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Properties

Label 16-26e8-1.1-c7e8-0-0
Degree 1616
Conductor 208827064576208827064576
Sign 11
Analytic cond. 1.89368×1071.89368\times 10^{7}
Root an. cond. 2.849912.84991
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  + 32·2-s + 384·4-s + 556·5-s − 548·7-s + 1.76e3·9-s + 1.77e4·10-s − 7.39e3·11-s − 2.58e4·13-s − 1.75e4·14-s − 6.14e4·16-s + 2.83e4·17-s + 5.65e4·18-s − 9.98e4·19-s + 2.13e5·20-s − 2.36e5·22-s − 3.33e4·23-s − 7.10e4·25-s − 8.26e5·26-s + 7.08e4·27-s − 2.10e5·28-s + 9.31e4·29-s + 6.22e5·31-s − 7.86e5·32-s + 9.06e5·34-s − 3.04e5·35-s + 6.78e5·36-s − 9.63e3·37-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s + 1.98·5-s − 0.603·7-s + 0.807·9-s + 5.62·10-s − 1.67·11-s − 3.25·13-s − 1.70·14-s − 3.75·16-s + 1.39·17-s + 2.28·18-s − 3.34·19-s + 5.96·20-s − 4.73·22-s − 0.572·23-s − 0.909·25-s − 9.21·26-s + 0.692·27-s − 1.81·28-s + 0.709·29-s + 3.75·31-s − 4.24·32-s + 3.95·34-s − 1.20·35-s + 2.42·36-s − 0.0312·37-s + ⋯

Functional equation

Λ(s)=((28⋅138)s/2 ΓC(s)8 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((28⋅138)s/2 ΓC(s+7/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅1382^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 1.89368×1071.89368\times 10^{7}
Root analytic conductor: 2.849912.84991
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅138, ( :[7/2]8), 1)(16,\ 2^{8} \cdot 13^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )

Particular Values

L(4)L(4) ≈\approx 12.7465610812.74656108
L(12)L(\frac12) ≈\approx 12.7465610812.74656108
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)
bad2 (1−p3T+p6T2)4 ( 1 - p^{3} T + p^{6} T^{2} )^{4}
13 1+1986pT+1099853p2T2−476713734p3T3−4584001548p6T4−476713734p10T5+1099853p16T6+1986p22T7+p28T8 1 + 1986 p T + 1099853 p^{2} T^{2} - 476713734 p^{3} T^{3} - 4584001548 p^{6} T^{4} - 476713734 p^{10} T^{5} + 1099853 p^{16} T^{6} + 1986 p^{22} T^{7} + p^{28} T^{8}
good3 1−589pT2−2624p3T3−247679p2T4+1729216p4T5+107431462p4T6−115526848p6T7−16347737066p6T8−115526848p13T9+107431462p18T10+1729216p25T11−247679p30T12−2624p38T13−589p43T14+p56T16 1 - 589 p T^{2} - 2624 p^{3} T^{3} - 247679 p^{2} T^{4} + 1729216 p^{4} T^{5} + 107431462 p^{4} T^{6} - 115526848 p^{6} T^{7} - 16347737066 p^{6} T^{8} - 115526848 p^{13} T^{9} + 107431462 p^{18} T^{10} + 1729216 p^{25} T^{11} - 247679 p^{30} T^{12} - 2624 p^{38} T^{13} - 589 p^{43} T^{14} + p^{56} T^{16}
