Dirichlet series
| L(s) = 1 | + 4·3-s + 14·4-s + 4·7-s + 8·9-s + 8·11-s + 56·12-s + 93·16-s − 14·17-s + 4·19-s + 16·21-s + 8·23-s + 9·25-s + 16·27-s + 56·28-s + 32·33-s + 112·36-s + 8·37-s − 38·41-s − 32·43-s + 112·44-s − 40·47-s + 372·48-s − 80·49-s − 56·51-s − 10·53-s + 16·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 7·4-s + 1.51·7-s + 8/3·9-s + 2.41·11-s + 16.1·12-s + 93/4·16-s − 3.39·17-s + 0.917·19-s + 3.49·21-s + 1.66·23-s + 9/5·25-s + 3.07·27-s + 10.5·28-s + 5.57·33-s + 56/3·36-s + 1.31·37-s − 5.93·41-s − 4.87·43-s + 16.8·44-s − 5.83·47-s + 53.6·48-s − 11.4·49-s − 7.84·51-s − 1.37·53-s + 2.11·57-s + 1.04·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 13^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.82539\times 10^{16}\) |
| Root analytic conductor: | \(2.59756\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{20} \cdot 13^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(89.66922774\) |
| \(L(\frac12)\) | \(\approx\) | \(89.66922774\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 - 9 T^{2} + 33 T^{4} + 16 T^{5} - 24 T^{6} - 48 T^{7} - 906 T^{8} - 96 T^{9} + 7346 T^{10} - 96 p T^{11} - 906 p^{2} T^{12} - 48 p^{3} T^{13} - 24 p^{4} T^{14} + 16 p^{5} T^{15} + 33 p^{6} T^{16} - 9 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - 7 p T^{2} + 103 T^{4} - 133 p^{2} T^{6} + 2153 T^{8} - 3619 p T^{10} + 21009 T^{12} - 6773 p^{3} T^{14} + 127183 T^{16} - 138645 p T^{18} + 569745 T^{20} - 138645 p^{3} T^{22} + 127183 p^{4} T^{24} - 6773 p^{9} T^{26} + 21009 p^{8} T^{28} - 3619 p^{11} T^{30} + 2153 p^{12} T^{32} - 133 p^{16} T^{34} + 103 p^{16} T^{36} - 7 p^{19} T^{38} + p^{20} T^{40} \) |
| 3 | \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 22 T^{4} - 16 T^{5} + 16 T^{6} - 8 p T^{7} - 5 T^{8} + 148 T^{9} - 392 T^{10} + 1076 T^{11} - 2168 T^{12} + 628 p T^{13} - 784 T^{14} - 1984 T^{15} + 15533 T^{16} - 37604 T^{17} + 776 p^{4} T^{18} - 13196 p^{2} T^{19} + 222694 T^{20} - 13196 p^{3} T^{21} + 776 p^{6} T^{22} - 37604 p^{3} T^{23} + 15533 p^{4} T^{24} - 1984 p^{5} T^{25} - 784 p^{6} T^{26} + 628 p^{8} T^{27} - 2168 p^{8} T^{28} + 1076 p^{9} T^{29} - 392 p^{10} T^{30} + 148 p^{11} T^{31} - 5 p^{12} T^{32} - 8 p^{14} T^{33} + 16 p^{14} T^{34} - 16 p^{15} T^{35} + 22 p^{16} T^{36} - 16 p^{17} T^{37} + 8 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( ( 1 - 2 T + 46 T^{2} - 80 T^{3} + 1027 T^{4} - 1552 T^{5} + 14876 T^{6} - 19686 T^{7} + 156185 T^{8} - 182088 T^{9} + 1245978 T^{10} - 182088 p T^{11} + 156185 p^{2} T^{12} - 19686 p^{3} T^{13} + 14876 p^{4} T^{14} - 1552 p^{5} T^{15} + 1027 p^{6} T^{16} - 80 p^{7} T^{17} + 46 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 11 | \( 1 - 8 T + 32 T^{2} - 72 T^{3} + 6 T^{4} + 120 T^{5} + 1440 T^{6} - 10232 T^{7} - 237 T^{8} + 220864 T^{9} - 1024064 T^{10} + 2441248 T^{11} - 1582312 T^{12} + 331952 T^{13} - 42455232 T^{14} + 217277600 T^{15} - 118748739 T^{16} - 2793785704 T^{17} + 12248253536 T^{18} - 28788271480 T^{19} + 65183319798 T^{20} - 28788271480 p T^{21} + 12248253536 p^{2} T^{22} - 