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Properties

Label 16-63e8-1.1-c7e8-0-1
Degree 1616
Conductor 2.482×10142.482\times 10^{14}
Sign 11
Analytic cond. 2.25032×10102.25032\times 10^{10}
Root an. cond. 4.436244.43624
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 206·4-s + 196·5-s + 154·7-s + 607·8-s − 196·10-s − 5.21e3·11-s − 7.58e3·13-s − 154·14-s + 2.89e4·16-s − 1.14e4·17-s + 6.91e4·19-s + 4.03e4·20-s + 5.21e3·22-s − 1.46e5·23-s + 1.72e5·25-s + 7.58e3·26-s + 3.17e4·28-s + 1.41e5·29-s + 2.88e5·31-s + 1.72e5·32-s + 1.14e4·34-s + 3.01e4·35-s + 4.48e5·37-s − 6.91e4·38-s + 1.18e5·40-s + 1.32e6·41-s + ⋯
L(s)  = 1  − 0.0883·2-s + 1.60·4-s + 0.701·5-s + 0.169·7-s + 0.419·8-s − 0.0619·10-s − 1.18·11-s − 0.957·13-s − 0.0149·14-s + 1.76·16-s − 0.564·17-s + 2.31·19-s + 1.12·20-s + 0.104·22-s − 2.50·23-s + 2.20·25-s + 0.0846·26-s + 0.273·28-s + 1.07·29-s + 1.74·31-s + 0.932·32-s + 0.0498·34-s + 0.118·35-s + 1.45·37-s − 0.204·38-s + 0.293·40-s + 3.00·41-s + ⋯

Functional equation

Λ(s)=((316⋅78)s/2 ΓC(s)8 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((316⋅78)s/2 ΓC(s+7/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 316⋅783^{16} \cdot 7^{8}
Sign: 11
Analytic conductor: 2.25032×10102.25032\times 10^{10}
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 316⋅78, ( :[7/2]8), 1)(16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )

Particular Values

L(4)L(4) ≈\approx 62.6805409962.68054099
L(12)L(\frac12) ≈\approx 62.6805409962.68054099
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1−22pT−53948p2T2+44152p4T3+25832189p6T4+44152p11T5−53948p16T6−22p22T7+p28T8 1 - 22 p T - 53948 p^{2} T^{2} + 44152 p^{4} T^{3} + 25832189 p^{6} T^{4} + 44152 p^{11} T^{5} - 53948 p^{16} T^{6} - 22 p^{22} T^{7} + p^{28} T^{8}
good2 1+T−205T2−509pT3+1463p3T4+18021p3T5+26303p5T6−261523p5T7−1566247p6T8−261523p12T9+26303p19T10+18021p24T11+1463p31T12−509p36T13−205p42T14+p49T15+p56T16 1 + T - 205 T^{2} - 509 p T^{3} + 1463 p^{3} T^{4} + 18021 p^{3} T^{5} + 26303 p^{5} T^{6} - 261523 p^{5} T^{7} - 1566247 p^{6} T^{8} - 261523 p^{12} T^{9} + 26303 p^{19} T^{10} + 18021 p^{24} T^{11} + 1463 p^{31} T^{12} - 509 p^{36} T^{13} - 205 p^{42} T^{14} + p^{49} T^{15} + p^{56} T^{16}