5 (1−278T+151453T2−10092158pT3+507231524p2T4−10092158p8T5+151453p14T6−278p21T7+p28T8)2 ( 1 - 278 T + 151453 T^{2} - 10092158 p T^{3} + 507231524 p^{2} T^{4} - 10092158 p^{8} T^{5} + 151453 p^{14} T^{6} - 278 p^{21} T^{7} + p^{28} T^{8} )^{2}
7 1+548T−1138035T2−1222097964T3+107471171841T4+727605589451640T5+478925092223085470T6− 1 + 548 T - 1138035 T^{2} - 1222097964 T^{3} + 107471171841 T^{4} + 727605589451640 T^{5} + 478925092223085470 T^{6} - 12 ⁣⋯ ⁣2012\!\cdots\!20T7− T^{7} - 37 ⁣⋯ ⁣5037\!\cdots\!50T8− T^{8} - 12 ⁣⋯ ⁣2012\!\cdots\!20p7T9+478925092223085470p14T10+727605589451640p21T11+107471171841p28T12−1222097964p35T13−1138035p42T14+548p49T15+p56T16 p^{7} T^{9} + 478925092223085470 p^{14} T^{10} + 727605589451640 p^{21} T^{11} + 107471171841 p^{28} T^{12} - 1222097964 p^{35} T^{13} - 1138035 p^{42} T^{14} + 548 p^{49} T^{15} + p^{56} T^{16}
11 1+672pT−15015695T2−209572703216T3+202098762294505T4+3132174508300450192T5− 1 + 672 p T - 15015695 T^{2} - 209572703216 T^{3} + 202098762294505 T^{4} + 3132174508300450192 T^{5} - 54 ⁣⋯ ⁣5054\!\cdots\!50T6− T^{6} - 18 ⁣⋯ ⁣1218\!\cdots\!12T7+ T^{7} + 15 ⁣⋯ ⁣4615\!\cdots\!46T8− T^{8} - 18 ⁣⋯ ⁣1218\!\cdots\!12p7T9− p^{7} T^{9} - 54 ⁣⋯ ⁣5054\!\cdots\!50p14T10+3132174508300450192p21T11+202098762294505p28T12−209572703216p35T13−15015695p42T14+672p50T15+p56T16 p^{14} T^{10} + 3132174508300450192 p^{21} T^{11} + 202098762294505 p^{28} T^{12} - 209572703216 p^{35} T^{13} - 15015695 p^{42} T^{14} + 672 p^{50} T^{15} + p^{56} T^{16}
17 1−28316T−563111602T2+7342663983176T3+488082653800903993T4+ 1 - 28316 T - 563111602 T^{2} + 7342663983176 T^{3} + 488082653800903993 T^{4} + 16 ⁣⋯ ⁣3616\!\cdots\!36T5− T^{5} - 26 ⁣⋯ ⁣3026\!\cdots\!30T6+ T^{6} + 65 ⁣⋯ ⁣2465\!\cdots\!24T7+ T^{7} + 75 ⁣⋯ ⁣3675\!\cdots\!36T8+ T^{8} + 65 ⁣⋯ ⁣2465\!\cdots\!24p7T9− p^{7} T^{9} - 26 ⁣⋯ ⁣3026\!\cdots\!30p14T10+ p^{14} T^{10} + 16 ⁣⋯ ⁣3616\!\cdots\!36p21T11+488082653800903993p28T12+7342663983176p35T13−563111602p42T14−28316p49T15+p56T16 p^{21} T^{11} + 488082653800903993 p^{28} T^{12} + 7342663983176 p^{35} T^{13} - 563111602 p^{42} T^{14} - 28316 p^{49} T^{15} + p^{56} T^{16}