2793785704 p^{3} T^{23} - 118748739 p^{4} T^{24} + 217277600 p^{5} T^{25} - 42455232 p^{6} T^{26} + 331952 p^{7} T^{27} - 1582312 p^{8} T^{28} + 2441248 p^{9} T^{29} - 1024064 p^{10} T^{30} + 220864 p^{11} T^{31} - 237 p^{12} T^{32} - 10232 p^{13} T^{33} + 1440 p^{14} T^{34} + 120 p^{15} T^{35} + 6 p^{16} T^{36} - 72 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 14 T + 98 T^{2} + 524 T^{3} + 3347 T^{4} + 21532 T^{5} + 110730 T^{6} + 467702 T^{7} + 2277351 T^{8} + 13015084 T^{9} + 65306828 T^{10} + 270685184 T^{11} + 1250464570 T^{12} + 6751721184 T^{13} + 32677944676 T^{14} + 129337453748 T^{15} + 527582116957 T^{16} + 2526302027762 T^{17} + 11548682196118 T^{18} + 43268927461612 T^{19} + 161988731618493 T^{20} + 43268927461612 p T^{21} + 11548682196118 p^{2} T^{22} + 2526302027762 p^{3} T^{23} + 527582116957 p^{4} T^{24} + 129337453748 p^{5} T^{25} + 32677944676 p^{6} T^{26} + 6751721184 p^{7} T^{27} + 1250464570 p^{8} T^{28} + 270685184 p^{9} T^{29} + 65306828 p^{10} T^{30} + 13015084 p^{11} T^{31} + 2277351 p^{12} T^{32} + 467702 p^{13} T^{33} + 110730 p^{14} T^{34} + 21532 p^{15} T^{35} + 3347 p^{16} T^{36} + 524 p^{17} T^{37} + 98 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 242 T^{4} + 1668 T^{5} - 4536 T^{6} + 46716 T^{7} - 8007 p T^{8} - 278656 T^{9} + 1690768 T^{10} - 6259840 T^{11} + 49457664 T^{12} - 263060576 T^{13} + 1127262480 T^{14} - 7257936216 T^{15} + 21560809389 T^{16} + 64614421524 T^{17} - 223821487720 T^{18} + 2267258709836 T^{19} - 17967318422394 T^{20} + 2267258709836 p T^{21} - 223821487720 p^{2} T^{22} + 64614421524 p^{3} T^{23} + 21560809389 p^{4} T^{24} - 7257936216 p^{5} T^{25} + 1127262480 p^{6} T^{26} - 263060576 p^{7} T^{27} + 49457664 p^{8} T^{28} - 6259840 p^{9} T^{29} + 1690768 p^{10} T^{30} - 278656 p^{11} T^{31} - 8007 p^{13} T^{32} + 46716 p^{13} T^{33} - 4536 p^{14} T^{34} + 1668 p^{15} T^{35} - 242 p^{16} T^{36} - 20 p^{17} T^{37} + 8 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 8 T + 32 T^{2} - 32 T^{3} + 1918 T^{4} - 18956 T^{5} + 90784 T^{6} - 188052 T^{7} + 2028803 T^{8} - 21308784 T^{9} + 117107496 T^{10} - 318432352 T^{11} + 1817984576 T^{12} - 17168984572 T^{13} + 102182263384 T^{14} - 312313722108 T^{15} + 1309092718269 T^{16} - 10994969249788 T^{17} + 69814543980232 T^{18} - 232448387496180 T^{19} + 750697112344518 T^{20} - 232448387496180 p T^{21} + 69814543980232 p^{2} T^{22} - 10994969249788 p^{3} T^{23} + 1309092718269 p^{4} T^{24} - 312313722108 p^{5} T^{25} + 102182263384 p^{6} T^{26} - 17168984572 p^{7} T^{27} + 1817984576 p^{8} T^{28} - 318432352 p^{9} T^{29} + 117107496 p^{10} T^{30} - 21308784 p^{11} T^{31} + 2028803 p^{12} T^{32} - 188052 p^{13} T^{33} + 90784 p^{14} T^{34} - 18956 p^{15} T^{35} + 1918 p^{16} T^{36} - 32 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 234 T^{2} + 27327 T^{4} - 2132034 T^{6} + 126760071 T^{8} - 6247626956 T^{10} + 270144601698 T^{12} - 10569032340228 T^{14} + 378287039769453 T^{16} - 12424601805867678 T^{18} + 375402711432122049 T^{20} - 12424601805867678 p^{2} T^{22} + 378287039769453 p^{4} T^{24} - 10569032340228 p^{6} T^{26} + 270144601698 p^{8} T^{28} - 6247626956 p^{10} T^{30} + 126760071 p^{12} T^{32} - 2132034 p^{14} T^{34} + 27327 p^{16} T^{36} - 234 p^{18} T^{38} + p^{20} T^{40} \) | |
