Dirichlet series
| L(s) = 1 | + 4·7-s + 6·13-s + 12·17-s − 4·19-s − 18·25-s + 8·41-s + 2·43-s + 36·47-s − 46·49-s − 8·53-s − 8·59-s − 2·61-s − 30·67-s + 4·71-s − 22·73-s − 54·79-s − 32·89-s + 24·91-s + 12·97-s + 12·101-s + 24·107-s − 12·109-s + 48·113-s + 48·119-s + 120·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.66·13-s + 2.91·17-s − 0.917·19-s − 3.59·25-s + 1.24·41-s + 0.304·43-s + 5.25·47-s − 6.57·49-s − 1.09·53-s − 1.04·59-s − 0.256·61-s − 3.66·67-s + 0.474·71-s − 2.57·73-s − 6.07·79-s − 3.39·89-s + 2.51·91-s + 1.21·97-s + 1.19·101-s + 2.32·107-s − 1.14·109-s + 4.51·113-s + 4.40·119-s + 10.9·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 3^{40} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.13569\times 10^{26}\) |
| Root analytic conductor: | \(4.67408\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 3^{40} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03305341342\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03305341342\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 19 | \( 1 + 4 T + 10 T^{2} + 236 T^{3} + 861 T^{4} + 4448 T^{5} + 34472 T^{6} + 98720 T^{7} + 690522 T^{8} + 3506776 T^{9} + 9079356 T^{10} + 3506776 p T^{11} + 690522 p^{2} T^{12} + 98720 p^{3} T^{13} + 34472 p^{4} T^{14} + 4448 p^{5} T^{15} + 861 p^{6} T^{16} + 236 p^{7} T^{17} + 10 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| good | 5 | \( 1 + 18 T^{2} + 163 T^{4} - 24 T^{5} + 754 T^{6} - 576 T^{7} + 327 T^{8} - 6504 T^{9} - 4172 p T^{10} - 43608 T^{11} - 150398 T^{12} - 163776 T^{13} - 452884 T^{14} - 65736 T^{15} + 121149 p T^{16} + 748968 p T^{17} + 14982438 T^{18} + 6134784 p T^{19} + 95858849 T^{20} + 6134784 p^{2} T^{21} + 14982438 p^{2} T^{22} + 748968 p^{4} T^{23} + 121149 p^{5} T^{24} - 65736 p^{5} T^{25} - 452884 p^{6} T^{26} - 163776 p^{7} T^{27} - 150398 p^{8} T^{28} - 43608 p^{9} T^{29} - 4172 p^{11} T^{30} - 6504 p^{11} T^{31} + 327 p^{12} T^{32} - 576 p^{13} T^{33} + 754 p^{14} T^{34} - 24 p^{15} T^{35} + 163 p^{16} T^{36} + 18 p^{18} T^{38} + p^{20} T^{40} \) |
| 7 | \( ( 1 - 2 T + 29 T^{2} - 58 T^{3} + 431 T^{4} - 124 p T^{5} + 4722 T^{6} - 9136 T^{7} + 43705 T^{8} - 10926 p T^{9} + 339275 T^{10} - 10926 p^{2} T^{11} + 43705 p^{2} T^{12} - 9136 p^{3} T^{13} + 4722 p^{4} T^{14} - 124 p^{6} T^{15} + 431 p^{6} T^{16} - 58 p^{7} T^{17} + 29 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 11 | \( 1 - 120 T^{2} + 7210 T^{4} - 288808 T^{6} + 8674821 T^{8} - 18949792 p T^{10} + 4173251560 T^{12} - 71516498048 T^{14} + 1067982257466 T^{16} - 14055376108800 T^{18} + 164026893067196 T^{20} - 14055376108800 p^{2} T^{22} + 1067982257466 p^{4} T^{24} - 71516498048 p^{6} T^{26} + 4173251560 p^{8} T^{28} - 18949792 p^{11} T^{30} + 8674821 p^{12} T^{32} - 288808 p^{14} T^{34} + 7210 p^{16} T^{36} - 120 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 6 T + 7 p T^{2} - 474 T^{3} + 3910 T^{4} - 18030 T^{5} + 111145 T^{6} - 463794 T^{7} + 2456790 T^{8} - 9463842 T^{9} + 3490049 p T^{10} - 164505390 T^{11} + 