19 1+99888T+3675226065T2+82754874802800T3+3147787540762584233T4+ 1 + 99888 T + 3675226065 T^{2} + 82754874802800 T^{3} + 3147787540762584233 T^{4} + 61 ⁣⋯ ⁣9261\!\cdots\!92pT5+ p T^{5} + 20 ⁣⋯ ⁣8220\!\cdots\!82T6+ T^{6} + 26 ⁣⋯ ⁣7226\!\cdots\!72T7+ T^{7} + 88 ⁣⋯ ⁣0688\!\cdots\!06T8+ T^{8} + 26 ⁣⋯ ⁣7226\!\cdots\!72p7T9+ p^{7} T^{9} + 20 ⁣⋯ ⁣8220\!\cdots\!82p14T10+ p^{14} T^{10} + 61 ⁣⋯ ⁣9261\!\cdots\!92p22T11+3147787540762584233p28T12+82754874802800p35T13+3675226065p42T14+99888p49T15+p56T16 p^{22} T^{11} + 3147787540762584233 p^{28} T^{12} + 82754874802800 p^{35} T^{13} + 3675226065 p^{42} T^{14} + 99888 p^{49} T^{15} + p^{56} T^{16}
23 1+33388T−3382348979T2−387071200641348T3−4201788242576114527T4+ 1 + 33388 T - 3382348979 T^{2} - 387071200641348 T^{3} - 4201788242576114527 T^{4} + 10 ⁣⋯ ⁣3210\!\cdots\!32T5+ T^{5} + 76 ⁣⋯ ⁣9076\!\cdots\!90T6− T^{6} - 44 ⁣⋯ ⁣8844\!\cdots\!88pT7− p T^{7} - 28 ⁣⋯ ⁣8628\!\cdots\!86T8− T^{8} - 44 ⁣⋯ ⁣8844\!\cdots\!88p8T9+ p^{8} T^{9} + 76 ⁣⋯ ⁣9076\!\cdots\!90p14T10+ p^{14} T^{10} + 10 ⁣⋯ ⁣3210\!\cdots\!32p21T11−4201788242576114527p28T12−387071200641348p35T13−3382348979p42T14+33388p49T15+p56T16 p^{21} T^{11} - 4201788242576114527 p^{28} T^{12} - 387071200641348 p^{35} T^{13} - 3382348979 p^{42} T^{14} + 33388 p^{49} T^{15} + p^{56} T^{16}
29 1−93140T−883636642pT2+2243727152264568T3+80055595218391895489T4+ 1 - 93140 T - 883636642 p T^{2} + 2243727152264568 T^{3} + 80055595218391895489 T^{4} + 16 ⁣⋯ ⁣1216\!\cdots\!12T5− T^{5} - 54 ⁣⋯ ⁣3854\!\cdots\!38T6− T^{6} - 50 ⁣⋯ ⁣2450\!\cdots\!24T7+ T^{7} + 23 ⁣⋯ ⁣9623\!\cdots\!96T8− T^{8} - 50 ⁣⋯ ⁣2450\!\cdots\!24p7T9− p^{7} T^{9} - 54 ⁣⋯ ⁣3854\!\cdots\!38p14T10+ p^{14} T^{10} + 16 ⁣⋯ ⁣1216\!\cdots\!12p21T11+80055595218391895489p28T12+2243727152264568p35T13−883636642p43T14−93140p49T15+p56T16 p^{21} T^{11} + 80055595218391895489 p^{28} T^{12} + 2243727152264568 p^{35} T^{13} - 883636642 p^{43} T^{14} - 93140 p^{49} T^{15} + p^{56} T^{16}
31 (1−311160T+140644653260T2−26779795197492120T3+ ( 1 - 311160 T + 140644653260 T^{2} - 26779795197492120 T^{3} + 62 ⁣⋯ ⁣0662\!\cdots\!06T4−26779795197492120p7T5+140644653260p14T6−311160p21T7+p28T8)2 T^{4} - 26779795197492120 p^{7} T^{5} + 140644653260 p^{14} T^{6} - 311160 p^{21} T^{7} + p^{28} T^{8} )^{2}