5 1−196T−133927T2+582836p3T3+176610889p2T4−7538169316992T5+1573062976649806T6+69013603051990576pT7−7354408837532411194p2T8+69013603051990576p8T9+1573062976649806p14T10−7538169316992p21T11+176610889p30T12+582836p38T13−133927p42T14−196p49T15+p56T16 1 - 196 T - 133927 T^{2} + 582836 p^{3} T^{3} + 176610889 p^{2} T^{4} - 7538169316992 T^{5} + 1573062976649806 T^{6} + 69013603051990576 p T^{7} - 7354408837532411194 p^{2} T^{8} + 69013603051990576 p^{8} T^{9} + 1573062976649806 p^{14} T^{10} - 7538169316992 p^{21} T^{11} + 176610889 p^{30} T^{12} + 582836 p^{38} T^{13} - 133927 p^{42} T^{14} - 196 p^{49} T^{15} + p^{56} T^{16}
11 1+5210T−13550341T2+87815127118T3+813472841934301T4−1670782125020401848T5+ 1 + 5210 T - 13550341 T^{2} + 87815127118 T^{3} + 813472841934301 T^{4} - 1670782125020401848 T^{5} + 73 ⁣⋯ ⁣9873\!\cdots\!98T6+ T^{6} + 52 ⁣⋯ ⁣2852\!\cdots\!28T7− T^{7} - 17 ⁣⋯ ⁣3817\!\cdots\!38T8+ T^{8} + 52 ⁣⋯ ⁣2852\!\cdots\!28p7T9+ p^{7} T^{9} + 73 ⁣⋯ ⁣9873\!\cdots\!98p14T10−1670782125020401848p21T11+813472841934301p28T12+87815127118p35T13−13550341p42T14+5210p49T15+p56T16 p^{14} T^{10} - 1670782125020401848 p^{21} T^{11} + 813472841934301 p^{28} T^{12} + 87815127118 p^{35} T^{13} - 13550341 p^{42} T^{14} + 5210 p^{49} T^{15} + p^{56} T^{16}
13 (1+3794T+102976741T2+172660567546T3+8327318022182528T4+172660567546p7T5+102976741p14T6+3794p21T7+p28T8)2 ( 1 + 3794 T + 102976741 T^{2} + 172660567546 T^{3} + 8327318022182528 T^{4} + 172660567546 p^{7} T^{5} + 102976741 p^{14} T^{6} + 3794 p^{21} T^{7} + p^{28} T^{8} )^{2}
17 1+11436T−1114853444T2−158928045720T3+809471792809409914T4− 1 + 11436 T - 1114853444 T^{2} - 158928045720 T^{3} + 809471792809409914 T^{4} - 31 ⁣⋯ ⁣9231\!\cdots\!92T5− T^{5} - 38 ⁣⋯ ⁣6038\!\cdots\!60T6+ T^{6} + 57 ⁣⋯ ⁣5257\!\cdots\!52T7+ T^{7} + 14 ⁣⋯ ⁣5914\!\cdots\!59T8+ T^{8} + 57 ⁣⋯ ⁣5257\!\cdots\!52p7T9− p^{7} T^{9} - 38 ⁣⋯ ⁣6038\!\cdots\!60p14T10− p^{14} T^{10} - 31 ⁣⋯ ⁣9231\!\cdots\!92p21T11+809471792809409914p28T12−158928045720p35T13−1114853444p42T14+11436p49T15+p56T16 p^{21} T^{11} + 809471792809409914 p^{28} T^{12} - 158928045720 p^{35} T^{13} - 1114853444 p^{42} T^{14} + 11436 p^{49} T^{15} + p^{56} T^{16}
