↑

Properties

Label 16-1472e8-1.1-c3e8-0-1
Degree 1616
Conductor 2.204×10252.204\times 10^{25}
Sign 11
Analytic cond. 3.23736×10153.23736\times 10^{15}
Root an. cond. 9.319379.31937
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 88

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 12·5-s − 14·7-s + 9·9-s − 88·11-s + 30·13-s − 144·15-s + 58·17-s − 190·19-s + 168·21-s − 184·23-s − 414·25-s + 432·27-s − 190·29-s − 60·31-s + 1.05e3·33-s − 168·35-s + 156·37-s − 360·39-s + 282·41-s − 810·43-s + 108·45-s + 564·47-s − 898·49-s − 696·51-s + 230·53-s − 1.05e3·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.07·5-s − 0.755·7-s + 1/3·9-s − 2.41·11-s + 0.640·13-s − 2.47·15-s + 0.827·17-s − 2.29·19-s + 1.74·21-s − 1.66·23-s − 3.31·25-s + 3.07·27-s − 1.21·29-s − 0.347·31-s + 5.57·33-s − 0.811·35-s + 0.693·37-s − 1.47·39-s + 1.07·41-s − 2.87·43-s + 0.357·45-s + 1.75·47-s − 2.61·49-s − 1.91·51-s + 0.596·53-s − 2.58·55-s + ⋯

Functional equation

Λ(s)=((248⋅238)s/2 ΓC(s)8 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((248⋅238)s/2 ΓC(s+3/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 248⋅2382^{48} \cdot 23^{8}
Sign: 11
Analytic conductor: 3.23736×10153.23736\times 10^{15}
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 88
Selberg data: (16, 248⋅238, ( :[3/2]8), 1)(16,\ 2^{48} \cdot 23^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad2 1 1
23 (1+pT)8 ( 1 + p T )^{8}
good3 1+4pT+5p3T2+40p3T3+965p2T4+6244p2T5+371930T6+686036pT7+3831022pT8+686036p4T9+371930p6T10+6244p11T11+965p14T12+40p18T13+5p21T14+4p22T15+p24T16 1 + 4 p T + 5 p^{3} T^{2} + 40 p^{3} T^{3} + 965 p^{2} T^{4} + 6244 p^{2} T^{5} + 371930 T^{6} + 686036 p T^{7} + 3831022 p T^{8} + 686036 p^{4} T^{9} + 371930 p^{6} T^{10} + 6244 p^{11} T^{11} + 965 p^{14} T^{12} + 40 p^{18} T^{13} + 5 p^{21} T^{14} + 4 p^{22} T^{15} + p^{24} T^{16}
5 1−12T+558T2−6792T3+171396T4−1905872T5+35327514T6−342929052T7+5210867942T8−342929052p3T9+35327514p6T10−1905872p9T11+171396p12T12−6792p15T13+558p18T14−12p21T15+p24T16 1 - 12 T + 558 T^{2} - 6792 T^{3} + 171396 T^{4} - 1905872 T^{5} + 35327514 T^{6} - 342929052 T^{7} + 5210867942 T^{8} - 342929052 p^{3} T^{9} + 35327514 p^{6} T^{10} - 1905872 p^{9} T^{11} + 171396 p^{12} T^{12} - 6792 p^{15} T^{13} + 558 p^{18} T^{14} - 12 p^{21} T^{15} + p^{24} T^{16}
7 1+2pT+1094T2+3722pT3+831492T4+17743510T5+458102450T6+8894837758T7+172843238790T8+8894837758p3T9+458102450p6T10+17743510p9T11+831492p12T12+3722p16T13+1094p18T14+2p22T15+p24T16 1 + 2 p T + 1094 T^{2} + 3722 p T^{3} + 831492 T^{4} + 17743510 T^{5} + 458102450 T^{6} + 8894837758 T^{7} + 172843238790 T^{8} + 8894837758 p^{3} T^{9} + 458102450 p^{6} T^{10} + 17743510 p^{9} T^{11} + 831492 p^{12} T^{12} + 3722 p^{16} T^{13} + 1094 p^{18} T^{14} + 2 p^{22} T^{15} + p^{24} T^{16}
