# Properties

 Degree $28$ Conductor $7.206\times 10^{30}$ Sign $1$ Motivic weight $5$ Primitive no Self-dual yes Analytic rank $0$

# Learn more about

## Dirichlet series

 L(s)  = 1 + 10·3-s + 42·5-s + 66·7-s + 50·9-s − 414·13-s + 420·15-s + 1.22e3·17-s + 5.67e3·19-s + 660·21-s + 2.90e3·23-s − 1.35e3·25-s + 254·27-s + 2.77e3·35-s − 1.79e3·37-s − 4.14e3·39-s + 1.16e4·41-s − 3.98e3·43-s + 2.10e3·45-s − 1.27e3·47-s + 2.17e3·49-s + 1.22e4·51-s + 5.88e3·53-s + 5.67e4·57-s − 8.50e3·59-s + 2.05e4·61-s + 3.30e3·63-s − 1.73e4·65-s + ⋯
 L(s)  = 1 + 0.641·3-s + 0.751·5-s + 0.509·7-s + 0.205·9-s − 0.679·13-s + 0.481·15-s + 1.02·17-s + 3.60·19-s + 0.326·21-s + 1.14·23-s − 0.432·25-s + 0.0670·27-s + 0.382·35-s − 0.214·37-s − 0.435·39-s + 1.08·41-s − 0.328·43-s + 0.154·45-s − 0.0843·47-s + 0.129·49-s + 0.657·51-s + 0.287·53-s + 2.31·57-s − 0.318·59-s + 0.707·61-s + 0.104·63-s − 0.510·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$28$$ Conductor: $$2^{70} \cdot 5^{14}$$ Sign: $1$ Motivic weight: $$5$$ Character: induced by $\chi_{160} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(28,\ 2^{70} \cdot 5^{14} ,\ ( \ : [5/2]^{14} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$9.265857704$$ $$L(\frac12)$$ $$\approx$$ $$9.265857704$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - 42 T + 623 p T^{2} - 15532 p T^{3} - 64003 p^{2} T^{4} + 24826 p^{4} T^{5} - 5225877 p^{5} T^{6} + 11846872 p^{7} T^{7} - 5225877 p^{10} T^{8} + 24826 p^{14} T^{9} - 64003 p^{17} T^{10} - 15532 p^{21} T^{11} + 623 p^{26} T^{12} - 42 p^{30} T^{13} + p^{35} T^{14}$$
good3 $$1 - 10 T + 50 T^{2} - 254 T^{3} - 91369 T^{4} + 1354100 T^{5} - 8940292 T^{6} + 156709276 T^{7} + 3134936461 T^{8} - 12907304306 p T^{9} + 32288446142 p^{2} T^{10} - 27718466434 p^{4} T^{11} + 116231885755 p^{4} T^{12} - 1947008143736 p^{5} T^{13} + 26412951129032 p^{6} T^{14} - 1947008143736 p^{10} T^{15} + 116231885755 p^{14} T^{16} - 27718466434 p^{19} T^{17} + 32288446142 p^{22} T^{18} - 12907304306 p^{26} T^{19} + 3134936461 p^{30} T^{20} + 156709276 p^{35} T^{21} - 8940292 p^{40} T^{22} + 1354100 p^{45} T^{23} - 91369 p^{50} T^{24} - 254 p^{55} T^{25} + 50 p^{60} T^{26} - 10 p^{65} T^{27} + p^{70} T^{28}$$