726598952 T^{12} - 2540600934 T^{13} + 10659959069 T^{14} - 36679560090 T^{15} + 150051802689 T^{16} - 510839413020 T^{17} + 2034927746082 T^{18} - 6870580554228 T^{19} + 26724060606452 T^{20} - 6870580554228 p T^{21} + 2034927746082 p^{2} T^{22} - 510839413020 p^{3} T^{23} + 150051802689 p^{4} T^{24} - 36679560090 p^{5} T^{25} + 10659959069 p^{6} T^{26} - 2540600934 p^{7} T^{27} + 726598952 p^{8} T^{28} - 164505390 p^{9} T^{29} + 3490049 p^{11} T^{30} - 9463842 p^{11} T^{31} + 2456790 p^{12} T^{32} - 463794 p^{13} T^{33} + 111145 p^{14} T^{34} - 18030 p^{15} T^{35} + 3910 p^{16} T^{36} - 474 p^{17} T^{37} + 7 p^{19} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10243 T^{4} - 64272 T^{5} + 377100 T^{6} - 1860300 T^{7} + 8527911 T^{8} - 33185448 T^{9} + 119808680 T^{10} - 358074720 T^{11} + 1021283178 T^{12} - 2292778368 T^{13} + 350209208 p T^{14} - 10318755624 T^{15} + 19782723021 T^{16} + 156350063148 T^{17} - 1397284423180 T^{18} + 9728709421296 T^{19} - 42980701569299 T^{20} + 9728709421296 p T^{21} - 1397284423180 p^{2} T^{22} + 156350063148 p^{3} T^{23} + 19782723021 p^{4} T^{24} - 10318755624 p^{5} T^{25} + 350209208 p^{7} T^{26} - 2292778368 p^{7} T^{27} + 1021283178 p^{8} T^{28} - 358074720 p^{9} T^{29} + 119808680 p^{10} T^{30} - 33185448 p^{11} T^{31} + 8527911 p^{12} T^{32} - 1860300 p^{13} T^{33} + 377100 p^{14} T^{34} - 64272 p^{15} T^{35} + 10243 p^{16} T^{36} - 1296 p^{17} T^{37} + 156 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 116 T^{2} + 6999 T^{4} + 816 T^{5} + 303860 T^{6} + 106896 T^{7} + 10739927 T^{8} + 7131024 T^{9} + 325749200 T^{10} + 292087200 T^{11} + 8902570986 T^{12} + 8248355136 T^{13} + 230492374712 T^{14} + 174026300352 T^{15} + 5778997111249 T^{16} + 2718294569520 T^{17} + 140338952549820 T^{18} + 32834606661504 T^{19} + 3288584209291773 T^{20} + 32834606661504 p T^{21} + 140338952549820 p^{2} T^{22} + 2718294569520 p^{3} T^{23} + 5778997111249 p^{4} T^{24} + 174026300352 p^{5} T^{25} + 230492374712 p^{6} T^{26} + 8248355136 p^{7} T^{27} + 8902570986 p^{8} T^{28} + 292087200 p^{9} T^{29} + 325749200 p^{10} T^{30} + 7131024 p^{11} T^{31} + 10739927 p^{12} T^{32} + 106896 p^{13} T^{33} + 303860 p^{14} T^{34} + 816 p^{15} T^{35} + 6999 p^{16} T^{36} + 116 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( 1 - 156 T^{2} - 64 T^{3} + 11979 T^{4} + 12640 T^{5} - 611628 T^{6} - 1347136 T^{7} + 23535447 T^{8} + 92836640 T^{9} - 713511976 T^{10} - 4485356832 T^{11} + 16278966586 T^{12} + 162261547392 T^{13} - 196675124536 T^{14} - 4561664032160 T^{15} - 4326661101203 T^{16} + 95587763960928 T^{17} + 12476280996412 p T^{18} - 1009598744454080 T^{19} - 13069426534367483 T^{20} - 1009598744454080 p T^{21} + 12476280996412 p^{3} T^{22} + 95587763960928 p^{3} T^{23} - 4326661101203 p^{4} T^{24} - 4561664032160 p^{5} T^{25} - 196675124536 p^{6} T^{26} + 162261547392 p^{7} T^{27} + 16278966586 p^{8} T^{28} - 4485356832 p^{9} T^{29} - 713511976 p^{10} T^{30} + 92836640 p^{11} T^{31} + 23535447 p^{12} T^{32} - 1347136 p^{13} T^{33} - 611628 p^{14} T^{34} + 12640 p^{15} T^{35} + 11979 p^{16} T^{36} - 64 p^{17} T^{37} - 156 p^{18} T^{38} + p^{20} T^{40} \) | |