11 1+8pT+8134T2+527068T3+29522140T4+1476743188T5+66227765058T6+2673749809776T7+103788190691478T8+2673749809776p3T9+66227765058p6T10+1476743188p9T11+29522140p12T12+527068p15T13+8134p18T14+8p22T15+p24T16 1 + 8 p T + 8134 T^{2} + 527068 T^{3} + 29522140 T^{4} + 1476743188 T^{5} + 66227765058 T^{6} + 2673749809776 T^{7} + 103788190691478 T^{8} + 2673749809776 p^{3} T^{9} + 66227765058 p^{6} T^{10} + 1476743188 p^{9} T^{11} + 29522140 p^{12} T^{12} + 527068 p^{15} T^{13} + 8134 p^{18} T^{14} + 8 p^{22} T^{15} + p^{24} T^{16}
13 1−30T+8623T2−24962pT3+38745457T4−1778226588T5+118720437366T6−6007223069192T7+287733902451010T8−6007223069192p3T9+118720437366p6T10−1778226588p9T11+38745457p12T12−24962p16T13+8623p18T14−30p21T15+p24T16 1 - 30 T + 8623 T^{2} - 24962 p T^{3} + 38745457 T^{4} - 1778226588 T^{5} + 118720437366 T^{6} - 6007223069192 T^{7} + 287733902451010 T^{8} - 6007223069192 p^{3} T^{9} + 118720437366 p^{6} T^{10} - 1778226588 p^{9} T^{11} + 38745457 p^{12} T^{12} - 24962 p^{16} T^{13} + 8623 p^{18} T^{14} - 30 p^{21} T^{15} + p^{24} T^{16}
17 1−58T+24714T2−739646T3+265388300T4−3013008610T5+1862751072702T6−3359705913894T7+10169183353995862T8−3359705913894p3T9+1862751072702p6T10−3013008610p9T11+265388300p12T12−739646p15T13+24714p18T14−58p21T15+p24T16 1 - 58 T + 24714 T^{2} - 739646 T^{3} + 265388300 T^{4} - 3013008610 T^{5} + 1862751072702 T^{6} - 3359705913894 T^{7} + 10169183353995862 T^{8} - 3359705913894 p^{3} T^{9} + 1862751072702 p^{6} T^{10} - 3013008610 p^{9} T^{11} + 265388300 p^{12} T^{12} - 739646 p^{15} T^{13} + 24714 p^{18} T^{14} - 58 p^{21} T^{15} + p^{24} T^{16}
19 1+10pT+46416T2+5768430T3+878213356T4+86547684862T5+10282296471280T6+848607523793086T7+83599454050384390T8+848607523793086p3T9+10282296471280p6T10+86547684862p9T11+878213356p12T12+5768430p15T13+46416p18T14+10p22T15+p24T16 1 + 10 p T + 46416 T^{2} + 5768430 T^{3} + 878213356 T^{4} + 86547684862 T^{5} + 10282296471280 T^{6} + 848607523793086 T^{7} + 83599454050384390 T^{8} + 848607523793086 p^{3} T^{9} + 10282296471280 p^{6} T^{10} + 86547684862 p^{9} T^{11} + 878213356 p^{12} T^{12} + 5768430 p^{15} T^{13} + 46416 p^{18} T^{14} + 10 p^{22} T^{15} + p^{24} T^{16}
29 1+190T+81903T2+12668238T3+3348797321T4+443232784680T5+99790588535430T6+11545386938443436T7+86407098224178002pT8+11545386938443436p3T9+99790588535430p6T10+443232784680p9T11+3348797321p12T12+12668238p15T13+81903p18T14+190p21T15+p24T16 1 + 190 T + 81903 T^{2} + 12668238 T^{3} + 3348797321 T^{4} + 443232784680 T^{5} + 99790588535430 T^{6} + 11545386938443436 T^{7} + 86407098224178002 p T^{8} + 11545386938443436 p^{3} T^{9} + 99790588535430 p^{6} T^{10} + 443232784680 p^{9} T^{11} + 3348797321 p^{12} T^{12} + 12668238 p^{15} T^{13} + 81903 p^{18} T^{14} + 190 p^{21} T^{15} + p^{24} T^{16}
