Dirichlet series
| L(s) = 1 | + 8·2-s + 4·3-s + 18·4-s − 6·5-s + 32·6-s − 28·8-s + 8·9-s − 48·10-s + 8·11-s + 72·12-s − 24·15-s − 195·16-s + 14·17-s + 64·18-s − 4·19-s − 108·20-s + 64·22-s − 8·23-s − 112·24-s + 9·25-s + 16·27-s − 192·30-s − 204·32-s + 32·33-s + 112·34-s + 144·36-s − 32·38-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 2.30·3-s + 9·4-s − 2.68·5-s + 13.0·6-s − 9.89·8-s + 8/3·9-s − 15.1·10-s + 2.41·11-s + 20.7·12-s − 6.19·15-s − 48.7·16-s + 3.39·17-s + 15.0·18-s − 0.917·19-s − 24.1·20-s + 13.6·22-s − 1.66·23-s − 22.8·24-s + 9/5·25-s + 3.07·27-s − 35.0·30-s − 36.0·32-s + 5.57·33-s + 19.2·34-s + 24·36-s − 5.19·38-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
| \(L(1)\) | \(\approx\) | \(1414.679868\) |
| \(L(\frac12)\) | \(\approx\) | \(1414.679868\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 6 T + 27 T^{2} + 114 T^{3} + 393 T^{4} + 1244 T^{5} + 3672 T^{6} + 10116 T^{7} + 25998 T^{8} + 63264 T^{9} + 146954 T^{10} + 63264 p T^{11} + 25998 p^{2} T^{12} + 10116 p^{3} T^{13} + 3672 p^{4} T^{14} + 1244 p^{5} T^{15} + 393 p^{6} T^{16} + 114 p^{7} T^{17} + 27 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) |
| 13 | \( 1 \) | |
| good | 2 | \( ( 1 - p^{2} T + 15 T^{2} - 19 p T^{3} + 91 T^{4} - 45 p^{2} T^{5} + 343 T^{6} - 145 p^{2} T^{7} + 951 T^{8} - 715 p T^{9} + 2101 T^{10} - 715 p^{2} T^{11} + 951 p^{2} T^{12} - 145 p^{5} T^{13} + 343 p^{4} T^{14} - 45 p^{7} T^{15} + 91 p^{6} T^{16} - 19 p^{8} T^{17} + 15 p^{8} T^{18} - p^{11} T^{19} + p^{10} T^{20} )^{2} \) |
| 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 | \( 1 - 88 T^{2} + 550 p T^{4} - 111628 T^{6} + 2408627 T^{8} - 41129464 T^{10} + 576697584 T^{12} - 6797883964 T^{14} + 68400845293 T^{16} - 593173330564 T^{18} + 4456980309874 T^{20} - 593173330564 p^{2} T^{22} + 68400845293 p^{4} T^{24} - 6797883964 p^{6} T^{26} + 576697584 p^{8} T^{28} - 41129464 p^{10} T^{30} + 2408627 p^{12} T^{32} - 111628 p^{14} T^{34} + 550 p^{17} T^{36} - 88 p^{18} T^{38} + p^{20} T^{40} \) | |
| 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 | \( 1 - 526 T^{2} + 136487 T^{4} - 23221046 T^{6} + 2903604335 T^{8} - 283491122180 T^{10} + 22412512573818 T^{12} - 1468572264080108 T^{14} + 80971225626368461 T^{16} - 3792670952536990506 T^{18} + \)\(15\!\cdots\!05\)\( T^{20} - 3792670952536990506 p^{2} T^{22} + 80971225626368461 p^{4} T^{24} - 1468572264080108 p^{6} T^{26} + 22412512573818 p^{8} T^{28} - 283491122180 p^{10} T^{30} + 2903604335 p^{12} T^{32} - 23221046 p^{14} T^{34} + 136487 p^{16} T^{36} - 526 p^{18} T^{38} + p^{20} T^{40} \) | |
| 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 | \( 1 - 572 T^{2} + 161342 T^{4} - 30069852 T^{6} + 4176817325 T^{8} - 461218805808 T^{10} + 895219768856 p T^{12} - 3248720168760816 T^{14} + 215624745949324178 T^{16} - 12422061007474931400 T^{18} + \)\(62\!\cdots\!44\)\( T^{20} - 12422061007474931400 p^{2} T^{22} + 215624745949324178 p^{4} T^{24} - 3248720168760816 p^{6} T^{26} + 895219768856 p^{9} T^{28} - 461218805808 p^{10} T^{30} + 4176817325 p^{12} T^{32} - 30069852 p^{14} T^{34} + 161342 p^{16} T^{36} - 572 p^{18} T^{38} + p^{20} T^{40} \) | |