7 $$1 - 66 T + 2178 T^{2} + 3104786 T^{3} - 7972673 p^{2} T^{4} - 28598040732 T^{5} + 7558178349116 T^{6} - 730117525814980 T^{7} - 100452887411613603 T^{8} + 13091475719468548578 T^{9} + 64214231979007274450 p T^{10} -$$$$38\!\cdots\!06$$$$T^{11} +$$$$37\!\cdots\!31$$$$T^{12} +$$$$47\!\cdots\!84$$$$T^{13} -$$$$77\!\cdots\!76$$$$T^{14} +$$$$47\!\cdots\!84$$$$p^{5} T^{15} +$$$$37\!\cdots\!31$$$$p^{10} T^{16} -$$$$38\!\cdots\!06$$$$p^{15} T^{17} + 64214231979007274450 p^{21} T^{18} + 13091475719468548578 p^{25} T^{19} - 100452887411613603 p^{30} T^{20} - 730117525814980 p^{35} T^{21} + 7558178349116 p^{40} T^{22} - 28598040732 p^{45} T^{23} - 7972673 p^{52} T^{24} + 3104786 p^{55} T^{25} + 2178 p^{60} T^{26} - 66 p^{65} T^{27} + p^{70} T^{28}$$
11 $$1 - 137970 p T^{2} + 1118279059571 T^{4} - 531239593921187868 T^{6} +$$$$18\!\cdots\!61$$$$T^{8} -$$$$48\!\cdots\!18$$$$T^{10} +$$$$10\!\cdots\!67$$$$T^{12} -$$$$18\!\cdots\!88$$$$T^{14} +$$$$10\!\cdots\!67$$$$p^{10} T^{16} -$$$$48\!\cdots\!18$$$$p^{20} T^{18} +$$$$18\!\cdots\!61$$$$p^{30} T^{20} - 531239593921187868 p^{40} T^{22} + 1118279059571 p^{50} T^{24} - 137970 p^{61} T^{26} + p^{70} T^{28}$$
13 $$1 + 414 T + 85698 T^{2} + 485371686 T^{3} + 407510115667 T^{4} - 25330305055892 T^{5} + 72383288599671444 T^{6} +$$$$13\!\cdots\!64$$$$T^{7} +$$$$19\!\cdots\!85$$$$T^{8} -$$$$26\!\cdots\!74$$$$T^{9} +$$$$24\!\cdots\!22$$$$T^{10} +$$$$12\!\cdots\!02$$$$T^{11} -$$$$59\!\cdots\!93$$$$p^{2} T^{12} -$$$$21\!\cdots\!88$$$$p T^{13} +$$$$44\!\cdots\!64$$$$T^{14} -$$$$21\!\cdots\!88$$$$p^{6} T^{15} -$$$$59\!\cdots\!93$$$$p^{12} T^{16} +$$$$12\!\cdots\!02$$$$p^{15} T^{17} +$$$$24\!\cdots\!22$$$$p^{20} T^{18} -$$$$26\!\cdots\!74$$$$p^{25} T^{19} +$$$$19\!\cdots\!85$$$$p^{30} T^{20} +$$$$13\!\cdots\!64$$$$p^{35} T^{21} + 72383288599671444 p^{40} T^{22} - 25330305055892 p^{45} T^{23} + 407510115667 p^{50} T^{24} + 485371686 p^{55} T^{25} + 85698 p^{60} T^{26} + 414 p^{65} T^{27} + p^{70} T^{28}$$
17 $$1 - 1222 T + 746642 T^{2} - 1023656774 T^{3} - 7517818418661 T^{4} + 9805314295428420 T^{5} - 5845038493789599340 T^{6} +$$$$85\!\cdots\!80$$$$T^{7} +$$$$21\!\cdots\!05$$$$T^{8} -$$$$32\!\cdots\!70$$$$T^{9} +$$$$19\!\cdots\!70$$$$T^{10} -$$$$33\!\cdots\!50$$$$T^{11} -$$$$20\!\cdots\!37$$$$T^{12} +$$$$66\!\cdots\!24$$$$T^{13} -$$$$44\!\cdots\!04$$$$T^{14} +$$$$66\!\cdots\!24$$$$p^{5} T^{15} -$$$$20\!\cdots\!37$$$$p^{10} T^{16} -$$$$33\!\cdots\!50$$$$p^{15} T^{17} +$$$$19\!\cdots\!70$$$$p^{20} T^{18} -$$$$32\!\cdots\!70$$$$p^{25} T^{19} +$$$$21\!\cdots\!05$$$$p^{30} T^{20} +$$$$85\!\cdots\!80$$$$p^{35} T^{21} - 5845038493789599340 p^{40} T^{22} + 9805314295428420 p^{45} T^{23} - 7517818418661 p^{50} T^{24} - 1023656774 p^{55} T^{25} + 746642 p^{60} T^{26} - 1222 p^{65} T^{27} + p^{70} T^{28}$$