31 1+60T+88331T2−212924T3+3775918869T4−218356055120T5+126547839290226T6−9799382872505480T7+4033569334997275226T8−9799382872505480p3T9+126547839290226p6T10−218356055120p9T11+3775918869p12T12−212924p15T13+88331p18T14+60p21T15+p24T16 1 + 60 T + 88331 T^{2} - 212924 T^{3} + 3775918869 T^{4} - 218356055120 T^{5} + 126547839290226 T^{6} - 9799382872505480 T^{7} + 4033569334997275226 T^{8} - 9799382872505480 p^{3} T^{9} + 126547839290226 p^{6} T^{10} - 218356055120 p^{9} T^{11} + 3775918869 p^{12} T^{12} - 212924 p^{15} T^{13} + 88331 p^{18} T^{14} + 60 p^{21} T^{15} + p^{24} T^{16}
37 1−156T+305414T2−48304008T3+44269746164T4−6510880736624T5+3993208351902882T6−511794502952013804T7+ 1 - 156 T + 305414 T^{2} - 48304008 T^{3} + 44269746164 T^{4} - 6510880736624 T^{5} + 3993208351902882 T^{6} - 511794502952013804 T^{7} + 24 ⁣⋯ ⁣3024\!\cdots\!30T8−511794502952013804p3T9+3993208351902882p6T10−6510880736624p9T11+44269746164p12T12−48304008p15T13+305414p18T14−156p21T15+p24T16 T^{8} - 511794502952013804 p^{3} T^{9} + 3993208351902882 p^{6} T^{10} - 6510880736624 p^{9} T^{11} + 44269746164 p^{12} T^{12} - 48304008 p^{15} T^{13} + 305414 p^{18} T^{14} - 156 p^{21} T^{15} + p^{24} T^{16}
41 1−282T+406967T2−111531986T3+79032706225T4−19801930698600T5+9602070331297422T6−2091298025211225364T7+ 1 - 282 T + 406967 T^{2} - 111531986 T^{3} + 79032706225 T^{4} - 19801930698600 T^{5} + 9602070331297422 T^{6} - 2091298025211225364 T^{7} + 79 ⁣⋯ ⁣8279\!\cdots\!82T8−2091298025211225364p3T9+9602070331297422p6T10−19801930698600p9T11+79032706225p12T12−111531986p15T13+406967p18T14−282p21T15+p24T16 T^{8} - 2091298025211225364 p^{3} T^{9} + 9602070331297422 p^{6} T^{10} - 19801930698600 p^{9} T^{11} + 79032706225 p^{12} T^{12} - 111531986 p^{15} T^{13} + 406967 p^{18} T^{14} - 282 p^{21} T^{15} + p^{24} T^{16}
43 1+810T+748836T2+400852146T3+219302352916T4+88392716782050T5+35440160183076028T6+11264147007704519274T7+ 1 + 810 T + 748836 T^{2} + 400852146 T^{3} + 219302352916 T^{4} + 88392716782050 T^{5} + 35440160183076028 T^{6} + 11264147007704519274 T^{7} + 35 ⁣⋯ ⁣5435\!\cdots\!54T8+11264147007704519274p3T9+35440160183076028p6T10+88392716782050p9T11+219302352916p12T12+400852146p15T13+748836p18T14+810p21T15+p24T16 T^{8} + 11264147007704519274 p^{3} T^{9} + 35440160183076028 p^{6} T^{10} + 88392716782050 p^{9} T^{11} + 219302352916 p^{12} T^{12} + 400852146 p^{15} T^{13} + 748836 p^{18} T^{14} + 810 p^{21} T^{15} + p^{24} T^{16}
47 1−12pT+357203T2−119277468T3+57768525893T4−12628648199520T5+4563365674672178T6−578225878401858360T7+ 1 - 12 p T + 357203 T^{2} - 119277468 T^{3} + 57768525893 T^{4} - 12628648199520 T^{5} + 4563365674672178 T^{6} - 578225878401858360 T^{7} + 38 ⁣⋯ ⁣7038\!\cdots\!70T8−578225878401858360p3T9+4563365674672178p6T10−12628648199520p9T11+57768525893p12T12−119277468p15T13+357203p18T14−12p22T15+p24T16 T^{8} - 578225878401858360 p^{3} T^{9} + 4563365674672178 p^{6} T^{10} - 12628648199520 p^{9} T^{11} + 57768525893 p^{12} T^{12} - 119277468 p^{15} T^{13} + 357203 p^{18} T^{14} - 12 p^{22} T^{15} + p^{24} T^{16}