19 1−69158T+1595596935T2−18293280666090T3−81905563961069391T4+ 1 - 69158 T + 1595596935 T^{2} - 18293280666090 T^{3} - 81905563961069391 T^{4} + 37 ⁣⋯ ⁣9637\!\cdots\!96T5− T^{5} - 13 ⁣⋯ ⁣7413\!\cdots\!74T6+ T^{6} + 11 ⁣⋯ ⁣3611\!\cdots\!36pT7− p T^{7} - 48 ⁣⋯ ⁣9848\!\cdots\!98T8+ T^{8} + 11 ⁣⋯ ⁣3611\!\cdots\!36p8T9− p^{8} T^{9} - 13 ⁣⋯ ⁣7413\!\cdots\!74p14T10+ p^{14} T^{10} + 37 ⁣⋯ ⁣9637\!\cdots\!96p21T11−81905563961069391p28T12−18293280666090p35T13+1595596935p42T14−69158p49T15+p56T16 p^{21} T^{11} - 81905563961069391 p^{28} T^{12} - 18293280666090 p^{35} T^{13} + 1595596935 p^{42} T^{14} - 69158 p^{49} T^{15} + p^{56} T^{16}
23 1+146220T+1817735620T2−252982184818344T3+44908344735632175994T4+ 1 + 146220 T + 1817735620 T^{2} - 252982184818344 T^{3} + 44908344735632175994 T^{4} + 35 ⁣⋯ ⁣0435\!\cdots\!04T5− T^{5} - 53 ⁣⋯ ⁣3653\!\cdots\!36T6+ T^{6} + 30 ⁣⋯ ⁣7230\!\cdots\!72T7+ T^{7} + 10 ⁣⋯ ⁣2710\!\cdots\!27T8+ T^{8} + 30 ⁣⋯ ⁣7230\!\cdots\!72p7T9− p^{7} T^{9} - 53 ⁣⋯ ⁣3653\!\cdots\!36p14T10+ p^{14} T^{10} + 35 ⁣⋯ ⁣0435\!\cdots\!04p21T11+44908344735632175994p28T12−252982184818344p35T13+1817735620p42T14+146220p49T15+p56T16 p^{21} T^{11} + 44908344735632175994 p^{28} T^{12} - 252982184818344 p^{35} T^{13} + 1817735620 p^{42} T^{14} + 146220 p^{49} T^{15} + p^{56} T^{16}
29 (1−70664T+48199790963T2−1247069390276928T3+ ( 1 - 70664 T + 48199790963 T^{2} - 1247069390276928 T^{3} + 10 ⁣⋯ ⁣8010\!\cdots\!80T4−1247069390276928p7T5+48199790963p14T6−70664p21T7+p28T8)2 T^{4} - 1247069390276928 p^{7} T^{5} + 48199790963 p^{14} T^{6} - 70664 p^{21} T^{7} + p^{28} T^{8} )^{2}
31 1−288618T−50722994584T2+9961277652025592T3+ 1 - 288618 T - 50722994584 T^{2} + 9961277652025592 T^{3} + 50 ⁣⋯ ⁣5550\!\cdots\!55T4− T^{4} - 54 ⁣⋯ ⁣3254\!\cdots\!32T5− T^{5} - 18 ⁣⋯ ⁣8418\!\cdots\!84T6+ T^{6} + 15 ⁣⋯ ⁣0615\!\cdots\!06T7+ T^{7} + 73 ⁣⋯ ⁣6473\!\cdots\!64T8+ T^{8} + 15 ⁣⋯ ⁣0615\!\cdots\!06p7T9− p^{7} T^{9} - 18 ⁣⋯ ⁣8418\!\cdots\!84p14T10− p^{14} T^{10} - 54 ⁣⋯ ⁣3254\!\cdots\!32p21T11+ p^{21} T^{11} + 50 ⁣⋯ ⁣5550\!\cdots\!55p28T12+9961277652025592p35T13−50722994584p42T14−288618p49T15+p56T16 p^{28} T^{12} + 9961277652025592 p^{35} T^{13} - 50722994584 p^{42} T^{14} - 288618 p^{49} T^{15} + p^{56} T^{16}
