771538600015 p^{7} T^{27} + 170654313648350 p^{8} T^{28} - 14406388234225 p^{9} T^{29} + 1601693140998 p^{10} T^{30} - 122983674302 p^{11} T^{31} + 12233558357 p^{12} T^{32} - 824255236 p^{13} T^{33} + 72254961 p^{14} T^{34} - 4047653 p^{15} T^{35} + 305961 p^{16} T^{36} - 12903 p^{17} T^{37} + 814 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 63 T + 2590 T^{2} - 78606 T^{3} + 1970879 T^{4} - 42229125 T^{5} + 800998266 T^{6} - 13664206408 T^{7} + 213204472880 T^{8} - 3070903631126 T^{9} + 41242902030686 T^{10} - 519647975394003 T^{11} + 6184547496360173 T^{12} - 69833932774379894 T^{13} + 751885537417647114 T^{14} - 7744065320700227311 T^{15} + 76579426796513361691 T^{16} - \)\(72\!\cdots\!30\)\( T^{17} + \)\(66\!\cdots\!44\)\( T^{18} - \)\(59\!\cdots\!46\)\( T^{19} + \)\(50\!\cdots\!04\)\( T^{20} - \)\(59\!\cdots\!46\)\( p T^{21} + \)\(66\!\cdots\!44\)\( p^{2} T^{22} - \)\(72\!\cdots\!30\)\( p^{3} T^{23} + 76579426796513361691 p^{4} T^{24} - 7744065320700227311 p^{5} T^{25} + 751885537417647114 p^{6} T^{26} - 69833932774379894 p^{7} T^{27} + 6184547496360173 p^{8} T^{28} - 519647975394003 p^{9} T^{29} + 41242902030686 p^{10} T^{30} - 3070903631126 p^{11} T^{31} + 213204472880 p^{12} T^{32} - 13664206408 p^{13} T^{33} + 800998266 p^{14} T^{34} - 42229125 p^{15} T^{35} + 1970879 p^{16} T^{36} - 78606 p^{17} T^{37} + 2590 p^{18} T^{38} - 63 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 19 T + 769 T^{2} - 10982 T^{3} + 276918 T^{4} - 3345467 T^{5} + 66999737 T^{6} - 718002144 T^{7} + 12387233437 T^{8} - 120457948023 T^{9} + 1864921663172 T^{10} - 16704013939955 T^{11} + 237570269110792 T^{12} - 1979196282339905 T^{13} + 26221041043135092 T^{14} - 204595515703267829 T^{15} + 2546979484335337090 T^{16} - 18708000306500559833 T^{17} + \)\(21\!\cdots\!82\)\( T^{18} - \)\(15\!\cdots\!55\)\( T^{19} + \)\(16\!\cdots\!56\)\( T^{20} - \)\(15\!\cdots\!55\)\( p T^{21} + \)\(21\!\cdots\!82\)\( p^{2} T^{22} - 18708000306500559833 p^{3} T^{23} + 2546979484335337090 p^{4} T^{24} - 204595515703267829 p^{5} T^{25} + 26221041043135092 p^{6} T^{26} - 1979196282339905 p^{7} T^{27} + 237570269110792 p^{8} T^{28} - 16704013939955 p^{9} T^{29} + 1864921663172 p^{10} T^{30} - 120457948023 p^{11} T^{31} + 12387233437 p^{12} T^{32} - 718002144 p^{13} T^{33} + 66999737 p^{14} T^{34} - 3345467 p^{15} T^{35} + 276918 p^{16} T^{36} - 10982 p^{17} T^{37} + 769 p^{18} T^{38} - 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 32 T + 1430 T^{2} - 33799 T^{3} + 909433 T^{4} - 17532202 T^{5} + 360187910 T^{6} - 5951638450 T^{7} + 101789545993 T^{8} - 1483644502631 T^{9} + 22048084895066 T^{10} - 288634171288938 T^{11} + 3819726063800976 T^{12} - 45443454703136376 T^{13} + 543762219529712722 T^{14} - 5924597129880906313 T^{15} + 64725468439977681990 T^{16} - \)\(64\!\cdots\!24\)\( T^{17} + 82443301352125850056 p T^{18} - \)\(60\!\cdots\!41\)\( T^{19} + \)\(55\!\cdots\!06\)\( T^{20} - \)\(60\!\cdots\!41\)\( p T^{21} + 82443301352125850056 p^{3} T^{22} - \)\(64\!\cdots\!24\)\( p^{3} T^{23} + 64725468439977681990 p^{4} T^{24} - 5924597129880906313 p^{5} T^{25} + 543762219529712722 p^{6} T^{26} - 45443454703136376 p^{7} T^{27} + 3819726063800976 p^{8} T^{28} - 288634171288938 p^{9} T^{29} + 22048084895066 p^{10} T^{30} - 1483644502631 p^{11} T^{31} + 101789545993 p^{12} T^{32} - 5951638450 p^{13} T^{33} + 360187910 p^{14} T^{34} - 17532202 p^{15} T^{35} + 909433 p^{16} T^{36} - 33799 p^{17} T^{37} + 1430 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 21 T + 1119 T^{2} + 19471 T^{3} + 599759 T^{4} + 108882 p T^{5} + 208286349 T^{6} + 2790710326 T^{7} + 53051147853 T^{8} + 643058468719 T^{9} + 10596342405190 T^{10} + 117601394018569 T^{11} + 1729819168571668 T^{12} + 17729018166575289 T^{13} + 237214420638684586 T^{14} + 2259038398021528647 T^{15} + 27841006306951857538 T^{16} + \)\(24\!\cdots\!61\)\( T^{17} + \)\(28\!\cdots\!36\)\( T^{18} + \)\(23\!\cdots\!27\)\( T^{19} + \)\(25\!\cdots\!66\)\( T^{20} + \)\(23\!\cdots\!27\)\( p T^{21} + \)\(28\!\cdots\!36\)\( p^{2} T^{22} + \)\(24\!