| 31 | \( 1 - 104 T^{3} + 1794 T^{4} - 3320 T^{5} + 5408 T^{6} - 552640 T^{7} + 1315037 T^{8} + 1490240 T^{9} + 53283808 T^{10} - 779482432 T^{11} + 2015402168 T^{12} + 9179164672 T^{13} + 126121189280 T^{14} - 557942017984 T^{15} - 69210310750 T^{16} - 4465408437184 T^{17} + 158380681056 p^{2} T^{18} - 15988767100848 p T^{19} - 2929717194662260 T^{20} - 15988767100848 p^{2} T^{21} + 158380681056 p^{4} T^{22} - 4465408437184 p^{3} T^{23} - 69210310750 p^{4} T^{24} - 557942017984 p^{5} T^{25} + 126121189280 p^{6} T^{26} + 9179164672 p^{7} T^{27} + 2015402168 p^{8} T^{28} - 779482432 p^{9} T^{29} + 53283808 p^{10} T^{30} + 1490240 p^{11} T^{31} + 1315037 p^{12} T^{32} - 552640 p^{13} T^{33} + 5408 p^{14} T^{34} - 3320 p^{15} T^{35} + 1794 p^{16} T^{36} - 104 p^{17} T^{37} + p^{20} T^{40} \) | |
| 37 | \( ( 1 - 4 T + 271 T^{2} - 856 T^{3} + 34947 T^{4} - 90596 T^{5} + 2868638 T^{6} - 6283256 T^{7} + 166438065 T^{8} - 313103556 T^{9} + 7125252257 T^{10} - 313103556 p T^{11} + 166438065 p^{2} T^{12} - 6283256 p^{3} T^{13} + 2868638 p^{4} T^{14} - 90596 p^{5} T^{15} + 34947 p^{6} T^{16} - 856 p^{7} T^{17} + 271 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 41 | \( 1 + 38 T + 722 T^{2} + 9712 T^{3} + 109171 T^{4} + 1106552 T^{5} + 10388986 T^{6} + 91886454 T^{7} + 778524071 T^{8} + 6359893836 T^{9} + 50035501500 T^{10} + 379983206888 T^{11} + 2806220265194 T^{12} + 20299186572832 T^{13} + 144287013415828 T^{14} + 1008576584591724 T^{15} + 6937305353423693 T^{16} + 46936853511728354 T^{17} + 312118609508364086 T^{18} + 2041213011526541200 T^{19} + 13156875859223582685 T^{20} + 2041213011526541200 p T^{21} + 312118609508364086 p^{2} T^{22} + 46936853511728354 p^{3} T^{23} + 6937305353423693 p^{4} T^{24} + 1008576584591724 p^{5} T^{25} + 144287013415828 p^{6} T^{26} + 20299186572832 p^{7} T^{27} + 2806220265194 p^{8} T^{28} + 379983206888 p^{9} T^{29} + 50035501500 p^{10} T^{30} + 6359893836 p^{11} T^{31} + 778524071 p^{12} T^{32} + 91886454 p^{13} T^{33} + 10388986 p^{14} T^{34} + 1106552 p^{15} T^{35} + 109171 p^{16} T^{36} + 9712 p^{17} T^{37} + 722 p^{18} T^{38} + 38 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 32 T + 512 T^{2} + 6092 T^{3} + 59158 T^{4} + 457964 T^{5} + 2922184 T^{6} + 15866780 T^{7} + 73561771 T^{8} + 361195596 T^{9} + 2550007656 T^{10} + 23038901324 T^{11} + 217896051656 T^{12} + 1889811262092 T^{13} + 14741719374112 T^{14} + 107615676041552 T^{15} + 731431984149549 T^{16} + 4564979872508656 T^{17} + 27024641563084528 T^{18} + 158222420660822272 T^{19} + 979624989628247430 T^{20} + 158222420660822272 p T^{21} + 27024641563084528 p^{2} T^{22} + 4564979872508656 p^{3} T^{23} + 731431984149549 p^{4} T^{24} + 107615676041552 p^{5} T^{25} + 14741719374112 p^{6} T^{26} + 1889811262092 p^{7} T^{27} + 217896051656 p^{8} T^{28} + 