37 1−448902T−84053579929T2+1683019295379758T3+ 1 - 448902 T - 84053579929 T^{2} + 1683019295379758 T^{3} + 60 ⁣⋯ ⁣0960\!\cdots\!09pT4+ p T^{4} + 20 ⁣⋯ ⁣3620\!\cdots\!36T5− T^{5} - 19 ⁣⋯ ⁣3019\!\cdots\!30T6+ T^{6} + 44 ⁣⋯ ⁣0444\!\cdots\!04T7+ T^{7} + 18 ⁣⋯ ⁣2618\!\cdots\!26T8+ T^{8} + 44 ⁣⋯ ⁣0444\!\cdots\!04p7T9− p^{7} T^{9} - 19 ⁣⋯ ⁣3019\!\cdots\!30p14T10+ p^{14} T^{10} + 20 ⁣⋯ ⁣3620\!\cdots\!36p21T11+ p^{21} T^{11} + 60 ⁣⋯ ⁣0960\!\cdots\!09p29T12+1683019295379758p35T13−84053579929p42T14−448902p49T15+p56T16 p^{29} T^{12} + 1683019295379758 p^{35} T^{13} - 84053579929 p^{42} T^{14} - 448902 p^{49} T^{15} + p^{56} T^{16}
41 (1−663316T+871076913224T2−384262293935086044T3+ ( 1 - 663316 T + 871076913224 T^{2} - 384262293935086044 T^{3} + 26 ⁣⋯ ⁣2626\!\cdots\!26T4−384262293935086044p7T5+871076913224p14T6−663316p21T7+p28T8)2 T^{4} - 384262293935086044 p^{7} T^{5} + 871076913224 p^{14} T^{6} - 663316 p^{21} T^{7} + p^{28} T^{8} )^{2}
43 (1−554T−15077285pT2+4664521444356754T3+ ( 1 - 554 T - 15077285 p T^{2} + 4664521444356754 T^{3} + 13 ⁣⋯ ⁣2813\!\cdots\!28T4+4664521444356754p7T5−15077285p15T6−554p21T7+p28T8)2 T^{4} + 4664521444356754 p^{7} T^{5} - 15077285 p^{15} T^{6} - 554 p^{21} T^{7} + p^{28} T^{8} )^{2}
47 1−762180T−81239846636T2−48896398946940456T3− 1 - 762180 T - 81239846636 T^{2} - 48896398946940456 T^{3} - 68 ⁣⋯ ⁣4268\!\cdots\!42T4+ T^{4} + 24 ⁣⋯ ⁣9224\!\cdots\!92T5+ T^{5} + 83 ⁣⋯ ⁣2483\!\cdots\!24T6− T^{6} - 15 ⁣⋯ ⁣8415\!\cdots\!84T7+ T^{7} + 57 ⁣⋯ ⁣2757\!\cdots\!27T8− T^{8} - 15 ⁣⋯ ⁣8415\!\cdots\!84p7T9+ p^{7} T^{9} + 83 ⁣⋯ ⁣2483\!\cdots\!24p14T10+ p^{14} T^{10} + 24 ⁣⋯ ⁣9224\!\cdots\!92p21T11− p^{21} T^{11} - 68 ⁣⋯ ⁣4268\!\cdots\!42p28T12−48896398946940456p35T13−81239846636p42T14−762180p49T15+p56T16 p^{28} T^{12} - 48896398946940456 p^{35} T^{13} - 81239846636 p^{42} T^{14} - 762180 p^{49} T^{15} + p^{56} T^{16}
53 1+2761920T+1375852346317T2−146941467225378768T3+ 1 + 2761920 T + 1375852346317 T^{2} - 146941467225378768 T^{3} + 39 ⁣⋯ ⁣2939\!\cdots\!29T4+ T^{4} + 32 ⁣⋯ ⁣8032\!\cdots\!80T5− T^{5} - 47 ⁣⋯ ⁣5047\!\cdots\!50T6− T^{6} - 27 ⁣⋯ ⁣6827\!\cdots\!68T7+ T^{7} + 36 ⁣⋯ ⁣4636\!\cdots\!46T8− T^{8} - 27 ⁣⋯ ⁣6827\!\cdots\!68p7T9− p^{7} T^{9} - 47 ⁣⋯ ⁣5047\!\cdots\!50p14T10+ p^{14} T^{10} + 32 ⁣⋯ ⁣8032\!\cdots\!80p21T11+ p^{21} T^{11} + 39 ⁣⋯ ⁣2939\!\cdots\!29p28T12−146941467225378768p35T13+1375852346317p42T14+2761920p49T15+p56T16 p^{28} T^{12} - 146941467225378768 p^{35} T^{13} + 1375852346317 p^{42} T^{14} + 2761920 p^{49} T^{15} + p^{56} T^{16}