| 31 | \( 1 - 322 T^{2} + 51871 T^{4} - 5572050 T^{6} + 448823511 T^{8} - 28911858604 T^{10} + 1552343642306 T^{12} - 71552067738540 T^{14} + 2895091186811901 T^{16} - 104551239994358782 T^{18} + 3406072487314635857 T^{20} - 104551239994358782 p^{2} T^{22} + 2895091186811901 p^{4} T^{24} - 71552067738540 p^{6} T^{26} + 1552343642306 p^{8} T^{28} - 28911858604 p^{10} T^{30} + 448823511 p^{12} T^{32} - 5572050 p^{14} T^{34} + 51871 p^{16} T^{36} - 322 p^{18} T^{38} + p^{20} T^{40} \) | |
| 37 | \( 1 - 290 T^{2} + 44231 T^{4} - 124530 p T^{6} + 265551 p^{2} T^{8} - 22927464316 T^{10} + 1197037757602 T^{12} - 53326432181780 T^{14} + 2104309370874381 T^{16} - 77576780641573414 T^{18} + 2838888488552328521 T^{20} - 77576780641573414 p^{2} T^{22} + 2104309370874381 p^{4} T^{24} - 53326432181780 p^{6} T^{26} + 1197037757602 p^{8} T^{28} - 22927464316 p^{10} T^{30} + 265551 p^{14} T^{32} - 124530 p^{15} T^{34} + 44231 p^{16} T^{36} - 290 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( 1 - 8 T - 114 T^{2} + 896 T^{3} + 5127 T^{4} - 23712 T^{5} - 235834 T^{6} - 889608 T^{7} + 16213575 T^{8} + 69641904 T^{9} - 573323516 T^{10} - 3628610784 T^{11} - 542730110 T^{12} + 264603149376 T^{13} + 483250809612 T^{14} - 11920248159504 T^{15} - 24748612467475 T^{16} + 228983295969960 T^{17} + 2620055796614602 T^{18} - 1102311975605056 T^{19} - 158452850651580183 T^{20} - 1102311975605056 p T^{21} + 2620055796614602 p^{2} T^{22} + 228983295969960 p^{3} T^{23} - 24748612467475 p^{4} T^{24} - 11920248159504 p^{5} T^{25} + 483250809612 p^{6} T^{26} + 264603149376 p^{7} T^{27} - 542730110 p^{8} T^{28} - 3628610784 p^{9} T^{29} - 573323516 p^{10} T^{30} + 69641904 p^{11} T^{31} + 16213575 p^{12} T^{32} - 889608 p^{13} T^{33} - 235834 p^{14} T^{34} - 23712 p^{15} T^{35} + 5127 p^{16} T^{36} + 896 p^{17} T^{37} - 114 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 - 2 T - 171 T^{2} - 1130 T^{3} + 18898 T^{4} + 214882 T^{5} - 402981 T^{6} - 23169570 T^{7} - 87874094 T^{8} + 1179592102 T^{9} + 13382735767 T^{10} - 10920696786 T^{11} - 866124474332 T^{12} - 4104068024366 T^{13} + 29646445762071 T^{14} + 361246858831618 T^{15} + 344349793991509 T^{16} - 16817093302073412 T^{17} - 94821904688064318 T^{18} + 300472514955093024 T^{19} + 5790731117648943268 T^{20} + 300472514955093024 p T^{21} - 94821904688064318 p^{2} T^{22} - 16817093302073412 p^{3} T^{23} + 344349793991509 p^{4} T^{24} + 361246858831618 p^{5} T^{25} + 29646445762071 p^{6} T^{26} - 4104068024366 p^{7} T^{27} - 866124474332 p^{8} T^{28} - 10920696786 p^{9} T^{29} + 13382735767 p^{10} T^{30} + 1179592102 p^{11} T^{31} - 87874094 p^{12} T^{32} - 23169570 p^{13} T^{33} - 402981 p^{14} T^{34} + 214882 p^{15} T^{35} + 18898 p^{16} T^{36} - 1130 p^{17} T^{37} - 171 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 36 T + 920 T^{2} - 17568 T^{3} + 284395 T^{4} - 