\cdots\!61\)\( p^{3} T^{23} + 27841006306951857538 p^{4} T^{24} + 2259038398021528647 p^{5} T^{25} + 237214420638684586 p^{6} T^{26} + 17729018166575289 p^{7} T^{27} + 1729819168571668 p^{8} T^{28} + 117601394018569 p^{9} T^{29} + 10596342405190 p^{10} T^{30} + 643058468719 p^{11} T^{31} + 53051147853 p^{12} T^{32} + 2790710326 p^{13} T^{33} + 208286349 p^{14} T^{34} + 108882 p^{16} T^{35} + 599759 p^{16} T^{36} + 19471 p^{17} T^{37} + 1119 p^{18} T^{38} + 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 13 T + 1227 T^{2} + 13857 T^{3} + 731036 T^{4} + 7298054 T^{5} + 283553347 T^{6} + 2535583981 T^{7} + 80777199578 T^{8} + 653711712400 T^{9} + 18035127148077 T^{10} + 133173693213556 T^{11} + 3282443098361722 T^{12} + 22259188527282398 T^{13} + 499356299606160985 T^{14} + 3125596297317945832 T^{15} + 64535473127533893589 T^{16} + \)\(37\!\cdots\!66\)\( T^{17} + \)\(71\!\cdots\!52\)\( T^{18} + \)\(38\!\cdots\!95\)\( T^{19} + \)\(68\!\cdots\!28\)\( T^{20} + \)\(38\!\cdots\!95\)\( p T^{21} + \)\(71\!\cdots\!52\)\( p^{2} T^{22} + \)\(37\!\cdots\!66\)\( p^{3} T^{23} + 64535473127533893589 p^{4} T^{24} + 3125596297317945832 p^{5} T^{25} + 499356299606160985 p^{6} T^{26} + 22259188527282398 p^{7} T^{27} + 3282443098361722 p^{8} T^{28} + 133173693213556 p^{9} T^{29} + 18035127148077 p^{10} T^{30} + 653711712400 p^{11} T^{31} + 80777199578 p^{12} T^{32} + 2535583981 p^{13} T^{33} + 283553347 p^{14} T^{34} + 7298054 p^{15} T^{35} + 731036 p^{16} T^{36} + 13857 p^{17} T^{37} + 1227 p^{18} T^{38} + 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 9 T + 772 T^{2} + 3956 T^{3} + 295553 T^{4} + 6921 p T^{5} + 79467812 T^{6} + 499520 T^{7} + 17138414889 T^{8} - 30145470491 T^{9} + 3144529120192 T^{10} - 9528963923781 T^{11} + 505253771785252 T^{12} - 1941916199657855 T^{13} + 72070436975111872 T^{14} - 308292019346329785 T^{15} + 9183383407813211734 T^{16} - 40631474133499742421 T^{17} + \)\(10\!\cdots\!52\)\( T^{18} - \)\(45\!\cdots\!29\)\( T^{19} + \)\(10\!\cdots\!22\)\( T^{20} - \)\(45\!\cdots\!29\)\( p T^{21} + \)\(10\!\cdots\!52\)\( p^{2} T^{22} - 40631474133499742421 p^{3} T^{23} + 9183383407813211734 p^{4} T^{24} - 308292019346329785 p^{5} T^{25} + 72070436975111872 p^{6} T^{26} - 1941916199657855 p^{7} T^{27} + 505253771785252 p^{8} T^{28} - 9528963923781 p^{9} T^{29} + 3144529120192 p^{10} T^{30} - 30145470491 p^{11} T^{31} + 17138414889 p^{12} T^{32} + 499520 p^{13} T^{33} + 79467812 p^{14} T^{34} + 6921 p^{16} T^{35} + 295553 p^{16} T^{36} + 3956 p^{17} T^{37} + 772 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.00924852733625435686250778000, −1.95396464788166910518470597010, −1.93100420637295074867136751316, −1.88361616150578850373202807044, −1.75720142666994672631958217711, −1.69266637221762260836288658916, −1.56758925097933240637642518541, −1.56299332592609180941386181846, −1.48675419053447724075631121365, −1.47320495248686258520529750090, −1.34477859443964746919688062157, −1.25413669415418901006054657701, −1.22279830034674351542939807868, −1.19825533744524992424105785928, −1.10745230094542969779487931233, −1.08512967675122435718368959560, −1.02777379449661887524363401483, −0.973548566113093204979476594524, −0.841246478205455246607994161795, −0.826454854217917746203835466043, −0.822193626087309992395528299991, −0.75881951106758693244342627331, −0.55379191183042260852570291370, −0.35008251248634018234921751290, −0.25704280602051626612447957810, 0.25704280602051626612447957810, 0.35008251248634018234921751290, 0.55379191183042260852570291370, 0.75881951106758693244342627331, 0.822193626087309992395528299991, 0.826454854217917746203835466043, 0.841246478205455246607994161795, 0.973548566113093204979476594524, 1.02777379449661887524363401483, 1.08512967675122435718368959560, 1.10745230094542969779487931233, 1.19825533744524992424105785928, 1.22279830034674351542939807868, 1.25413669415418901006054657701, 1.34477859443964746919688062157, 1.47320495248686258520529750090, 1.48675419053447724075631121365, 1.56299332592609180941386181846, 1.56758925097933240637642518541, 1.69266637221762260836288658916, 1.75720142666994672631958217711, 1.88361616150578850373202807044, 1.93100420637295074867136751316, 1.95396464788166910518470597010, 2.00924852733625435686250778000
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.