Dirichlet series
| L(s) = 1 | − 2·3-s + 2·5-s − 23·9-s − 9·11-s + 7·13-s − 4·15-s − 8·17-s − 19-s + 11·23-s + 22·25-s + 50·27-s + 21·29-s + 5·31-s + 18·33-s − 4·37-s − 14·39-s + 9·41-s + 17·43-s − 46·45-s − 15·47-s + 5·49-s + 16·51-s + 10·53-s − 18·55-s + 2·57-s + 24·59-s + 15·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 7.66·9-s − 2.71·11-s + 1.94·13-s − 1.03·15-s − 1.94·17-s − 0.229·19-s + 2.29·23-s + 22/5·25-s + 9.62·27-s + 3.89·29-s + 0.898·31-s + 3.13·33-s − 0.657·37-s − 2.24·39-s + 1.40·41-s + 2.59·43-s − 6.85·45-s − 2.18·47-s + 5/7·49-s + 2.24·51-s + 1.37·53-s − 2.42·55-s + 0.264·57-s + 3.12·59-s + 1.92·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 41^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 41^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 41^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.41890\times 10^{14}\) |
| Root analytic conductor: | \(2.28870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 41^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.422239910\) |
| \(L(\frac12)\) | \(\approx\) | \(8.422239910\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 - 9 T + 3 T^{2} - 62 T^{3} + 5701 T^{4} - 42350 T^{5} + 126209 T^{6} - 959478 T^{7} + 12371337 T^{8} - 78853201 T^{9} + 424461698 T^{10} - 78853201 p T^{11} + 12371337 p^{2} T^{12} - 959478 p^{3} T^{13} + 126209 p^{4} T^{14} - 42350 p^{5} T^{15} + 5701 p^{6} T^{16} - 62 p^{7} T^{17} + 3 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| good | 3 | \( ( 1 + T + 13 T^{2} + 4 p T^{3} + 95 T^{4} + 8 p^{2} T^{5} + 53 p^{2} T^{6} + 302 T^{7} + 1873 T^{8} + 113 p^{2} T^{9} + 6098 T^{10} + 113 p^{3} T^{11} + 1873 p^{2} T^{12} + 302 p^{3} T^{13} + 53 p^{6} T^{14} + 8 p^{7} T^{15} + 95 p^{6} T^{16} + 4 p^{8} T^{17} + 13 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) |
| 5 | \( 1 - 2 T - 18 T^{2} + 31 T^{3} + 177 T^{4} - 152 T^{5} - 1469 T^{6} - 41 T^{7} + 10133 T^{8} + 4097 T^{9} - 52209 T^{10} - 26463 T^{11} + 232411 T^{12} + 2003 p^{2} T^{13} - 2027 p^{4} T^{14} + 5084 p^{3} T^{15} + 345503 p^{2} T^{16} - 43909 p^{3} T^{17} - 92126 p^{4} T^{18} + 21292 p^{4} T^{19} + 515408 p^{4} T^{20} + 21292 p^{5} T^{21} - 92126 p^{6} T^{22} - 43909 p^{6} T^{23} + 345503 p^{6} T^{24} + 5084 p^{8} T^{25} - 2027 p^{10} T^{26} + 2003 p^{9} T^{27} + 232411 p^{8} T^{28} - 26463 p^{9} T^{29} - 52209 p^{10} T^{30} + 4097 p^{11} T^{31} + 10133 p^{12} T^{32} - 41 p^{13} T^{33} - 1469 p^{14} T^{34} - 152 p^{15} T^{35} + 177 p^{16} T^{36} + 31 p^{17} T^{37} - 18 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 - 5 T^{2} - 6 T^{3} + 57 T^{4} - 34 T^{5} + 145 T^{6} - 4 p T^{7} + 332 T^{8} - 2545 T^{9} - 7671 T^{10} - 36177 T^{11} + 166132 T^{12} + 206918 T^{13} - 1151819 T^{14} + 663154 T^{15} + 3494025 T^{16} - 8647316 T^{17} - 21930507 T^{18} + 117280208 T^{19} + 11924732 T^{20} + 117280208 p T^{21} - 21930507 p^{2} T^{22} - 8647316 p^{3} T^{23} + 3494025 p^{4} T^{24} + 663154 p^{5} T^{25} - 1151819 p^{6} T^{26} + 206918 p^{7} T^{27} + 166132 p^{8} T^{28} - 36177 p^{9} T^{29} - 7671 p^{10} T^{30} - 2545 p^{11} T^{31} + 332 p^{12} T^{32} - 4 p^{14} T^{33} + 145 p^{14} T^{34} - 34 p^{15} T^{35} + 57 p^{16} T^{36} - 6 p^{17} T^{37} - 5 p^{18} T^{38} + p^{20} T^{40} \) | |