3964944 T^{5} + 49375288 T^{6} - 553041564 T^{7} + 5651400463 T^{8} - 52757074344 T^{9} + 451430255888 T^{10} - 3519117256080 T^{11} + 24732765267138 T^{12} - 152118344545536 T^{13} + 760467646861552 T^{14} - 2260082920495704 T^{15} - 9378099753178963 T^{16} + 241935591534952596 T^{17} - 2725790043399683512 T^{18} + 504004064313880224 p T^{19} - 3714669872762704525 p T^{20} + 504004064313880224 p^{2} T^{21} - 2725790043399683512 p^{2} T^{22} + 241935591534952596 p^{3} T^{23} - 9378099753178963 p^{4} T^{24} - 2260082920495704 p^{5} T^{25} + 760467646861552 p^{6} T^{26} - 152118344545536 p^{7} T^{27} + 24732765267138 p^{8} T^{28} - 3519117256080 p^{9} T^{29} + 451430255888 p^{10} T^{30} - 52757074344 p^{11} T^{31} + 5651400463 p^{12} T^{32} - 553041564 p^{13} T^{33} + 49375288 p^{14} T^{34} - 3964944 p^{15} T^{35} + 284395 p^{16} T^{36} - 17568 p^{17} T^{37} + 920 p^{18} T^{38} - 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 8 T - 144 T^{2} - 1824 T^{3} + 4203 T^{4} + 154232 T^{5} + 853920 T^{6} - 2714920 T^{7} - 95772173 T^{8} - 643624408 T^{9} + 2551506288 T^{10} + 60692467368 T^{11} + 277957924654 T^{12} - 1626714951344 T^{13} - 28904270159608 T^{14} - 119764701731752 T^{15} + 792165784591161 T^{16} + 12519949363815280 T^{17} + 52111347357452488 T^{18} - 342628086157743152 T^{19} - 5366428302390766235 T^{20} - 342628086157743152 p T^{21} + 52111347357452488 p^{2} T^{22} + 12519949363815280 p^{3} T^{23} + 792165784591161 p^{4} T^{24} - 119764701731752 p^{5} T^{25} - 28904270159608 p^{6} T^{26} - 1626714951344 p^{7} T^{27} + 277957924654 p^{8} T^{28} + 60692467368 p^{9} T^{29} + 2551506288 p^{10} T^{30} - 643624408 p^{11} T^{31} - 95772173 p^{12} T^{32} - 2714920 p^{13} T^{33} + 853920 p^{14} T^{34} + 154232 p^{15} T^{35} + 4203 p^{16} T^{36} - 1824 p^{17} T^{37} - 144 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 8 T - 382 T^{2} - 3312 T^{3} + 74091 T^{4} + 665656 T^{5} - 10211710 T^{6} - 86944968 T^{7} + 1183330895 T^{8} + 8487337432 T^{9} - 124101906116 T^{10} - 678244049288 T^{11} + 11809048164234 T^{12} + 46374251496048 T^{13} - 1003318555664844 T^{14} - 2668356278297336 T^{15} + 75932798638004049 T^{16} + 117268504512620032 T^{17} - 5167894728391365490 T^{18} - 2575959546793047744 T^{19} + \)\(31\!\cdots\!01\)\( T^{20} - 2575959546793047744 p T^{21} - 5167894728391365490 p^{2} T^{22} + 117268504512620032 p^{3} T^{23} + 75932798638004049 p^{4} T^{24} - 2668356278297336 p^{5} T^{25} - 1003318555664844 p^{6} T^{26} + 46374251496048 p^{7} T^{27} + 11809048164234 p^{8} T^{28} - 678244049288 p^{9} T^{29} - 124101906116 p^{10} T^{30} + 8487337432 p^{11} T^{31} + 1183330895 p^{12} T^{32} - 86944968 p^{13} T^{33} - 10211710 p^{14} T^{34} + 665656 p^{15} T^{35} + 74091 p^{16} T^{36} - 3312 p^{17} T^{37} - 382 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 2 T - 325 T^{2} + 254 T^{3} + 51006 T^{4} - 163798 T^{5} - 5067791 T^{6} + 24629462 T^{7} + 380519774 T^{8} - 1925400810 T^{9} - 27967590115 T^{10} + 96608557530 T^{11} + 