59 1−3410898T+868027314559T2+5620025039380348794T3+ 1 - 3410898 T + 868027314559 T^{2} + 5620025039380348794 T^{3} + 49 ⁣⋯ ⁣2149\!\cdots\!21T4− T^{4} - 16 ⁣⋯ ⁣6016\!\cdots\!60T5− T^{5} - 14 ⁣⋯ ⁣9014\!\cdots\!90T6+ T^{6} + 35 ⁣⋯ ⁣1235\!\cdots\!12T7− T^{7} - 15 ⁣⋯ ⁣8615\!\cdots\!86T8+ T^{8} + 35 ⁣⋯ ⁣1235\!\cdots\!12p7T9− p^{7} T^{9} - 14 ⁣⋯ ⁣9014\!\cdots\!90p14T10− p^{14} T^{10} - 16 ⁣⋯ ⁣6016\!\cdots\!60p21T11+ p^{21} T^{11} + 49 ⁣⋯ ⁣2149\!\cdots\!21p28T12+5620025039380348794p35T13+868027314559p42T14−3410898p49T15+p56T16 p^{28} T^{12} + 5620025039380348794 p^{35} T^{13} + 868027314559 p^{42} T^{14} - 3410898 p^{49} T^{15} + p^{56} T^{16}
61 1+300892T−86494008900pT2+5853008758722988296T3+ 1 + 300892 T - 86494008900 p T^{2} + 5853008758722988296 T^{3} + 12 ⁣⋯ ⁣5812\!\cdots\!58T4− T^{4} - 28 ⁣⋯ ⁣4428\!\cdots\!44T5+ T^{5} + 20 ⁣⋯ ⁣0820\!\cdots\!08T6+ T^{6} + 60 ⁣⋯ ⁣4860\!\cdots\!48T7− T^{7} - 94 ⁣⋯ ⁣0994\!\cdots\!09T8+ T^{8} + 60 ⁣⋯ ⁣4860\!\cdots\!48p7T9+ p^{7} T^{9} + 20 ⁣⋯ ⁣0820\!\cdots\!08p14T10− p^{14} T^{10} - 28 ⁣⋯ ⁣4428\!\cdots\!44p21T11+ p^{21} T^{11} + 12 ⁣⋯ ⁣5812\!\cdots\!58p28T12+5853008758722988296p35T13−86494008900p43T14+300892p49T15+p56T16 p^{28} T^{12} + 5853008758722988296 p^{35} T^{13} - 86494008900 p^{43} T^{14} + 300892 p^{49} T^{15} + p^{56} T^{16}
67 1−4222478T−9315307834005T2+34395757506970837806T3+ 1 - 4222478 T - 9315307834005 T^{2} + 34395757506970837806 T^{3} + 16 ⁣⋯ ⁣3316\!\cdots\!33T4− T^{4} - 33 ⁣⋯ ⁣9233\!\cdots\!92T5− T^{5} - 12 ⁣⋯ ⁣4212\!\cdots\!42T6+ T^{6} + 46 ⁣⋯ ⁣2846\!\cdots\!28T7+ T^{7} + 10 ⁣⋯ ⁣1410\!\cdots\!14T8+ T^{8} + 46 ⁣⋯ ⁣2846\!\cdots\!28p7T9− p^{7} T^{9} - 12 ⁣⋯ ⁣4212\!\cdots\!42p14T10− p^{14} T^{10} - 33 ⁣⋯ ⁣9233\!\cdots\!92p21T11+ p^{21} T^{11} + 16 ⁣⋯ ⁣3316\!\cdots\!33p28T12+34395757506970837806p35T13−9315307834005p42T14−4222478p49T15+p56T16 p^{28} T^{12} + 34395757506970837806 p^{35} T^{13} - 9315307834005 p^{42} T^{14} - 4222478 p^{49} T^{15} + p^{56} T^{16}
71 (1−380964T+17482423832204T2+29051119312520145372T3+ ( 1 - 380964 T + 17482423832204 T^{2} + 29051119312520145372 T^{3} + 13 ⁣⋯ ⁣7413\!\cdots\!74T4+29051119312520145372p7T5+17482423832204p14T6−380964p21T7+p28T8)2 T^{4} + 29051119312520145372 p^{7} T^{5} + 17482423832204 p^{14} T^{6} - 380964 p^{21} T^{7} + p^{28} T^{8} )^{2}