| 11 | \( 1 + 9 T + 12 T^{2} - 14 p T^{3} - 647 T^{4} - 258 T^{5} + 897 T^{6} - 10907 T^{7} - 6605 T^{8} + 334223 T^{9} + 157556 p T^{10} + 4278811 T^{11} - 19395 p T^{12} - 54736288 T^{13} - 215945745 T^{14} - 380066338 T^{15} - 996175385 T^{16} - 474891651 p T^{17} - 4478607372 T^{18} + 96104331757 T^{19} + 521679218674 T^{20} + 96104331757 p T^{21} - 4478607372 p^{2} T^{22} - 474891651 p^{4} T^{23} - 996175385 p^{4} T^{24} - 380066338 p^{5} T^{25} - 215945745 p^{6} T^{26} - 54736288 p^{7} T^{27} - 19395 p^{9} T^{28} + 4278811 p^{9} T^{29} + 157556 p^{11} T^{30} + 334223 p^{11} T^{31} - 6605 p^{12} T^{32} - 10907 p^{13} T^{33} + 897 p^{14} T^{34} - 258 p^{15} T^{35} - 647 p^{16} T^{36} - 14 p^{18} T^{37} + 12 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 - 7 T - 29 T^{2} + 168 T^{3} + 1091 T^{4} - 3566 T^{5} - 13682 T^{6} + 20278 T^{7} + 123295 T^{8} - 442372 T^{9} + 526880 T^{10} + 9579823 T^{11} + 9814767 T^{12} - 182881084 T^{13} - 454796647 T^{14} + 1532008330 T^{15} + 4611142464 T^{16} - 113792348 T^{17} + 6664867233 T^{18} + 12056501818 T^{19} - 978876403738 T^{20} + 12056501818 p T^{21} + 6664867233 p^{2} T^{22} - 113792348 p^{3} T^{23} + 4611142464 p^{4} T^{24} + 1532008330 p^{5} T^{25} - 454796647 p^{6} T^{26} - 182881084 p^{7} T^{27} + 9814767 p^{8} T^{28} + 9579823 p^{9} T^{29} + 526880 p^{10} T^{30} - 442372 p^{11} T^{31} + 123295 p^{12} T^{32} + 20278 p^{13} T^{33} - 13682 p^{14} T^{34} - 3566 p^{15} T^{35} + 1091 p^{16} T^{36} + 168 p^{17} T^{37} - 29 p^{18} T^{38} - 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 8 T - 8 T^{2} - 375 T^{3} - 1237 T^{4} + 5396 T^{5} + 43192 T^{6} - 13681 T^{7} - 862854 T^{8} - 1163196 T^{9} + 16428002 T^{10} + 62032288 T^{11} - 162202641 T^{12} - 1465601788 T^{13} - 771422918 T^{14} + 19859445592 T^{15} + 39370285687 T^{16} - 283972044360 T^{17} - 1288307036053 T^{18} + 2624256486096 T^{19} + 31164019493538 T^{20} + 2624256486096 p T^{21} - 1288307036053 p^{2} T^{22} - 283972044360 p^{3} T^{23} + 39370285687 p^{4} T^{24} + 19859445592 p^{5} T^{25} - 771422918 p^{6} T^{26} - 1465601788 p^{7} T^{27} - 162202641 p^{8} T^{28} + 62032288 p^{9} T^{29} + 16428002 p^{10} T^{30} - 1163196 p^{11} T^{31} - 862854 p^{12} T^{32} - 13681 p^{13} T^{33} + 43192 p^{14} T^{34} + 5396 p^{15} T^{35} - 1237 p^{16} T^{36} - 375 p^{17} T^{37} - 8 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + T - 102 T^{2} - 85 T^{3} + 4512 T^{4} + 2576 T^{5} - 106824 T^{6} + 28374 T^{7} + 1140303 T^{8} - 5084126 T^{9} + 7940112 T^{10} + 179952736 T^{11} - 411641923 T^{12} - 2683035874 T^{13} + 3557740836 T^{14} - 9611859148 T^{15} + 57822058812 T^{16} + 1059894977663 T^{17} - 2083718489062 T^{18} - 11644488072997 T^{19} + 40587003299886 T^{20} - 11644488072997 p T^{21} - 2083718489062 p^{2} T^{22} + 1059894977663 p^{3} T^{23} + 57822058812 p^{4} T^{24} - 9611859148 