53 1−230T+570128T2−141594314T3+128952985292T4−37826221869878T5+14782294503604016T6−6446592419167632138T7+ 1 - 230 T + 570128 T^{2} - 141594314 T^{3} + 128952985292 T^{4} - 37826221869878 T^{5} + 14782294503604016 T^{6} - 6446592419167632138 T^{7} + 14 ⁣⋯ ⁣7014\!\cdots\!70T8−6446592419167632138p3T9+14782294503604016p6T10−37826221869878p9T11+128952985292p12T12−141594314p15T13+570128p18T14−230p21T15+p24T16 T^{8} - 6446592419167632138 p^{3} T^{9} + 14782294503604016 p^{6} T^{10} - 37826221869878 p^{9} T^{11} + 128952985292 p^{12} T^{12} - 141594314 p^{15} T^{13} + 570128 p^{18} T^{14} - 230 p^{21} T^{15} + p^{24} T^{16}
59 1+1916T+2405664T2+2321229932T3+1839565138892T4+1249730100885324T5+747053166046109216T6+ 1 + 1916 T + 2405664 T^{2} + 2321229932 T^{3} + 1839565138892 T^{4} + 1249730100885324 T^{5} + 747053166046109216 T^{6} + 39 ⁣⋯ ⁣2039\!\cdots\!20T7+ T^{7} + 18 ⁣⋯ ⁣9018\!\cdots\!90T8+ T^{8} + 39 ⁣⋯ ⁣2039\!\cdots\!20p3T9+747053166046109216p6T10+1249730100885324p9T11+1839565138892p12T12+2321229932p15T13+2405664p18T14+1916p21T15+p24T16 p^{3} T^{9} + 747053166046109216 p^{6} T^{10} + 1249730100885324 p^{9} T^{11} + 1839565138892 p^{12} T^{12} + 2321229932 p^{15} T^{13} + 2405664 p^{18} T^{14} + 1916 p^{21} T^{15} + p^{24} T^{16}
61 1+22T+656584T2+56665610T3+289690058876T4+25759150848710T5+91312341450809528T6+7654640322361851626T7+ 1 + 22 T + 656584 T^{2} + 56665610 T^{3} + 289690058876 T^{4} + 25759150848710 T^{5} + 91312341450809528 T^{6} + 7654640322361851626 T^{7} + 23 ⁣⋯ ⁣2623\!\cdots\!26T8+7654640322361851626p3T9+91312341450809528p6T10+25759150848710p9T11+289690058876p12T12+56665610p15T13+656584p18T14+22p21T15+p24T16 T^{8} + 7654640322361851626 p^{3} T^{9} + 91312341450809528 p^{6} T^{10} + 25759150848710 p^{9} T^{11} + 289690058876 p^{12} T^{12} + 56665610 p^{15} T^{13} + 656584 p^{18} T^{14} + 22 p^{21} T^{15} + p^{24} T^{16}
67 1+2292T+3793886T2+4344144896T3+4198586189676T4+3337397232457784T5+2382188962605544170T6+ 1 + 2292 T + 3793886 T^{2} + 4344144896 T^{3} + 4198586189676 T^{4} + 3337397232457784 T^{5} + 2382188962605544170 T^{6} + 14 ⁣⋯ ⁣6014\!\cdots\!60T7+ T^{7} + 86 ⁣⋯ ⁣2686\!\cdots\!26T8+ T^{8} + 14 ⁣⋯ ⁣6014\!\cdots\!60p3T9+2382188962605544170p6T10+3337397232457784p9T11+4198586189676p12T12+4344144896p15T13+3793886p18T14+2292p21T15+p24T16 p^{3} T^{9} + 2382188962605544170 p^{6} T^{10} + 3337397232457784 p^{9} T^{11} + 4198586189676 p^{12} T^{12} + 4344144896 p^{15} T^{13} + 3793886 p^{18} T^{14} + 2292 p^{21} T^{15} + p^{24} T^{16}