73 1−451674T−6076887842341T2−71774769404461846774T3− 1 - 451674 T - 6076887842341 T^{2} - 71774769404461846774 T^{3} - 23 ⁣⋯ ⁣3523\!\cdots\!35T4+ T^{4} + 66 ⁣⋯ ⁣8066\!\cdots\!80T5+ T^{5} + 24 ⁣⋯ ⁣2224\!\cdots\!22T6− T^{6} - 30 ⁣⋯ ⁣8030\!\cdots\!80T7− T^{7} - 26 ⁣⋯ ⁣3826\!\cdots\!38T8− T^{8} - 30 ⁣⋯ ⁣8030\!\cdots\!80p7T9+ p^{7} T^{9} + 24 ⁣⋯ ⁣2224\!\cdots\!22p14T10+ p^{14} T^{10} + 66 ⁣⋯ ⁣8066\!\cdots\!80p21T11− p^{21} T^{11} - 23 ⁣⋯ ⁣3523\!\cdots\!35p28T12−71774769404461846774p35T13−6076887842341p42T14−451674p49T15+p56T16 p^{28} T^{12} - 71774769404461846774 p^{35} T^{13} - 6076887842341 p^{42} T^{14} - 451674 p^{49} T^{15} + p^{56} T^{16}
79 1−12154822T+90170014725672T2− 1 - 12154822 T + 90170014725672 T^{2} - 60 ⁣⋯ ⁣6860\!\cdots\!68T3+ T^{3} + 31 ⁣⋯ ⁣0331\!\cdots\!03T4− T^{4} - 13 ⁣⋯ ⁣6813\!\cdots\!68T5+ T^{5} + 58 ⁣⋯ ⁣2458\!\cdots\!24T6− T^{6} - 22 ⁣⋯ ⁣3422\!\cdots\!34T7+ T^{7} + 87 ⁣⋯ ⁣3687\!\cdots\!36T8− T^{8} - 22 ⁣⋯ ⁣3422\!\cdots\!34p7T9+ p^{7} T^{9} + 58 ⁣⋯ ⁣2458\!\cdots\!24p14T10− p^{14} T^{10} - 13 ⁣⋯ ⁣6813\!\cdots\!68p21T11+ p^{21} T^{11} + 31 ⁣⋯ ⁣0331\!\cdots\!03p28T12− p^{28} T^{12} - 60 ⁣⋯ ⁣6860\!\cdots\!68p35T13+90170014725672p42T14−12154822p49T15+p56T16 p^{35} T^{13} + 90170014725672 p^{42} T^{14} - 12154822 p^{49} T^{15} + p^{56} T^{16}
83 (1−12087978T+133280044237529T2− ( 1 - 12087978 T + 133280044237529 T^{2} - 88 ⁣⋯ ⁣4288\!\cdots\!42T3+ T^{3} + 55 ⁣⋯ ⁣2055\!\cdots\!20T4− T^{4} - 88 ⁣⋯ ⁣4288\!\cdots\!42p7T5+133280044237529p14T6−12087978p21T7+p28T8)2 p^{7} T^{5} + 133280044237529 p^{14} T^{6} - 12087978 p^{21} T^{7} + p^{28} T^{8} )^{2}
89 1−4955752T−151120762805968T2+ 1 - 4955752 T - 151120762805968 T^{2} + 40 ⁣⋯ ⁣1640\!\cdots\!16T3+ T^{3} + 16 ⁣⋯ ⁣3416\!\cdots\!34T4− T^{4} - 25 ⁣⋯ ⁣2025\!\cdots\!20T5− T^{5} - 10 ⁣⋯ ⁣1610\!\cdots\!16T6+ T^{6} + 37 ⁣⋯ ⁣4837\!\cdots\!48T7+ T^{7} + 56 ⁣⋯ ⁣8756\!\cdots\!87T8+ T^{8} + 37 ⁣⋯ ⁣4837\!\cdots\!48p7T9− p^{7} T^{9} - 10 ⁣⋯ ⁣1610\!\cdots\!16p14T10− p^{14} T^{10} - 25 ⁣⋯ ⁣2025\!\cdots\!20p21T11+ p^{21} T^{11} + 16 ⁣⋯ ⁣3416\!\cdots\!34p28T12+ p^{28} T^{12} + 40 ⁣⋯ ⁣1640\!\cdots\!16p35T13−151120762805968p42T14−4955752p49T15+p56T16 p^{35} T^{13} - 151120762805968 p^{42} T^{14} - 4955752 p^{49} T^{15} + p^{56} T^{16}