p^{5} T^{25} + 3557740836 p^{6} T^{26} - 2683035874 p^{7} T^{27} - 411641923 p^{8} T^{28} + 179952736 p^{9} T^{29} + 7940112 p^{10} T^{30} - 5084126 p^{11} T^{31} + 1140303 p^{12} T^{32} + 28374 p^{13} T^{33} - 106824 p^{14} T^{34} + 2576 p^{15} T^{35} + 4512 p^{16} T^{36} - 85 p^{17} T^{37} - 102 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 11 T + 56 T^{2} + 181 T^{3} - 3461 T^{4} + 12313 T^{5} + 63586 T^{6} - 784903 T^{7} + 1375231 T^{8} + 17754984 T^{9} - 124635360 T^{10} + 15599494 T^{11} + 3200845849 T^{12} - 12049047547 T^{13} - 36170683354 T^{14} + 429906740185 T^{15} - 522289235355 T^{16} - 8359968625891 T^{17} + 39429302894940 T^{18} + 2888176713773 p T^{19} - 1087201183223882 T^{20} + 2888176713773 p^{2} T^{21} + 39429302894940 p^{2} T^{22} - 8359968625891 p^{3} T^{23} - 522289235355 p^{4} T^{24} + 429906740185 p^{5} T^{25} - 36170683354 p^{6} T^{26} - 12049047547 p^{7} T^{27} + 3200845849 p^{8} T^{28} + 15599494 p^{9} T^{29} - 124635360 p^{10} T^{30} + 17754984 p^{11} T^{31} + 1375231 p^{12} T^{32} - 784903 p^{13} T^{33} + 63586 p^{14} T^{34} + 12313 p^{15} T^{35} - 3461 p^{16} T^{36} + 181 p^{17} T^{37} + 56 p^{18} T^{38} - 11 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 21 T + 168 T^{2} - 590 T^{3} + 2083 T^{4} - 44386 T^{5} + 495893 T^{6} - 2078974 T^{7} - 1039907 T^{8} + 8467387 T^{9} + 465151540 T^{10} - 3887983447 T^{11} + 3085081822 T^{12} + 82015133509 T^{13} - 22884857967 T^{14} - 4426780619284 T^{15} + 21211538125789 T^{16} + 39158020192415 T^{17} - 432585091999847 T^{18} - 2154371242931429 T^{19} + 28710060631814386 T^{20} - 2154371242931429 p T^{21} - 432585091999847 p^{2} T^{22} + 39158020192415 p^{3} T^{23} + 21211538125789 p^{4} T^{24} - 4426780619284 p^{5} T^{25} - 22884857967 p^{6} T^{26} + 82015133509 p^{7} T^{27} + 3085081822 p^{8} T^{28} - 3887983447 p^{9} T^{29} + 465151540 p^{10} T^{30} + 8467387 p^{11} T^{31} - 1039907 p^{12} T^{32} - 2078974 p^{13} T^{33} + 495893 p^{14} T^{34} - 44386 p^{15} T^{35} + 2083 p^{16} T^{36} - 590 p^{17} T^{37} + 168 p^{18} T^{38} - 21 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 5 T - 105 T^{2} + 575 T^{3} + 2794 T^{4} - 15981 T^{5} + 59149 T^{6} - 468045 T^{7} - 3409364 T^{8} + 24232367 T^{9} - 35467033 T^{10} + 506845967 T^{11} + 5594818630 T^{12} - 44286124733 T^{13} - 169031171039 T^{14} + 286753892195 T^{15} + 2087180699352 T^{16} + 42757550415151 T^{17} - 10850266775609 T^{18} - 946742678949605 T^{19} + 373776643622176 T^{20} - 946742678949605 p T^{21} - 10850266775609 p^{2} T^{22} + 42757550415151 p^{3} T^{23} + 2087180699352 p^{4} T^{24} + 286753892195 p^{5} T^{25} - 169031171039 p^{6} T^{26} - 44286124733 p^{7} T^{27} + 5594818630 p^{8} T^{28} + 506845967 p^{9} T^{29} - 35467033 p^{10} T^{30} + 24232367 p^{11} T^{31} - 3409364 p^{12} T^{32} - 468045 p^{13} T^{33} + 59149 p^{14} T^{34} - 15981 p^{15} T^{35} + 2794 p^{16} T^{36} + 575 p^{17} T^{37} - 105 p^{18} T^{38} - 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + 4 T - 4 p T^{2} - 867 T^{3} + 8505 T^{4} + 89508 T^{5} - 115026 