23038901324 p^{9} T^{29} + 2550007656 p^{10} T^{30} + 361195596 p^{11} T^{31} + 73561771 p^{12} T^{32} + 15866780 p^{13} T^{33} + 2922184 p^{14} T^{34} + 457964 p^{15} T^{35} + 59158 p^{16} T^{36} + 6092 p^{17} T^{37} + 512 p^{18} T^{38} + 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( ( 1 + 20 T + 486 T^{2} + 6548 T^{3} + 93533 T^{4} + 960312 T^{5} + 10222280 T^{6} + 85662136 T^{7} + 747535234 T^{8} + 5352082240 T^{9} + 40243643940 T^{10} + 5352082240 p T^{11} + 747535234 p^{2} T^{12} + 85662136 p^{3} T^{13} + 10222280 p^{4} T^{14} + 960312 p^{5} T^{15} + 93533 p^{6} T^{16} + 6548 p^{7} T^{17} + 486 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 + 10 T + 50 T^{2} + 608 T^{3} + 8911 T^{4} + 42256 T^{5} + 161842 T^{6} + 1379418 T^{7} + 11864269 T^{8} + 42078400 T^{9} + 116867400 T^{10} - 342809968 T^{11} + 2993373476 T^{12} + 82242842224 T^{13} + 232250742760 T^{14} - 4408284842096 T^{15} - 104445179440542 T^{16} - 7400825808492 p T^{17} - 1433330906377060 T^{18} - 34029241052501104 T^{19} - 545288959843663782 T^{20} - 34029241052501104 p T^{21} - 1433330906377060 p^{2} T^{22} - 7400825808492 p^{4} T^{23} - 104445179440542 p^{4} T^{24} - 4408284842096 p^{5} T^{25} + 232250742760 p^{6} T^{26} + 82242842224 p^{7} T^{27} + 2993373476 p^{8} T^{28} - 342809968 p^{9} T^{29} + 116867400 p^{10} T^{30} + 42078400 p^{11} T^{31} + 11864269 p^{12} T^{32} + 1379418 p^{13} T^{33} + 161842 p^{14} T^{34} + 42256 p^{15} T^{35} + 8911 p^{16} T^{36} + 608 p^{17} T^{37} + 50 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 8 T + 32 T^{2} - 176 T^{3} + 15238 T^{4} - 115824 T^{5} + 454464 T^{6} - 1805392 T^{7} + 89857811 T^{8} - 651385592 T^{9} + 2435860832 T^{10} + 504315896 T^{11} + 134523639352 T^{12} - 1139267606520 T^{13} + 3841642163520 T^{14} + 95804041206976 T^{15} - 1232839826908291 T^{16} + 5248095573838584 T^{17} - 14755980062358496 T^{18} + 573944774118420680 T^{19} - 7715303843947901098 T^{20} + 573944774118420680 p T^{21} - 14755980062358496 p^{2} T^{22} + 5248095573838584 p^{3} T^{23} - 1232839826908291 p^{4} T^{24} + 95804041206976 p^{5} T^{25} + 3841642163520 p^{6} T^{26} - 1139267606520 p^{7} T^{27} + 134523639352 p^{8} T^{28} + 504315896 p^{9} T^{29} + 2435860832 p^{10} T^{30} - 651385592 p^{11} T^{31} + 89857811 p^{12} T^{32} - 1805392 p^{13} T^{33} + 454464 p^{14} T^{34} - 115824 p^{15} T^{35} + 15238 p^{16} T^{36} - 176 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( ( 1 - 16 T + 547 T^{2} - 6688 T^{3} + 129587 T^{4} - 1279840 T^{5} + 18216078 T^{6} - 150735376 T^{7} + 1744620469 T^{8} - 12405860224 T^{9} + 122649970081 T^{10} - 12405860224 p T^{11} + 1744620469 p^{2} T^{12} - 150735376 p^{3} T^{13} + 18216078 p^{4} T^{14} - 1279840 p^{5} T^{15} + 129587 p^{6} T^{16} - 6688 p^{7} T^{17} + 547 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 - 616 T^{2} + 184730 T^{4} - 35899364 T^{6} + 5088173715 T^{8} - 561962303896 T^{10} + 50491971828336 T^{12} - 3815319877899172 T^{14} + 252050635148222349 T^{16} - 15578928611431137564 T^{18} + \)\(99\!