19 $$( 1 - 2836 T + 12192485 T^{2} - 21773970856 T^{3} + 3075663164351 p T^{4} - 77435325554759820 T^{5} +$$$$17\!\cdots\!85$$$$T^{6} -$$$$20\!\cdots\!00$$$$T^{7} +$$$$17\!\cdots\!85$$$$p^{5} T^{8} - 77435325554759820 p^{10} T^{9} + 3075663164351 p^{16} T^{10} - 21773970856 p^{20} T^{11} + 12192485 p^{25} T^{12} - 2836 p^{30} T^{13} + p^{35} T^{14} )^{2}$$
23 $$1 - 2902 T + 4210802 T^{2} + 8585545830 T^{3} - 77137172452561 T^{4} + 5298422918089420 T^{5} +$$$$34\!\cdots\!32$$$$T^{6} -$$$$17\!\cdots\!48$$$$T^{7} +$$$$43\!\cdots\!33$$$$T^{8} +$$$$14\!\cdots\!50$$$$T^{9} -$$$$10\!\cdots\!54$$$$T^{10} +$$$$57\!\cdots\!98$$$$T^{11} -$$$$19\!\cdots\!21$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!40$$$$T^{14} -$$$$51\!\cdots\!28$$$$p^{5} T^{15} -$$$$19\!\cdots\!21$$$$p^{10} T^{16} +$$$$57\!\cdots\!98$$$$p^{15} T^{17} -$$$$10\!\cdots\!54$$$$p^{20} T^{18} +$$$$14\!\cdots\!50$$$$p^{25} T^{19} +$$$$43\!\cdots\!33$$$$p^{30} T^{20} -$$$$17\!\cdots\!48$$$$p^{35} T^{21} +$$$$34\!\cdots\!32$$$$p^{40} T^{22} + 5298422918089420 p^{45} T^{23} - 77137172452561 p^{50} T^{24} + 8585545830 p^{55} T^{25} + 4210802 p^{60} T^{26} - 2902 p^{65} T^{27} + p^{70} T^{28}$$
29 $$1 - 127109206 T^{2} + 8400089254893139 T^{4} -$$$$38\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!57$$$$T^{8} -$$$$39\!\cdots\!02$$$$T^{10} +$$$$98\!\cdots\!23$$$$T^{12} -$$$$21\!\cdots\!92$$$$T^{14} +$$$$98\!\cdots\!23$$$$p^{10} T^{16} -$$$$39\!\cdots\!02$$$$p^{20} T^{18} +$$$$13\!\cdots\!57$$$$p^{30} T^{20} -$$$$38\!\cdots\!76$$$$p^{40} T^{22} + 8400089254893139 p^{50} T^{24} - 127109206 p^{60} T^{26} + p^{70} T^{28}$$
31 $$1 - 265679454 T^{2} + 34933996110970539 T^{4} -$$$$30\!\cdots\!64$$$$T^{6} +$$$$19\!\cdots\!17$$$$T^{8} -$$$$92\!\cdots\!38$$$$T^{10} +$$$$36\!\cdots\!43$$$$T^{12} -$$$$11\!\cdots\!88$$$$T^{14} +$$$$36\!\cdots\!43$$$$p^{10} T^{16} -$$$$92\!\cdots\!38$$$$p^{20} T^{18} +$$$$19\!\cdots\!17$$$$p^{30} T^{20} -$$$$30\!\cdots\!64$$$$p^{40} T^{22} + 34933996110970539 p^{50} T^{24} - 265679454 p^{60} T^{26} + p^{70} T^{28}$$
37 $$1 + 1790 T + 1602050 T^{2} - 1085215470730 T^{3} - 5589943426343837 T^{4} - 5189725886098796788 T^{5} +$$$$58\!\cdots\!80$$$$T^{6} +$$$$37\!\cdots\!60$$$$T^{7} +$$$$20\!\cdots\!01$$$$T^{8} -$$$$14\!\cdots\!14$$$$T^{9} -$$$$11\!\cdots\!78$$$$T^{10} -$$$$12\!\cdots\!50$$$$T^{11} -$$$$40\!\cdots\!45$$$$T^{12} +$$$$67\!\cdots\!72$$$$T^{13} +$$$$47\!\cdots\!96$$$$T^{14} +$$$$67\!\cdots\!72$$$$p^{5} T^{15} -$$$$40\!\cdots\!45$$$$p^{10} T^{16} -$$$$12\!\cdots\!50$$$$p^{15} T^{17} -$$$$11\!\cdots\!78$$$$p^{20} T^{18} -$$$$14\!\cdots\!14$$$$p^{25} T^{19} +$$$$20\!\cdots\!01$$$$p^{30} T^{20} +$$$$37\!\cdots\!60$$$$p^{35} T^{21} +$$$$58\!\cdots\!80$$$$p^{40} T^{22} - 5189725886098796788 p^{45} T^{23} - 5589943426343837 p^{50} T^{24} - 1085215470730 p^{55} T^{25} + 1602050 p^{60} T^{26} + 1790 p^{65} T^{27} + p^{70} T^{28}$$