T^{6} - 5420147 T^{7} - 17646002 T^{8} + 4897908 p T^{9} + 1730237336 T^{10} + 141441986 T^{11} - 87348970675 T^{12} - 416841270618 T^{13} + 2231110154048 T^{14} + 27760185431994 T^{15} + 25120048276491 T^{16} - 1011690284547368 T^{17} - 5097637591967643 T^{18} + 15680755071579124 T^{19} + 252012500174482418 T^{20} + 15680755071579124 p T^{21} - 5097637591967643 p^{2} T^{22} - 1011690284547368 p^{3} T^{23} + 25120048276491 p^{4} T^{24} + 27760185431994 p^{5} T^{25} + 2231110154048 p^{6} T^{26} - 416841270618 p^{7} T^{27} - 87348970675 p^{8} T^{28} + 141441986 p^{9} T^{29} + 1730237336 p^{10} T^{30} + 4897908 p^{12} T^{31} - 17646002 p^{12} T^{32} - 5420147 p^{13} T^{33} - 115026 p^{14} T^{34} + 89508 p^{15} T^{35} + 8505 p^{16} T^{36} - 867 p^{17} T^{37} - 4 p^{19} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 17 T - 36 T^{2} + 2377 T^{3} - 12415 T^{4} - 102209 T^{5} + 1717682 T^{6} - 6062487 T^{7} - 75863519 T^{8} + 913896794 T^{9} - 1505680532 T^{10} - 40560497410 T^{11} + 344088497373 T^{12} - 105103915073 T^{13} - 16417561572904 T^{14} + 101736055024095 T^{15} + 156543640509561 T^{16} - 5440534831341429 T^{17} + 24392933069947746 T^{18} + 102839798359529371 T^{19} - 1634332613736760874 T^{20} + 102839798359529371 p T^{21} + 24392933069947746 p^{2} T^{22} - 5440534831341429 p^{3} T^{23} + 156543640509561 p^{4} T^{24} + 101736055024095 p^{5} T^{25} - 16417561572904 p^{6} T^{26} - 105103915073 p^{7} T^{27} + 344088497373 p^{8} T^{28} - 40560497410 p^{9} T^{29} - 1505680532 p^{10} T^{30} + 913896794 p^{11} T^{31} - 75863519 p^{12} T^{32} - 6062487 p^{13} T^{33} + 1717682 p^{14} T^{34} - 102209 p^{15} T^{35} - 12415 p^{16} T^{36} + 2377 p^{17} T^{37} - 36 p^{18} T^{38} - 17 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 + 15 T + 14 T^{2} - 778 T^{3} - 3122 T^{4} - 26217 T^{5} - 448642 T^{6} + 227993 T^{7} + 33508946 T^{8} + 89010680 T^{9} - 219122573 T^{10} + 6889199614 T^{11} + 27165458910 T^{12} - 446901039505 T^{13} - 1650899369750 T^{14} + 1784008230263 T^{15} - 170433113129798 T^{16} - 1127447316423412 T^{17} + 6138915639658658 T^{18} + 53995016181882725 T^{19} + 129298613469657512 T^{20} + 53995016181882725 p T^{21} + 6138915639658658 p^{2} T^{22} - 1127447316423412 p^{3} T^{23} - 170433113129798 p^{4} T^{24} + 1784008230263 p^{5} T^{25} - 1650899369750 p^{6} T^{26} - 446901039505 p^{7} T^{27} + 27165458910 p^{8} T^{28} + 6889199614 p^{9} T^{29} - 219122573 p^{10} T^{30} + 89010680 p^{11} T^{31} + 33508946 p^{12} T^{32} + 227993 p^{13} T^{33} - 448642 p^{14} T^{34} - 26217 p^{15} T^{35} - 3122 p^{16} T^{36} - 778 p^{17} T^{37} + 14 p^{18} T^{38} + 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 - 10 T - 157 T^{2} + 1842 T^{3} + 11706 T^{4} - 152032 T^{5} - 845326 T^{6} + 9726736 T^{7} + 63832201 T^{8} - 688404876 T^{9} - 2944423460 T^{10} + 43580002348 T^{11} + 47572033191 T^{12} - 2201064504992 T^{13} + 2438114016298 T^{14} + 102504622851664 T^{15} - 307747219971574 T^{16} - 4254626963047626 T^{17} + 25975897369848197 T^{18} + 89475969424245922 T^{19} - 