2364702196792 T^{12} - 4671954132414 T^{13} - 196667698942243 T^{14} + 334855360044926 T^{15} + 13838907207056825 T^{16} - 21862026017150396 T^{17} - 839078315082831318 T^{18} + 610787900971293612 T^{19} + 49687284414440816868 T^{20} + 610787900971293612 p T^{21} - 839078315082831318 p^{2} T^{22} - 21862026017150396 p^{3} T^{23} + 13838907207056825 p^{4} T^{24} + 334855360044926 p^{5} T^{25} - 196667698942243 p^{6} T^{26} - 4671954132414 p^{7} T^{27} + 2364702196792 p^{8} T^{28} + 96608557530 p^{9} T^{29} - 27967590115 p^{10} T^{30} - 1925400810 p^{11} T^{31} + 380519774 p^{12} T^{32} + 24629462 p^{13} T^{33} - 5067791 p^{14} T^{34} - 163798 p^{15} T^{35} + 51006 p^{16} T^{36} + 254 p^{17} T^{37} - 325 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 30 T + 721 T^{2} + 12630 T^{3} + 187006 T^{4} + 2417778 T^{5} + 27956163 T^{6} + 303389214 T^{7} + 3115514622 T^{8} + 31269018534 T^{9} + 307064242291 T^{10} + 2958581944206 T^{11} + 27894939150404 T^{12} + 255941309881746 T^{13} + 2296913065522251 T^{14} + 20197055474380098 T^{15} + 175416929020165737 T^{16} + 22508121048063156 p T^{17} + 12834830767634955790 T^{18} + 1609350677619238992 p T^{19} + \)\(89\!\cdots\!12\)\( T^{20} + 1609350677619238992 p^{2} T^{21} + 12834830767634955790 p^{2} T^{22} + 22508121048063156 p^{4} T^{23} + 175416929020165737 p^{4} T^{24} + 20197055474380098 p^{5} T^{25} + 2296913065522251 p^{6} T^{26} + 255941309881746 p^{7} T^{27} + 27894939150404 p^{8} T^{28} + 2958581944206 p^{9} T^{29} + 307064242291 p^{10} T^{30} + 31269018534 p^{11} T^{31} + 3115514622 p^{12} T^{32} + 303389214 p^{13} T^{33} + 27956163 p^{14} T^{34} + 2417778 p^{15} T^{35} + 187006 p^{16} T^{36} + 12630 p^{17} T^{37} + 721 p^{18} T^{38} + 30 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 4 T - 226 T^{2} - 624 T^{3} + 39455 T^{4} + 193104 T^{5} - 3168378 T^{6} - 40834844 T^{7} + 139907103 T^{8} + 3869021160 T^{9} + 17036808884 T^{10} - 266039596480 T^{11} - 2552049558902 T^{12} - 287162233344 T^{13} + 222335603528268 T^{14} + 1119373490666904 T^{15} - 3454550199826643 T^{16} - 135021089374855868 T^{17} - 563219169064041958 T^{18} + 45645382661379696 p T^{19} + 1233007891988697471 p T^{20} + 45645382661379696 p^{2} T^{21} - 563219169064041958 p^{2} T^{22} - 135021089374855868 p^{3} T^{23} - 3454550199826643 p^{4} T^{24} + 1119373490666904 p^{5} T^{25} + 222335603528268 p^{6} T^{26} - 287162233344 p^{7} T^{27} - 2552049558902 p^{8} T^{28} - 266039596480 p^{9} T^{29} + 17036808884 p^{10} T^{30} + 3869021160 p^{11} T^{31} + 139907103 p^{12} T^{32} - 40834844 p^{13} T^{33} - 3168378 p^{14} T^{34} + 193104 p^{15} T^{35} + 39455 p^{16} T^{36} - 624 p^{17} T^{37} - 226 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 22 T - 163 T^{2} - 3746 T^{3} + 76518 T^{4} + 783570 T^{5} - 13437617 T^{6} - 28690526 T^{7} + 2437292686 T^{8} - 4570021038 T^{9} - 243308019589 T^{10} + 2142167473402 T^{11} + 19117992937840 T^{12} - 292396407293526 T^{13} - 110708551321261 T^{14} + 31001204950468722 T^{15} - 123191445946156631 T^{16} - 1879996127620441244 T^{17} + 21207000341014394982 T^{18} + 63537869627362622476 T^{19} - \)\(17\!