41 $$( 1 - 142 p T + 527400583 T^{2} - 3511555483012 T^{3} + 138605008918596701 T^{4} -$$$$95\!\cdots\!50$$$$T^{5} +$$$$23\!\cdots\!95$$$$T^{6} -$$$$14\!\cdots\!20$$$$T^{7} +$$$$23\!\cdots\!95$$$$p^{5} T^{8} -$$$$95\!\cdots\!50$$$$p^{10} T^{9} + 138605008918596701 p^{15} T^{10} - 3511555483012 p^{20} T^{11} + 527400583 p^{25} T^{12} - 142 p^{31} T^{13} + p^{35} T^{14} )^{2}$$
43 $$1 + 3982 T + 7928162 T^{2} + 5090675647610 T^{3} + 44126507369726119 T^{4} + 95654814508201483332 T^{5} +$$$$12\!\cdots\!96$$$$T^{6} +$$$$15\!\cdots\!12$$$$T^{7} +$$$$46\!\cdots\!93$$$$T^{8} +$$$$33\!\cdots\!78$$$$T^{9} +$$$$35\!\cdots\!38$$$$T^{10} +$$$$10\!\cdots\!70$$$$T^{11} +$$$$78\!\cdots\!59$$$$T^{12} +$$$$84\!\cdots\!64$$$$T^{13} +$$$$18\!\cdots\!08$$$$T^{14} +$$$$84\!\cdots\!64$$$$p^{5} T^{15} +$$$$78\!\cdots\!59$$$$p^{10} T^{16} +$$$$10\!\cdots\!70$$$$p^{15} T^{17} +$$$$35\!\cdots\!38$$$$p^{20} T^{18} +$$$$33\!\cdots\!78$$$$p^{25} T^{19} +$$$$46\!\cdots\!93$$$$p^{30} T^{20} +$$$$15\!\cdots\!12$$$$p^{35} T^{21} +$$$$12\!\cdots\!96$$$$p^{40} T^{22} + 95654814508201483332 p^{45} T^{23} + 44126507369726119 p^{50} T^{24} + 5090675647610 p^{55} T^{25} + 7928162 p^{60} T^{26} + 3982 p^{65} T^{27} + p^{70} T^{28}$$
47 $$1 + 1278 T + 816642 T^{2} + 351784473730 T^{3} + 13143771115043551 T^{4} - 55735540634425394844 T^{5} -$$$$20\!\cdots\!24$$$$T^{6} -$$$$35\!\cdots\!24$$$$T^{7} +$$$$21\!\cdots\!21$$$$T^{8} +$$$$27\!\cdots\!06$$$$T^{9} +$$$$23\!\cdots\!58$$$$T^{10} +$$$$14\!\cdots\!22$$$$T^{11} -$$$$33\!\cdots\!25$$$$T^{12} -$$$$57\!\cdots\!44$$$$T^{13} -$$$$35\!\cdots\!52$$$$T^{14} -$$$$57\!\cdots\!44$$$$p^{5} T^{15} -$$$$33\!\cdots\!25$$$$p^{10} T^{16} +$$$$14\!\cdots\!22$$$$p^{15} T^{17} +$$$$23\!\cdots\!58$$$$p^{20} T^{18} +$$$$27\!\cdots\!06$$$$p^{25} T^{19} +$$$$21\!\cdots\!21$$$$p^{30} T^{20} -$$$$35\!\cdots\!24$$$$p^{35} T^{21} -$$$$20\!\cdots\!24$$$$p^{40} T^{22} - 55735540634425394844 p^{45} T^{23} + 13143771115043551 p^{50} T^{24} + 351784473730 p^{55} T^{25} + 816642 p^{60} T^{26} + 1278 p^{65} T^{27} + p^{70} T^{28}$$
53 $$1 - 5882 T + 17298962 T^{2} + 13844528488830 T^{3} - 174311180941592381 T^{4} -$$$$81\!\cdots\!52$$$$T^{5} +$$$$14\!\cdots\!36$$$$T^{6} -$$$$35\!\cdots\!12$$$$T^{7} -$$$$10\!\cdots\!67$$$$T^{8} +$$$$15\!\cdots\!82$$$$T^{9} +$$$$26\!\cdots\!38$$$$T^{10} -$$$$97\!\cdots\!30$$$$T^{11} +$$$$23\!\cdots\!39$$$$T^{12} +$$$$29\!\cdots\!36$$$$T^{13} -$$$$56\!\cdots\!72$$$$T^{14} +$$$$29\!\cdots\!36$$$$p^{5} T^{15} +$$$$23\!