| 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 | \( ( 1 - 58 T + 1990 T^{2} - 49220 T^{3} + 967075 T^{4} - 15803068 T^{5} + 221352576 T^{6} - 2707398734 T^{7} + 29291566337 T^{8} - 282571093816 T^{9} + 2441989941238 T^{10} - 282571093816 p T^{11} + 29291566337 p^{2} T^{12} - 2707398734 p^{3} T^{13} + 221352576 p^{4} T^{14} - 15803068 p^{5} T^{15} + 967075 p^{6} T^{16} - 49220 p^{7} T^{17} + 1990 p^{8} T^{18} - 58 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 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 | \( ( 1 - 36 T + 979 T^{2} - 17720 T^{3} + 269289 T^{4} - 3223420 T^{5} + 34128052 T^{6} - 304511620 T^{7} + 2578848294 T^{8} - 20162242116 T^{9} + 171086571074 T^{10} - 20162242116 p T^{11} + 2578848294 p^{2} T^{12} - 304511620 p^{3} T^{13} + 34128052 p^{4} T^{14} - 3223420 p^{5} T^{15} + 269289 p^{6} T^{16} - 17720 p^{7} T^{17} + 979 p^{8} T^{18} - 36 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 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 | \( 1 - 1212 T^{2} + 716166 T^{4} - 274705788 T^{6} + 76836687021 T^{8} - 16688569970480 T^{10} + 2926110414583464 T^{12} - 424954841088102384 T^{14} + 52013437736047347810 T^{16} - \)\(54\!\cdots\!00\)\( T^{18} + \)\(48\!\cdots\!20\)\( T^{20} - \)\(54\!\cdots\!00\)\( p^{2} T^{22} + 52013437736047347810 p^{4} T^{24} - 424954841088102384 p^{6} T^{26} + 2926110414583464 p^{8} T^{28} - 16688569970480 p^{10} T^{30} + 76836687021 p^{12} T^{32} - 274705788 p^{14} T^{34} + 716166 p^{16} T^{36} - 1212 p^{18} T^{38} + p^{20} T^{40} \) | |
| 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 | \( ( 1 - 22 T + 658 T^{2} - 10280 T^{3} + 186235 T^{4} - 2419240 T^{5} + 34456800 T^{6} - 393505958 T^{7} + 4778835449 T^{8} - 48517541800 T^{9} + 519469066090 T^{10} - 48517541800 p T^{11} + 4778835449 p^{2} T^{12} - 393505958 p^{3} T^{13} + 34456800 p^{4} T^{14} - 2419240 p^{5} T^{15} + 186235 p^{6} T^{16} - 10280 p^{7} T^{17} + 658 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
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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
−2.42691643794287687900865320546, −2.24366788226193488482794795472, −2.23931131370059297619630127484, −2.12157071871462693764474433882, −2.11665266770997857182875952248, −2.07974022916678249273426651688, −2.05386121911034814085845787400, −1.98501025214272925962554524279, −1.80624434059292900147578979342, −1.72663156679998842550003340225, −1.61522639887666798210915044635, −1.54370526048976370639273261832, −1.52115755725723743964478592339, −1.47622912672679144649674115559, −1.15610720811125059206782150672, −0.825100048309182728727591129713, −0.75564977346626196480193070224, −0.71291579136849210281351019815, −0.68957768887780218804810015583, −0.68309187421222990544042880235, −0.66293687872714399891496719814, −0.64916717224313389081073270238, −0.60090887773180096648024736323, −0.56922096412556422196100136927, −0.56566506969114947084191991713, 0.56566506969114947084191991713, 0.56922096412556422196100136927, 0.60090887773180096648024736323, 0.64916717224313389081073270238, 0.66293687872714399891496719814, 0.68309187421222990544042880235, 0.68957768887780218804810015583, 0.71291579136849210281351019815, 0.75564977346626196480193070224, 0.825100048309182728727591129713, 1.15610720811125059206782150672, 1.47622912672679144649674115559, 1.52115755725723743964478592339, 1.54370526048976370639273261832, 1.61522639887666798210915044635, 1.72663156679998842550003340225, 1.80624434059292900147578979342, 1.98501025214272925962554524279, 2.05386121911034814085845787400, 2.07974022916678249273426651688, 2.11665266770997857182875952248, 2.12157071871462693764474433882, 2.23931131370059297619630127484, 2.24366788226193488482794795472, 2.42691643794287687900865320546
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.