\cdots\!96\)\( T^{20} + 63537869627362622476 p T^{21} + 21207000341014394982 p^{2} T^{22} - 1879996127620441244 p^{3} T^{23} - 123191445946156631 p^{4} T^{24} + 31001204950468722 p^{5} T^{25} - 110708551321261 p^{6} T^{26} - 292396407293526 p^{7} T^{27} + 19117992937840 p^{8} T^{28} + 2142167473402 p^{9} T^{29} - 243308019589 p^{10} T^{30} - 4570021038 p^{11} T^{31} + 2437292686 p^{12} T^{32} - 28690526 p^{13} T^{33} - 13437617 p^{14} T^{34} + 783570 p^{15} T^{35} + 76518 p^{16} T^{36} - 3746 p^{17} T^{37} - 163 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 54 T + 1795 T^{2} + 44442 T^{3} + 900962 T^{4} + 15725910 T^{5} + 244072605 T^{6} + 3452910174 T^{7} + 45341560746 T^{8} + 561053653518 T^{9} + 6617033467433 T^{10} + 75034117085514 T^{11} + 823002475220740 T^{12} + 8765265971187678 T^{13} + 90857062779365713 T^{14} + 917874716796403842 T^{15} + 9046008996309118365 T^{16} + 87041376478321686012 T^{17} + \)\(81\!\cdots\!78\)\( T^{18} + \)\(75\!\cdots\!52\)\( T^{19} + \)\(67\!\cdots\!68\)\( T^{20} + \)\(75\!\cdots\!52\)\( p T^{21} + \)\(81\!\cdots\!78\)\( p^{2} T^{22} + 87041376478321686012 p^{3} T^{23} + 9046008996309118365 p^{4} T^{24} + 917874716796403842 p^{5} T^{25} + 90857062779365713 p^{6} T^{26} + 8765265971187678 p^{7} T^{27} + 823002475220740 p^{8} T^{28} + 75034117085514 p^{9} T^{29} + 6617033467433 p^{10} T^{30} + 561053653518 p^{11} T^{31} + 45341560746 p^{12} T^{32} + 3452910174 p^{13} T^{33} + 244072605 p^{14} T^{34} + 15725910 p^{15} T^{35} + 900962 p^{16} T^{36} + 44442 p^{17} T^{37} + 1795 p^{18} T^{38} + 54 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 976 T^{2} + 470378 T^{4} - 149986448 T^{6} + 35707021533 T^{8} - 6775587323776 T^{10} + 1066068068635320 T^{12} - 142608421165709056 T^{14} + 16482315843780806706 T^{16} - \)\(16\!\cdots\!80\)\( T^{18} + \)\(14\!\cdots\!36\)\( T^{20} - \)\(16\!\cdots\!80\)\( p^{2} T^{22} + 16482315843780806706 p^{4} T^{24} - 142608421165709056 p^{6} T^{26} + 1066068068635320 p^{8} T^{28} - 6775587323776 p^{10} T^{30} + 35707021533 p^{12} T^{32} - 149986448 p^{14} T^{34} + 470378 p^{16} T^{36} - 976 p^{18} T^{38} + p^{20} T^{40} \) | |
| 89 | \( 1 + 32 T + 82 T^{2} - 3024 T^{3} + 76687 T^{4} + 1219160 T^{5} - 9476646 T^{6} - 21010416 T^{7} + 2576112123 T^{8} - 1898883224 T^{9} - 156622499196 T^{10} + 2129638146968 T^{11} + 7219864179566 T^{12} - 111986222873392 T^{13} + 1287602536381172 T^{14} - 530942712192520 T^{15} - 22904390633941719 T^{16} + 1815710546095324808 T^{17} - 9322927802980704562 T^{18} - 55267080555448992352 T^{19} + \)\(21\!