\cdots\!54\)\( T^{20} - 15578928611431137564 p^{2} T^{22} + 252050635148222349 p^{4} T^{24} - 3815319877899172 p^{6} T^{26} + 50491971828336 p^{8} T^{28} - 561962303896 p^{10} T^{30} + 5088173715 p^{12} T^{32} - 35899364 p^{14} T^{34} + 184730 p^{16} T^{36} - 616 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( 1 + 40 T + 800 T^{2} + 13044 T^{3} + 199598 T^{4} + 2637256 T^{5} + 30884808 T^{6} + 346032792 T^{7} + 3673298187 T^{8} + 37042664708 T^{9} + 369733475152 T^{10} + 3657833497180 T^{11} + 35756754031920 T^{12} + 347488354002576 T^{13} + 3320075992969848 T^{14} + 436312991566820 p T^{15} + 281380928011102701 T^{16} + 2490634648428888636 T^{17} + 21663850115572509080 T^{18} + \)\(18\!\cdots\!20\)\( T^{19} + \)\(15\!\cdots\!30\)\( T^{20} + \)\(18\!\cdots\!20\)\( p T^{21} + 21663850115572509080 p^{2} T^{22} + 2490634648428888636 p^{3} T^{23} + 281380928011102701 p^{4} T^{24} + 436312991566820 p^{6} T^{25} + 3320075992969848 p^{6} T^{26} + 347488354002576 p^{7} T^{27} + 35756754031920 p^{8} T^{28} + 3657833497180 p^{9} T^{29} + 369733475152 p^{10} T^{30} + 37042664708 p^{11} T^{31} + 3673298187 p^{12} T^{32} + 346032792 p^{13} T^{33} + 30884808 p^{14} T^{34} + 2637256 p^{15} T^{35} + 199598 p^{16} T^{36} + 13044 p^{17} T^{37} + 800 p^{18} T^{38} + 40 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 662 T^{2} + 221179 T^{4} - 49439326 T^{6} + 8334146485 T^{8} - 1138166350304 T^{10} + 132447463647444 T^{12} - 13610776696011280 T^{14} + 1261445493668289578 T^{16} - \)\(10\!\cdots\!72\)\( T^{18} + \)\(81\!\cdots\!98\)\( T^{20} - \)\(10\!\cdots\!72\)\( p^{2} T^{22} + 1261445493668289578 p^{4} T^{24} - 13610776696011280 p^{6} T^{26} + 132447463647444 p^{8} T^{28} - 1138166350304 p^{10} T^{30} + 8334146485 p^{12} T^{32} - 49439326 p^{14} T^{34} + 221179 p^{16} T^{36} - 662 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( 1 - 772 T^{2} + 300974 T^{4} - 79250180 T^{6} + 15881528397 T^{8} - 2583290881840 T^{10} + 354721609143528 T^{12} - 42178231844928304 T^{14} + 4416281458244224626 T^{16} - \)\(41\!\cdots\!84\)\( T^{18} + \)\(34\!\cdots\!44\)\( T^{20} - \)\(41\!\cdots\!84\)\( p^{2} T^{22} + 4416281458244224626 p^{4} T^{24} - 42178231844928304 p^{6} T^{26} + 354721609143528 p^{8} T^{28} - 2583290881840 p^{10} T^{30} + 15881528397 p^{12} T^{32} - 79250180 p^{14} T^{34} + 300974 p^{16} T^{36} - 772 p^{18} T^{38} + p^{20} T^{40} \) | |
| 83 | \( ( 1 - 24 T + 894 T^{2} - 15600 T^{3} + 332865 T^{4} - 4597176 T^{5} + 71783808 T^{6} - 818073720 T^{7} + 10193779926 T^{8} - 97546659000 T^{9} + 1006811916964 T^{10} - 97546659000 p T^{11} + 10193779926 p^{2} T^{12} - 818073720 p^{3} T^{13} + 71783808 p^{4} T^{14} - 4597176 p^{5} T^{15} + 332865 p^{6} T^{16} - 15600 p^{7} T^{17} + 894 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 - 12 T + 72 T^{2} + 1492 T^{3} + 15778 T^{4} - 442924 T^{5} + 5292104 T^{6} + 17208588 T^{7} + 10057651 T^{8} - 5445262608 T^{9} + 104886231696 T^{10} - 138193383488 T^{11} + 619396189520 T^{12} - 51101478295744 T^{13} + 1214220832814864 T^{14} - 3074009831656856 