97 (1+22840614T+516597355973305T2+ ( 1 + 22840614 T + 516597355973305 T^{2} + 62 ⁣⋯ ⁣7862\!\cdots\!78T3+ T^{3} + 71 ⁣⋯ ⁣4871\!\cdots\!48T4+ T^{4} + 62 ⁣⋯ ⁣7862\!\cdots\!78p7T5+516597355973305p14T6+22840614p21T7+p28T8)2 p^{7} T^{5} + 516597355973305 p^{14} T^{6} + 22840614 p^{21} T^{7} + p^{28} T^{8} )^{2}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.51042673816867756724809834414, −5.33910294454682421113255651658, −5.12364182304182091875626329614, −5.05306169097531726510222605106, −4.67664871950749303115676327737, −4.61714282873216234169376470025, −4.48779838242372634312711509977, −4.11723385155902819661124047961, −3.88249561024902311952116057478, −3.71436063776737489672777046960, −3.39054700781508067773144333223, −3.32015849081514474601455111043, −2.77558979408361996243685593576, −2.59319370732128406579054309435, −2.57269045227894597999459053235, −2.44242764082802598078629851268, −2.41977105557665952518090115422, −1.91766581453537347328269316365, −1.86369500206933907292338679609, −1.40630528192828701414920454954, −1.13000944792508839656636016535, −0.823450308388871391058056931353, −0.64934329427361870445977549321, −0.64860142078860972552518359003, −0.51056083655295912217857597277, 0.51056083655295912217857597277, 0.64860142078860972552518359003, 0.64934329427361870445977549321, 0.823450308388871391058056931353, 1.13000944792508839656636016535, 1.40630528192828701414920454954, 1.86369500206933907292338679609, 1.91766581453537347328269316365, 2.41977105557665952518090115422, 2.44242764082802598078629851268, 2.57269045227894597999459053235, 2.59319370732128406579054309435, 2.77558979408361996243685593576, 3.32015849081514474601455111043, 3.39054700781508067773144333223, 3.71436063776737489672777046960, 3.88249561024902311952116057478, 4.11723385155902819661124047961, 4.48779838242372634312711509977, 4.61714282873216234169376470025, 4.67664871950749303115676327737, 5.05306169097531726510222605106, 5.12364182304182091875626329614, 5.33910294454682421113255651658, 5.51042673816867756724809834414

Graph of the ZZ-function along the critical line

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