\cdots\!39$$$$p^{10} T^{16} -$$$$97\!\cdots\!30$$$$p^{15} T^{17} +$$$$26\!\cdots\!38$$$$p^{20} T^{18} +$$$$15\!\cdots\!82$$$$p^{25} T^{19} -$$$$10\!\cdots\!67$$$$p^{30} T^{20} -$$$$35\!\cdots\!12$$$$p^{35} T^{21} +$$$$14\!\cdots\!36$$$$p^{40} T^{22} -$$$$81\!\cdots\!52$$$$p^{45} T^{23} - 174311180941592381 p^{50} T^{24} + 13844528488830 p^{55} T^{25} + 17298962 p^{60} T^{26} - 5882 p^{65} T^{27} + p^{70} T^{28}$$
59 $$( 1 + 4252 T + 2005910877 T^{2} + 20518241068280 T^{3} + 2395790088022200413 T^{4} +$$$$22\!\cdots\!48$$$$T^{5} +$$$$23\!\cdots\!29$$$$T^{6} +$$$$16\!\cdots\!28$$$$T^{7} +$$$$23\!\cdots\!29$$$$p^{5} T^{8} +$$$$22\!\cdots\!48$$$$p^{10} T^{9} + 2395790088022200413 p^{15} T^{10} + 20518241068280 p^{20} T^{11} + 2005910877 p^{25} T^{12} + 4252 p^{30} T^{13} + p^{35} T^{14} )^{2}$$
61 $$( 1 - 10282 T + 2916253203 T^{2} + 3083320715268 T^{3} + 3046127491767085861 T^{4} +$$$$70\!\cdots\!38$$$$T^{5} +$$$$15\!\cdots\!39$$$$T^{6} +$$$$10\!\cdots\!44$$$$T^{7} +$$$$15\!\cdots\!39$$$$p^{5} T^{8} +$$$$70\!\cdots\!38$$$$p^{10} T^{9} + 3046127491767085861 p^{15} T^{10} + 3083320715268 p^{20} T^{11} + 2916253203 p^{25} T^{12} - 10282 p^{30} T^{13} + p^{35} T^{14} )^{2}$$
67 $$1 - 107926 T + 5824010738 T^{2} - 286091118212770 T^{3} + 7910959270847152663 T^{4} +$$$$10\!\cdots\!32$$$$T^{5} -$$$$16\!\cdots\!76$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} -$$$$49\!\cdots\!51$$$$T^{8} +$$$$12\!\cdots\!38$$$$T^{9} -$$$$20\!\cdots\!90$$$$T^{10} -$$$$24\!\cdots\!26$$$$T^{11} +$$$$60\!\cdots\!11$$$$T^{12} -$$$$31\!\cdots\!32$$$$T^{13} +$$$$11\!\cdots\!96$$$$T^{14} -$$$$31\!\cdots\!32$$$$p^{5} T^{15} +$$$$60\!\cdots\!11$$$$p^{10} T^{16} -$$$$24\!\cdots\!26$$$$p^{15} T^{17} -$$$$20\!\cdots\!90$$$$p^{20} T^{18} +$$$$12\!\cdots\!38$$$$p^{25} T^{19} -$$$$49\!\cdots\!51$$$$p^{30} T^{20} +$$$$10\!\cdots\!52$$$$p^{35} T^{21} -$$$$16\!\cdots\!76$$$$p^{40} T^{22} +$$$$10\!\cdots\!32$$$$p^{45} T^{23} + 7910959270847152663 p^{50} T^{24} - 286091118212770 p^{55} T^{25} + 5824010738 p^{60} T^{26} - 107926 p^{65} T^{27} + p^{70} T^{28}$$
71 $$1 - 12494857902 T^{2} + 77600272039468829243 T^{4} -$$$$32\!\cdots\!32$$$$T^{6} +$$$$99\!\cdots\!41$$$$T^{8} -$$$$25\!\cdots\!58$$$$T^{10} +$$$$55\!\cdots\!91$$$$T^{12} -$$$$10\!\cdots\!64$$$$T^{14} +$$$$55\!\cdots\!91$$$$p^{10} T^{16} -$$$$25\!\cdots\!58$$$$p^{20} T^{18} +$$$$99\!\cdots\!41$$$$p^{30} T^{20} -$$$$32\!\cdots\!32$$$$p^{40} T^{22} + 77600272039468829243 p^{50} T^{24} - 12494857902 p^{60} T^{26} + p^{70} T^{28}$$