1628919279204150634 T^{20} + 89475969424245922 p T^{21} + 25975897369848197 p^{2} T^{22} - 4254626963047626 p^{3} T^{23} - 307747219971574 p^{4} T^{24} + 102504622851664 p^{5} T^{25} + 2438114016298 p^{6} T^{26} - 2201064504992 p^{7} T^{27} + 47572033191 p^{8} T^{28} + 43580002348 p^{9} T^{29} - 2944423460 p^{10} T^{30} - 688404876 p^{11} T^{31} + 63832201 p^{12} T^{32} + 9726736 p^{13} T^{33} - 845326 p^{14} T^{34} - 152032 p^{15} T^{35} + 11706 p^{16} T^{36} + 1842 p^{17} T^{37} - 157 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 24 T - 40 T^{2} + 85 p T^{3} - 16185 T^{4} - 570520 T^{5} + 3515953 T^{6} + 44021055 T^{7} - 399021101 T^{8} - 2930373211 T^{9} + 36565794465 T^{10} + 174782742207 T^{11} - 2883870571045 T^{12} - 9471711136525 T^{13} + 205546235636407 T^{14} + 450329841851460 T^{15} - 13717690642568967 T^{16} - 19424720164309605 T^{17} + 903786858823379770 T^{18} + 429523301391659598 T^{19} - 55755823963647478540 T^{20} + 429523301391659598 p T^{21} + 903786858823379770 p^{2} T^{22} - 19424720164309605 p^{3} T^{23} - 13717690642568967 p^{4} T^{24} + 450329841851460 p^{5} T^{25} + 205546235636407 p^{6} T^{26} - 9471711136525 p^{7} T^{27} - 2883870571045 p^{8} T^{28} + 174782742207 p^{9} T^{29} + 36565794465 p^{10} T^{30} - 2930373211 p^{11} T^{31} - 399021101 p^{12} T^{32} + 44021055 p^{13} T^{33} + 3515953 p^{14} T^{34} - 570520 p^{15} T^{35} - 16185 p^{16} T^{36} + 85 p^{18} T^{37} - 40 p^{18} T^{38} - 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 15 T - 13 T^{2} + 719 T^{3} + 4545 T^{4} - 29316 T^{5} - 175701 T^{6} - 4363303 T^{7} + 25885135 T^{8} + 371155541 T^{9} - 503066209 T^{10} - 24619885542 T^{11} + 31934903590 T^{12} - 289417106442 T^{13} + 8186334603782 T^{14} + 31512565937344 T^{15} - 82137423212514 T^{16} - 5518038656488834 T^{17} + 5910441363554448 T^{18} + 38048774752677084 T^{19} + 1554836829753879760 T^{20} + 38048774752677084 p T^{21} + 5910441363554448 p^{2} T^{22} - 5518038656488834 p^{3} T^{23} - 82137423212514 p^{4} T^{24} + 31512565937344 p^{5} T^{25} + 8186334603782 p^{6} T^{26} - 289417106442 p^{7} T^{27} + 31934903590 p^{8} T^{28} - 24619885542 p^{9} T^{29} - 503066209 p^{10} T^{30} + 371155541 p^{11} T^{31} + 25885135 p^{12} T^{32} - 4363303 p^{13} T^{33} - 175701 p^{14} T^{34} - 29316 p^{15} T^{35} + 4545 p^{16} T^{36} + 719 p^{17} T^{37} - 13 p^{18} T^{38} - 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 26 T + 131 T^{2} + 2486 T^{3} - 26356 T^{4} - 28644 T^{5} - 479902 T^{6} + 25867620 T^{7} + 29847515 T^{8} - 3300020320 T^{9} + 9271541748 T^{10} + 95887541836 T^{11} + 15958172583 p T^{12} - 11303443652592 T^{13} - 156766945973614 T^{14} + 1530578550966668 T^{15} + 4925874502173840 T^{16} - 28882218573775350 T^{17} - 645750563442024843 T^{18} - 1466224572553512710 T^{19} + 80137359791953116966 T^{20} - 1466224572553512710 p T^{21} - 645750563442024843 p^{2} T^{22} - 28882218573775350 p^{3} T^{23} + 4925874502173840 p^{4} T^{24} + 1530578550966668 p^{5} T^{25} - 156766945973614 p^{6} T^{26} - 11303443652592 p^{7} T^{27} + 15958172583 