37 1+9636T−360499434930T2−281458294795032T3+ 1 + 9636 T - 360499434930 T^{2} - 281458294795032 T^{3} + 79 ⁣⋯ ⁣9379\!\cdots\!93T4− T^{4} - 11 ⁣⋯ ⁣9211\!\cdots\!92T5− T^{5} - 11 ⁣⋯ ⁣4211\!\cdots\!42T6+ T^{6} + 57 ⁣⋯ ⁣4857\!\cdots\!48T7+ T^{7} + 12 ⁣⋯ ⁣5212\!\cdots\!52T8+ T^{8} + 57 ⁣⋯ ⁣4857\!\cdots\!48p7T9− p^{7} T^{9} - 11 ⁣⋯ ⁣4211\!\cdots\!42p14T10− p^{14} T^{10} - 11 ⁣⋯ ⁣9211\!\cdots\!92p21T11+ p^{21} T^{11} + 79 ⁣⋯ ⁣9379\!\cdots\!93p28T12−281458294795032p35T13−360499434930p42T14+9636p49T15+p56T16 p^{28} T^{12} - 281458294795032 p^{35} T^{13} - 360499434930 p^{42} T^{14} + 9636 p^{49} T^{15} + p^{56} T^{16}
41 1−82892T−546055663858T2+146083811772340136T3+ 1 - 82892 T - 546055663858 T^{2} + 146083811772340136 T^{3} + 16 ⁣⋯ ⁣9716\!\cdots\!97T4− T^{4} - 49 ⁣⋯ ⁣7249\!\cdots\!72T5− T^{5} - 26 ⁣⋯ ⁣7026\!\cdots\!70T6+ T^{6} + 51 ⁣⋯ ⁣2451\!\cdots\!24T7+ T^{7} + 39 ⁣⋯ ⁣3239\!\cdots\!32T8+ T^{8} + 51 ⁣⋯ ⁣2451\!\cdots\!24p7T9− p^{7} T^{9} - 26 ⁣⋯ ⁣7026\!\cdots\!70p14T10− p^{14} T^{10} - 49 ⁣⋯ ⁣7249\!\cdots\!72p21T11+ p^{21} T^{11} + 16 ⁣⋯ ⁣9716\!\cdots\!97p28T12+146083811772340136p35T13−546055663858p42T14−82892p49T15+p56T16 p^{28} T^{12} + 146083811772340136 p^{35} T^{13} - 546055663858 p^{42} T^{14} - 82892 p^{49} T^{15} + p^{56} T^{16}
43 1+569264T−351575602143T2−362392294979896560T3+ 1 + 569264 T - 351575602143 T^{2} - 362392294979896560 T^{3} + 28 ⁣⋯ ⁣4528\!\cdots\!45T4+ T^{4} + 10 ⁣⋯ ⁣3610\!\cdots\!36T5+ T^{5} + 36 ⁣⋯ ⁣8636\!\cdots\!86T6− T^{6} - 17 ⁣⋯ ⁣8017\!\cdots\!80T7− T^{7} - 17 ⁣⋯ ⁣0617\!\cdots\!06T8− T^{8} - 17 ⁣⋯ ⁣8017\!\cdots\!80p7T9+ p^{7} T^{9} + 36 ⁣⋯ ⁣8636\!\cdots\!86p14T10+ p^{14} T^{10} + 10 ⁣⋯ ⁣3610\!\cdots\!36p21T11+ p^{21} T^{11} + 28 ⁣⋯ ⁣4528\!\cdots\!45p28T12−362392294979896560p35T13−351575602143p42T14+569264p49T15+p56T16 p^{28} T^{12} - 362392294979896560 p^{35} T^{13} - 351575602143 p^{42} T^{14} + 569264 p^{49} T^{15} + p^{56} T^{16}
47 (1+574200T+1181775847100T2+840326241178871192T3+ ( 1 + 574200 T + 1181775847100 T^{2} + 840326241178871192 T^{3} + 77 ⁣⋯ ⁣9077\!\cdots\!90T4+840326241178871192p7T5+1181775847100p14T6+574200p21T7+p28T8)2 T^{4} + 840326241178871192 p^{7} T^{5} + 1181775847100 p^{14} T^{6} + 574200 p^{21} T^{7} + p^{28} T^{8} )^{2}
53 (1−1235350T+2684236572461T2−3308983772395621830T3+ ( 1 - 1235350 T + 2684236572461 T^{2} - 3308983772395621830 T^{3} + 45 ⁣⋯ ⁣0845\!\cdots\!08T4−3308983772395621830p7T5+2684236572461p14T6−1235350p21T7+p28T8)2 T^{4} - 3308983772395621830 p^{7} T^{5} + 2684236572461 p^{14} T^{6} - 1235350 p^{21} T^{7} + p^{28} T^{8} )^{2}