73 $$1 + 16418 T + 134775362 T^{2} - 257655666175822 T^{3} - 4365114300655601877 T^{4} +$$$$39\!\cdots\!72$$$$T^{5} +$$$$40\!\cdots\!12$$$$T^{6} +$$$$72\!\cdots\!32$$$$T^{7} -$$$$10\!\cdots\!35$$$$T^{8} -$$$$45\!\cdots\!66$$$$T^{9} +$$$$74\!\cdots\!94$$$$T^{10} +$$$$14\!\cdots\!50$$$$T^{11} +$$$$46\!\cdots\!07$$$$T^{12} -$$$$13\!\cdots\!16$$$$T^{13} -$$$$14\!\cdots\!04$$$$T^{14} -$$$$13\!\cdots\!16$$$$p^{5} T^{15} +$$$$46\!\cdots\!07$$$$p^{10} T^{16} +$$$$14\!\cdots\!50$$$$p^{15} T^{17} +$$$$74\!\cdots\!94$$$$p^{20} T^{18} -$$$$45\!\cdots\!66$$$$p^{25} T^{19} -$$$$10\!\cdots\!35$$$$p^{30} T^{20} +$$$$72\!\cdots\!32$$$$p^{35} T^{21} +$$$$40\!\cdots\!12$$$$p^{40} T^{22} +$$$$39\!\cdots\!72$$$$p^{45} T^{23} - 4365114300655601877 p^{50} T^{24} - 257655666175822 p^{55} T^{25} + 134775362 p^{60} T^{26} + 16418 p^{65} T^{27} + p^{70} T^{28}$$
79 $$( 1 + 73272 T + 14910422889 T^{2} + 713614712327088 T^{3} + 91878682859930077941 T^{4} +$$$$38\!\cdots\!48$$$$p T^{5} +$$$$35\!\cdots\!29$$$$T^{6} +$$$$97\!\cdots\!08$$$$T^{7} +$$$$35\!\cdots\!29$$$$p^{5} T^{8} +$$$$38\!\cdots\!48$$$$p^{11} T^{9} + 91878682859930077941 p^{15} T^{10} + 713614712327088 p^{20} T^{11} + 14910422889 p^{25} T^{12} + 73272 p^{30} T^{13} + p^{35} T^{14} )^{2}$$
83 $$1 + 36398 T + 662407202 T^{2} - 351755925637942 T^{3} - 18959358682913357257 T^{4} +$$$$10\!\cdots\!44$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} +$$$$60\!\cdots\!36$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8} -$$$$15\!\cdots\!90$$$$T^{9} -$$$$10\!\cdots\!50$$$$T^{10} -$$$$86\!\cdots\!90$$$$T^{11} -$$$$70\!\cdots\!85$$$$T^{12} +$$$$38\!\cdots\!20$$$$T^{13} +$$$$31\!\cdots\!80$$$$T^{14} +$$$$38\!\cdots\!20$$$$p^{5} T^{15} -$$$$70\!\cdots\!85$$$$p^{10} T^{16} -$$$$86\!\cdots\!90$$$$p^{15} T^{17} -$$$$10\!\cdots\!50$$$$p^{20} T^{18} -$$$$15\!\cdots\!90$$$$p^{25} T^{19} +$$$$11\!\cdots\!21$$$$p^{30} T^{20} +$$$$60\!\cdots\!36$$$$p^{35} T^{21} +$$$$11\!\cdots\!08$$$$p^{40} T^{22} +$$$$10\!\cdots\!44$$$$p^{45} T^{23} - 18959358682913357257 p^{50} T^{24} - 351755925637942 p^{55} T^{25} + 662407202 p^{60} T^{26} + 36398 p^{65} T^{27} + p^{70} T^{28}$$
89 $$1 - 30334038558 T^{2} +$$$$36\!\cdots\!35$$$$T^{4} -$$$$20\!\cdots\!52$$$$T^{6} +$$$$39\!\cdots\!69$$$$T^{8} -$$$$12\!\cdots\!10$$$$T^{10} +$$$$42\!\cdots\!95$$$$T^{12} -$$$$38\!\cdots\!60$$$$T^{14} +$$$$42\!\cdots\!95$$$$p^{10} T^{16} -$$$$12\!\cdots\!10$$$$p^{20} T^{18} +$$$$39\!\cdots\!69$$$$p^{30} T^{20} -$$$$20\!\cdots\!52$$$$p^{40} T^{22} +$$$$36\!\cdots\!35$$$$p^{50} T^{24} - 30334038558 p^{60} T^{26} + p^{70} T^{28}$$