T^{15} + 14005106194965597 T^{16} - 509797157599010436 T^{17} + 1529274298428632 p^{2} T^{18} - 418775626564727116 p T^{19} + 90818494714915126122 T^{20} - 418775626564727116 p^{2} T^{21} + 1529274298428632 p^{4} T^{22} - 509797157599010436 p^{3} T^{23} + 14005106194965597 p^{4} T^{24} - 3074009831656856 p^{5} T^{25} + 1214220832814864 p^{6} T^{26} - 51101478295744 p^{7} T^{27} + 619396189520 p^{8} T^{28} - 138193383488 p^{9} T^{29} + 104886231696 p^{10} T^{30} - 5445262608 p^{11} T^{31} + 10057651 p^{12} T^{32} + 17208588 p^{13} T^{33} + 5292104 p^{14} T^{34} - 442924 p^{15} T^{35} + 15778 p^{16} T^{36} + 1492 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 832 T^{2} + 353114 T^{4} - 101873900 T^{6} + 22532458371 T^{8} - 4084033365784 T^{10} + 632229086908512 T^{12} - 85927117495647796 T^{14} + 10455343241984702541 T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(11\!\cdots\!74\)\( T^{20} - \)\(11\!\cdots\!56\)\( p^{2} T^{22} + 10455343241984702541 p^{4} T^{24} - 85927117495647796 p^{6} T^{26} + 632229086908512 p^{8} T^{28} - 4084033365784 p^{10} T^{30} + 22532458371 p^{12} T^{32} - 101873900 p^{14} T^{34} + 353114 p^{16} T^{36} - 832 p^{18} T^{38} + p^{20} T^{40} \) | |
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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
−2.35385907884777666550455733076, −2.32340587515524636454770963302, −2.19953246795819751736659412168, −2.18523734740445213715844425393, −1.82649452496407735372914830778, −1.81451616194069955192002821659, −1.78966985200671952604032968256, −1.77805904415179210057863097187, −1.73908522101519657724997704677, −1.69385448188718526518143445910, −1.65092760765377212907995331662, −1.63805660642004940165753746008, −1.52873350429499366914582966021, −1.52816204998040045852550528471, −1.52412774748482439328381123465, −1.38040221596263880350695047469, −1.29587105592671260920213878213, −1.17777456569643888648088585316, −1.15111953299883969595790158844, −1.00365832497831718354563189982, −0.896874218099087614999977722956, −0.47191025462445677248715061579, −0.37013332804376304816130690424, −0.28607700011836006863388936583, −0.11245504545881684419442684407, 0.11245504545881684419442684407, 0.28607700011836006863388936583, 0.37013332804376304816130690424, 0.47191025462445677248715061579, 0.896874218099087614999977722956, 1.00365832497831718354563189982, 1.15111953299883969595790158844, 1.17777456569643888648088585316, 1.29587105592671260920213878213, 1.38040221596263880350695047469, 1.52412774748482439328381123465, 1.52816204998040045852550528471, 1.52873350429499366914582966021, 1.63805660642004940165753746008, 1.65092760765377212907995331662, 1.69385448188718526518143445910, 1.73908522101519657724997704677, 1.77805904415179210057863097187, 1.78966985200671952604032968256, 1.81451616194069955192002821659, 1.82649452496407735372914830778, 2.18523734740445213715844425393, 2.19953246795819751736659412168, 2.32340587515524636454770963302, 2.35385907884777666550455733076
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.