71 1−2376T+3675615T2−4279744976T3+4188196909029T4−3572345035677032T5+2727549647711214498T6− 1 - 2376 T + 3675615 T^{2} - 4279744976 T^{3} + 4188196909029 T^{4} - 3572345035677032 T^{5} + 2727549647711214498 T^{6} - 18 ⁣⋯ ⁣7618\!\cdots\!76T7+ T^{7} + 11 ⁣⋯ ⁣1811\!\cdots\!18T8− T^{8} - 18 ⁣⋯ ⁣7618\!\cdots\!76p3T9+2727549647711214498p6T10−3572345035677032p9T11+4188196909029p12T12−4279744976p15T13+3675615p18T14−2376p21T15+p24T16 p^{3} T^{9} + 2727549647711214498 p^{6} T^{10} - 3572345035677032 p^{9} T^{11} + 4188196909029 p^{12} T^{12} - 4279744976 p^{15} T^{13} + 3675615 p^{18} T^{14} - 2376 p^{21} T^{15} + p^{24} T^{16}
73 1+630T+2074167T2+1112203230T3+2126075486833T4+984814771175032T5+1400980936204462734T6+ 1 + 630 T + 2074167 T^{2} + 1112203230 T^{3} + 2126075486833 T^{4} + 984814771175032 T^{5} + 1400980936204462734 T^{6} + 55 ⁣⋯ ⁣3255\!\cdots\!32T7+ T^{7} + 64 ⁣⋯ ⁣8264\!\cdots\!82T8+ T^{8} + 55 ⁣⋯ ⁣3255\!\cdots\!32p3T9+1400980936204462734p6T10+984814771175032p9T11+2126075486833p12T12+1112203230p15T13+2074167p18T14+630p21T15+p24T16 p^{3} T^{9} + 1400980936204462734 p^{6} T^{10} + 984814771175032 p^{9} T^{11} + 2126075486833 p^{12} T^{12} + 1112203230 p^{15} T^{13} + 2074167 p^{18} T^{14} + 630 p^{21} T^{15} + p^{24} T^{16}
79 1−892T+2843620T2−1648787116T3+3328960395652T4−1293342518366300T5+2388714781909334172T6− 1 - 892 T + 2843620 T^{2} - 1648787116 T^{3} + 3328960395652 T^{4} - 1293342518366300 T^{5} + 2388714781909334172 T^{6} - 67 ⁣⋯ ⁣3267\!\cdots\!32T7+ T^{7} + 13 ⁣⋯ ⁣7813\!\cdots\!78T8− T^{8} - 67 ⁣⋯ ⁣3267\!\cdots\!32p3T9+2388714781909334172p6T10−1293342518366300p9T11+3328960395652p12T12−1648787116p15T13+2843620p18T14−892p21T15+p24T16 p^{3} T^{9} + 2388714781909334172 p^{6} T^{10} - 1293342518366300 p^{9} T^{11} + 3328960395652 p^{12} T^{12} - 1648787116 p^{15} T^{13} + 2843620 p^{18} T^{14} - 892 p^{21} T^{15} + p^{24} T^{16}
83 1+2800T+6734734T2+11284901676T3+16191548396524T4+19398083944552692T5+20218543045915497978T6+ 1 + 2800 T + 6734734 T^{2} + 11284901676 T^{3} + 16191548396524 T^{4} + 19398083944552692 T^{5} + 20218543045915497978 T^{6} + 18 ⁣⋯ ⁣5218\!\cdots\!52T7+ T^{7} + 14 ⁣⋯ ⁣0614\!\cdots\!06T8+ T^{8} + 18 ⁣⋯ ⁣5218\!\cdots\!52p3T9+20218543045915497978p6T10+19398083944552692p9T11+16191548396524p12T12+11284901676p15T13+6734734p18T14+2800p21T15+p24T16 p^{3} T^{9} + 20218543045915497978 p^{6} T^{10} + 19398083944552692 p^{9} T^{11} + 16191548396524 p^{12} T^{12} + 11284901676 p^{15} T^{13} + 6734734 p^{18} T^{14} + 2800 p^{21} T^{15} + p^{24} T^{16}
89 1+308T+2580572T2+537416492T3+3737430388820T4+466567404321124T5+3957470808670947684T6+ 1 + 308 T + 2580572 T^{2} + 537416492 T^{3} + 3737430388820 T^{4} + 466567404321124 T^{5} + 3957470808670947684 T^{6} + 43 ⁣⋯ ⁣6443\!\cdots\!64T7+ T^{7} + 31 ⁣⋯ ⁣9031\!\cdots\!90T8+ T^{8} + 43 ⁣⋯ ⁣6443\!\cdots\!64p3T9+3957470808670947684p6T10+466567404321124p9T11+3737430388820p12T12+537416492p15T13+2580572p18T14+308p21T15+p24T16 p^{3} T^{9} + 3957470808670947684 p^{6} T^{10} + 466567404321124 p^{9} T^{11} + 3737430388820 p^{12} T^{12} + 537416492 p^{15} T^{13} + 2580572 p^{18} T^{14} + 308 p^{21} T^{15} + p^{24} T^{16}