97 $$1 + 60314 T + 1818889298 T^{2} - 623535693824102 T^{3} -$$$$16\!\cdots\!77$$$$T^{4} -$$$$33\!\cdots\!64$$$$T^{5} +$$$$29\!\cdots\!52$$$$T^{6} +$$$$51\!\cdots\!52$$$$T^{7} +$$$$13\!\cdots\!33$$$$p T^{8} -$$$$44\!\cdots\!50$$$$T^{9} +$$$$18\!\cdots\!50$$$$T^{10} +$$$$91\!\cdots\!50$$$$T^{11} +$$$$44\!\cdots\!75$$$$T^{12} -$$$$82\!\cdots\!00$$$$T^{13} -$$$$73\!\cdots\!00$$$$T^{14} -$$$$82\!\cdots\!00$$$$p^{5} T^{15} +$$$$44\!\cdots\!75$$$$p^{10} T^{16} +$$$$91\!\cdots\!50$$$$p^{15} T^{17} +$$$$18\!\cdots\!50$$$$p^{20} T^{18} -$$$$44\!\cdots\!50$$$$p^{25} T^{19} +$$$$13\!\cdots\!33$$$$p^{31} T^{20} +$$$$51\!\cdots\!52$$$$p^{35} T^{21} +$$$$29\!\cdots\!52$$$$p^{40} T^{22} -$$$$33\!\cdots\!64$$$$p^{45} T^{23} -$$$$16\!\cdots\!77$$$$p^{50} T^{24} - 623535693824102 p^{55} T^{25} + 1818889298 p^{60} T^{26} + 60314 p^{65} T^{27} + p^{70} T^{28}$$
show more
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−3.19914015631181393228383475479, −2.89753591575533337412428868412, −2.82907750635253006250112616136, −2.60870918179524455051715736523, −2.57115347528033955768273724667, −2.46320163188241689481934650008, −2.41371282080863118701772723750, −2.32772902192153018731103182030, −2.21796798113967994700151704038, −2.21195161939312623989040451545, −1.80642404621072424911334776632, −1.71961416883680084948431854474, −1.70130479412006016918873077545, −1.48185048105592344971770333785, −1.35857038319010044384302043991, −1.27371045554518024638722051706, −1.17902911520192284746270255018, −0.976546217881951214578717580638, −0.933270648023861040921065627100, −0.930462747598537041568561568035, −0.819059916348767244595269756700, −0.45600610741081031681833466976, −0.34058295467615935608739459195, −0.27544117574174641306384999406, −0.07853856617321036398884190249, 0.07853856617321036398884190249, 0.27544117574174641306384999406, 0.34058295467615935608739459195, 0.45600610741081031681833466976, 0.819059916348767244595269756700, 0.930462747598537041568561568035, 0.933270648023861040921065627100, 0.976546217881951214578717580638, 1.17902911520192284746270255018, 1.27371045554518024638722051706, 1.35857038319010044384302043991, 1.48185048105592344971770333785, 1.70130479412006016918873077545, 1.71961416883680084948431854474, 1.80642404621072424911334776632, 2.21195161939312623989040451545, 2.21796798113967994700151704038, 2.32772902192153018731103182030, 2.41371282080863118701772723750, 2.46320163188241689481934650008, 2.57115347528033955768273724667, 2.60870918179524455051715736523, 2.82907750635253006250112616136, 2.89753591575533337412428868412, 3.19914015631181393228383475479

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.