\cdots\!17\)\( T^{20} - 55267080555448992352 p T^{21} - 9322927802980704562 p^{2} T^{22} + 1815710546095324808 p^{3} T^{23} - 22904390633941719 p^{4} T^{24} - 530942712192520 p^{5} T^{25} + 1287602536381172 p^{6} T^{26} - 111986222873392 p^{7} T^{27} + 7219864179566 p^{8} T^{28} + 2129638146968 p^{9} T^{29} - 156622499196 p^{10} T^{30} - 1898883224 p^{11} T^{31} + 2576112123 p^{12} T^{32} - 21010416 p^{13} T^{33} - 9476646 p^{14} T^{34} + 1219160 p^{15} T^{35} + 76687 p^{16} T^{36} - 3024 p^{17} T^{37} + 82 p^{18} T^{38} + 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 12 T + 710 T^{2} - 7944 T^{3} + 253471 T^{4} - 2859024 T^{5} + 63754382 T^{6} - 734075148 T^{7} + 12858495535 T^{8} - 147595910208 T^{9} + 2208553441492 T^{10} - 24669791172600 T^{11} + 333554438148890 T^{12} - 3578852065785120 T^{13} + 44970680685635148 T^{14} - 462846432436475136 T^{15} + 5465479746532999949 T^{16} - 54155437726186975212 T^{17} + \)\(60\!\cdots\!54\)\( T^{18} - \)\(57\!\cdots\!84\)\( T^{19} + \)\(61\!\cdots\!41\)\( T^{20} - \)\(57\!\cdots\!84\)\( p T^{21} + \)\(60\!\cdots\!54\)\( p^{2} T^{22} - 54155437726186975212 p^{3} T^{23} + 5465479746532999949 p^{4} T^{24} - 462846432436475136 p^{5} T^{25} + 44970680685635148 p^{6} T^{26} - 3578852065785120 p^{7} T^{27} + 333554438148890 p^{8} T^{28} - 24669791172600 p^{9} T^{29} + 2208553441492 p^{10} T^{30} - 147595910208 p^{11} T^{31} + 12858495535 p^{12} T^{32} - 734075148 p^{13} T^{33} + 63754382 p^{14} T^{34} - 2859024 p^{15} T^{35} + 253471 p^{16} T^{36} - 7944 p^{17} T^{37} + 710 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−1.84206282955098073307477519495, −1.78349024314315077649565995822, −1.64474475890879295284449794728, −1.61277889763830021158106795532, −1.54566890365726290593487208123, −1.46019637170508103917449606704, −1.45259797032463009081918974617, −1.43568350599788797634226201158, −1.31131486163877452296161314748, −1.25043390292138805069141564276, −1.24872527802144282607840895464, −1.11191591416183468170113507299, −1.06086500345307531070965325055, −1.05317649022270313157159271212, −1.03793163926103233437462192521, −1.02021719745757179336912921535, −0.71530329685355441995805076448, −0.68983687966268637683409103708, −0.67046355232403493560418312585, −0.62925482168475008156541288018, −0.34852035853642008565442169298, −0.29552578297744359641799829530, −0.20921548040755070511681616962, −0.06763869677098977988408381267, −0.01725392564743959940587374473, 0.01725392564743959940587374473, 0.06763869677098977988408381267, 0.20921548040755070511681616962, 0.29552578297744359641799829530, 0.34852035853642008565442169298, 0.62925482168475008156541288018, 0.67046355232403493560418312585, 0.68983687966268637683409103708, 0.71530329685355441995805076448, 1.02021719745757179336912921535, 1.03793163926103233437462192521, 1.05317649022270313157159271212, 1.06086500345307531070965325055, 1.11191591416183468170113507299, 1.24872527802144282607840895464, 1.25043390292138805069141564276, 1.31131486163877452296161314748, 1.43568350599788797634226201158, 1.45259797032463009081918974617, 1.46019637170508103917449606704, 1.54566890365726290593487208123, 1.61277889763830021158106795532, 1.64474475890879295284449794728, 1.78349024314315077649565995822, 1.84206282955098073307477519495
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.