97 1−854T+3132426T2−2694403618T3+5557873802604T4−4455992353790526T5+7190741008072039166T6− 1 - 854 T + 3132426 T^{2} - 2694403618 T^{3} + 5557873802604 T^{4} - 4455992353790526 T^{5} + 7190741008072039166 T^{6} - 51 ⁣⋯ ⁣5451\!\cdots\!54T7+ T^{7} + 72 ⁣⋯ ⁣1472\!\cdots\!14T8− T^{8} - 51 ⁣⋯ ⁣5451\!\cdots\!54p3T9+7190741008072039166p6T10−4455992353790526p9T11+5557873802604p12T12−2694403618p15T13+3132426p18T14−854p21T15+p24T16 p^{3} T^{9} + 7190741008072039166 p^{6} T^{10} - 4455992353790526 p^{9} T^{11} + 5557873802604 p^{12} T^{12} - 2694403618 p^{15} T^{13} + 3132426 p^{18} T^{14} - 854 p^{21} T^{15} + p^{24} T^{16}
show more
show less
   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

−4.03630209144009521237940898566, −3.92394145637932955475008160990, −3.76633788999028068553047269402, −3.68627005356804579590516419731, −3.67598911253509922370088794556, −3.61467878951172875445477524100, −3.37125560688354240068018535012, −3.35774937188101063063408846775, −3.15555996760764495784993465668, −2.72668353865807251308029044285, −2.69766523348999238617496922337, −2.60147672248414712687019702612, −2.54145687691683855120552076672, −2.50877504333493819021099835409, −2.36628810954434616002290239018, −2.33256478868911951244009084323, −1.97277089098738476305300643542, −1.80421188699432600531460994462, −1.67442640170556094134764104086, −1.49183817081434813619177605511, −1.41701300076627980877848852270, −1.26383031288071616136291189976, −1.25033350857530458615822580454, −0.954681742448365421501653938920, −0.62261733248286535705815709966, 0, 0, 0, 0, 0, 0, 0, 0, 0.62261733248286535705815709966, 0.954681742448365421501653938920, 1.25033350857530458615822580454, 1.26383031288071616136291189976, 1.41701300076627980877848852270, 1.49183817081434813619177605511, 1.67442640170556094134764104086, 1.80421188699432600531460994462, 1.97277089098738476305300643542, 2.33256478868911951244009084323, 2.36628810954434616002290239018, 2.50877504333493819021099835409, 2.54145687691683855120552076672, 2.60147672248414712687019702612, 2.69766523348999238617496922337, 2.72668353865807251308029044285, 3.15555996760764495784993465668, 3.35774937188101063063408846775, 3.37125560688354240068018535012, 3.61467878951172875445477524100, 3.67598911253509922370088794556, 3.68627005356804579590516419731, 3.76633788999028068553047269402, 3.92394145637932955475008160990, 4.03630209144009521237940898566

Graph of the ZZ-function along the critical line

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