p^{9} T^{28} + 95887541836 p^{9} T^{29} + 9271541748 p^{10} T^{30} - 3300020320 p^{11} T^{31} + 29847515 p^{12} T^{32} + 25867620 p^{13} T^{33} - 479902 p^{14} T^{34} - 28644 p^{15} T^{35} - 26356 p^{16} T^{36} + 2486 p^{17} T^{37} + 131 p^{18} T^{38} - 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 16 T - 65 T^{2} - 3126 T^{3} - 12171 T^{4} + 270122 T^{5} + 2371176 T^{6} - 13021686 T^{7} - 242044431 T^{8} + 24107502 T^{9} + 15709259020 T^{10} + 46160258696 T^{11} - 670145982987 T^{12} - 3127425431038 T^{13} + 21332425074462 T^{14} - 10646895695510 T^{15} - 1726267818835135 T^{16} + 14878568241004562 T^{17} + 278158510629408771 T^{18} - 599268073317739658 T^{19} - 25871231225599861506 T^{20} - 599268073317739658 p T^{21} + 278158510629408771 p^{2} T^{22} + 14878568241004562 p^{3} T^{23} - 1726267818835135 p^{4} T^{24} - 10646895695510 p^{5} T^{25} + 21332425074462 p^{6} T^{26} - 3127425431038 p^{7} T^{27} - 670145982987 p^{8} T^{28} + 46160258696 p^{9} T^{29} + 15709259020 p^{10} T^{30} + 24107502 p^{11} T^{31} - 242044431 p^{12} T^{32} - 13021686 p^{13} T^{33} + 2371176 p^{14} T^{34} + 270122 p^{15} T^{35} - 12171 p^{16} T^{36} - 3126 p^{17} T^{37} - 65 p^{18} T^{38} + 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( ( 1 - 7 T + 555 T^{2} - 4306 T^{3} + 149805 T^{4} - 1157875 T^{5} + 25714283 T^{6} - 184563223 T^{7} + 3064477446 T^{8} - 19508812197 T^{9} + 262526452332 T^{10} - 19508812197 p T^{11} + 3064477446 p^{2} T^{12} - 184563223 p^{3} T^{13} + 25714283 p^{4} T^{14} - 1157875 p^{5} T^{15} + 149805 p^{6} T^{16} - 4306 p^{7} T^{17} + 555 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 13 T + 604 T^{2} - 6615 T^{3} + 168772 T^{4} - 1589456 T^{5} + 29353950 T^{6} - 241092840 T^{7} + 3585539259 T^{8} - 25812441618 T^{9} + 325643913868 T^{10} - 25812441618 p T^{11} + 3585539259 p^{2} T^{12} - 241092840 p^{3} T^{13} + 29353950 p^{4} T^{14} - 1589456 p^{5} T^{15} + 168772 p^{6} T^{16} - 6615 p^{7} T^{17} + 604 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 10 T + 422 T^{2} + 5622 T^{3} + 98270 T^{4} + 1384068 T^{5} + 17010962 T^{6} + 206641518 T^{7} + 27257262 p T^{8} + 21977980962 T^{9} + 221269601526 T^{10} + 21977980962 p T^{11} + 27257262 p^{3} T^{12} + 206641518 p^{3} T^{13} + 17010962 p^{4} T^{14} + 1384068 p^{5} T^{15} + 98270 p^{6} T^{16} + 5622 p^{7} T^{17} + 422 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 - 5 T - 285 T^{2} + 3604 T^{3} + 31521 T^{4} - 845204 T^{5} + 1321610 T^{6} + 98427762 T^{7} - 808587651 T^{8} - 4472085302 T^{9} + 92664468470 T^{10} - 277384981719 T^{11} - 2944379366453 T^{12} + 27262025584534 T^{13} - 319655106343753 T^{14} + 4374169857761404 T^{15} + 7466376517273106 T^{16} - 875414196891742620 T^{17} + 7233156046022809013 T^{18} + 42439429601894914062 T^{19} - \)\(10\!\cdots\!42\)\( T^{20} + 42439429601894914062 p T^{21} + 7233156046022809013 p^{2} T^{22} - 875414196891742620 p^{3} T^{23} + 7466376517273106 p^{4} T^{24} + 4374169857761404 p^{5} T^{25} - 319655106343753 p^{6} T^{26} + 27262025584534 p^{7} T^{27} - 2944379366453 p^{8} T^{28} - 277384981719 p^{9} T^{29} + 92664468470 p^{10} T^{30} - 4472085302 p^{11} T^{31} - 808587651 p^{12} T^{32} + 98427762 p^{13} T^{33} + 1321610 p^{14} T^{34} - 845204 p^{15} T^{35} + 31521 p^{16} T^{36} + 3604 p^{17} T^{37} - 285 p^{18} T^{38} - 5 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 + 22 T + 187 T^{2} + 2485 T^{3} + 48895 T^{4} + 460323 T^{5} + 4039236 T^{6} + 75926021 T^{7} + 1014920480 T^{8} + 9820451880 T^{9} + 116812908927 T^{10} + 1475491281544 T^{11} + 15152629754290 T^{12} + 147472221320687 T^{13} + 1714872411533062 T^{14} + 20097364620086825 T^{15} + 193328681050631775 T^{16} + 1921869540837751023 T^{17} + 22142884451167380571 T^{18} + \)\(21\!\cdots\!06\)\( T^{19} + \)\(18\!\cdots\!76\)\( T^{20} + \)\(21\!\cdots\!06\)\( p T^{21} + 22142884451167380571 p^{2} T^{22} + 1921869540837751023 p^{3} T^{23} + 193328681050631775 p^{4} T^{24} + 20097364620086825 p^{5} T^{25} + 1714872411533062 p^{6} T^{26} + 147472221320687 p^{7} T^{27} + 15152629754290 p^{8} T^{28} + 1475491281544 p^{9} T^{29} + 116812908927 p^{10} T^{30} + 9820451880 p^{11} T^{31} + 1014920480 p^{12} T^{32} + 75926021 p^{13} T^{33} + 4039236 p^{14} T^{34} + 460323 p^{15} T^{35} + 48895 p^{16} T^{36} + 2485 p^{17} T^{37} + 187 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.52443251632208118856145307384, −2.42042216180517074804255902083, −2.38885707183032828581072694733, −2.12691430181486114897440206344, −2.09065999420170122572852245290, −1.99431229100960238944871651614, −1.98491972107349440679184451059, −1.98043906591473476634180157163, −1.86200354267476556879475737439, −1.84903764064536166820881457726, −1.82941610684558306282709895713, −1.76919362130062796868993771677, −1.67371258755362721194115106498, −1.33473270280594034048871027614, −1.22794372099190346380664466422, −0.913791031569287057526341932107, −0.863538159458803055915782627921, −0.835177104322801241976803534020, −0.71540993628294091576548552797, −0.68651301590105194852754080027, −0.67220214277653431755854932151, −0.65401912803510182418708170458, −0.63050376820431687117536543072, −0.52132598782960532775009067010, −0.18639461152410136275564036044, 0.18639461152410136275564036044, 0.52132598782960532775009067010, 0.63050376820431687117536543072, 0.65401912803510182418708170458, 0.67220214277653431755854932151, 0.68651301590105194852754080027, 0.71540993628294091576548552797, 0.835177104322801241976803534020, 0.863538159458803055915782627921, 0.913791031569287057526341932107, 1.22794372099190346380664466422, 1.33473270280594034048871027614, 1.67371258755362721194115106498, 1.76919362130062796868993771677, 1.82941610684558306282709895713, 1.84903764064536166820881457726, 1.86200354267476556879475737439, 1.98043906591473476634180157163, 1.98491972107349440679184451059, 1.99431229100960238944871651614, 2.09065999420170122572852245290, 2.12691430181486114897440206344, 2.38885707183032828581072694733, 2.42042216180517074804255902083, 2.52443251632208118856145307384
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.