59 1−231504T−2356814530631T2+792722790467378240T3− 1 - 231504 T - 2356814530631 T^{2} + 792722790467378240 T^{3} - 33 ⁣⋯ ⁣1933\!\cdots\!19T4+ T^{4} + 84 ⁣⋯ ⁣5284\!\cdots\!52T5+ T^{5} + 82 ⁣⋯ ⁣8282\!\cdots\!82T6− T^{6} - 49 ⁣⋯ ⁣0849\!\cdots\!08T7+ T^{7} + 67 ⁣⋯ ⁣7867\!\cdots\!78T8− T^{8} - 49 ⁣⋯ ⁣0849\!\cdots\!08p7T9+ p^{7} T^{9} + 82 ⁣⋯ ⁣8282\!\cdots\!82p14T10+ p^{14} T^{10} + 84 ⁣⋯ ⁣5284\!\cdots\!52p21T11− p^{21} T^{11} - 33 ⁣⋯ ⁣1933\!\cdots\!19p28T12+792722790467378240p35T13−2356814530631p42T14−231504p49T15+p56T16 p^{28} T^{12} + 792722790467378240 p^{35} T^{13} - 2356814530631 p^{42} T^{14} - 231504 p^{49} T^{15} + p^{56} T^{16}
61 1−685684T+3433715225862T2+3646641375018060600T3− 1 - 685684 T + 3433715225862 T^{2} + 3646641375018060600 T^{3} - 13 ⁣⋯ ⁣0713\!\cdots\!07T4+ T^{4} + 29 ⁣⋯ ⁣2829\!\cdots\!28T5+ T^{5} + 50 ⁣⋯ ⁣0650\!\cdots\!06T6− T^{6} - 33 ⁣⋯ ⁣8433\!\cdots\!84T7+ T^{7} + 31 ⁣⋯ ⁣1231\!\cdots\!12T8− T^{8} - 33 ⁣⋯ ⁣8433\!\cdots\!84p7T9+ p^{7} T^{9} + 50 ⁣⋯ ⁣0650\!\cdots\!06p14T10+ p^{14} T^{10} + 29 ⁣⋯ ⁣2829\!\cdots\!28p21T11− p^{21} T^{11} - 13 ⁣⋯ ⁣0713\!\cdots\!07p28T12+3646641375018060600p35T13+3433715225862p42T14−685684p49T15+p56T16 p^{28} T^{12} + 3646641375018060600 p^{35} T^{13} + 3433715225862 p^{42} T^{14} - 685684 p^{49} T^{15} + p^{56} T^{16}
67 1−3271056T−14689835850767T2+35254481255509351008T3+ 1 - 3271056 T - 14689835850767 T^{2} + 35254481255509351008 T^{3} + 21 ⁣⋯ ⁣6521\!\cdots\!65T4− T^{4} - 32 ⁣⋯ ⁣9632\!\cdots\!96T5− T^{5} - 17 ⁣⋯ ⁣7417\!\cdots\!74T6+ T^{6} + 57 ⁣⋯ ⁣1257\!\cdots\!12T7+ T^{7} + 13 ⁣⋯ ⁣6613\!\cdots\!66T8+ T^{8} + 57 ⁣⋯ ⁣1257\!\cdots\!12p7T9− p^{7} T^{9} - 17 ⁣⋯ ⁣7417\!\cdots\!74p14T10− p^{14} T^{10} - 32 ⁣⋯ ⁣9632\!\cdots\!96p21T11+ p^{21} T^{11} + 21 ⁣⋯ ⁣6521\!\cdots\!65p28T12+35254481255509351008p35T13−14689835850767p42T14−3271056p49T15+p56T16 p^{28} T^{12} + 35254481255509351008 p^{35} T^{13} - 14689835850767 p^{42} T^{14} - 3271056 p^{49} T^{15} + p^{56} T^{16}
71 1+175012T−22406789187435T2−11058578082007940412T3+ 1 + 175012 T - 22406789187435 T^{2} - 11058578082007940412 T^{3} + 23 ⁣⋯ ⁣7723\!\cdots\!77T4+ T^{4} + 14 ⁣⋯ ⁣0414\!\cdots\!04T5− T^{5} - 22 ⁣⋯ ⁣7822\!\cdots\!78T6− T^{6} - 60 ⁣⋯ ⁣8060\!\cdots\!80T7+ T^{7} + 22 ⁣⋯ ⁣8222\!\cdots\!82T8− T^{8} - 60 ⁣⋯ ⁣8060\!\cdots\!80p7T9− p^{7} T^{9} - 22 ⁣⋯ ⁣7822\!\cdots\!78p14T10+ p^{14} T^{10} + 14 ⁣⋯ ⁣0414\!\cdots\!04p21T11+ p^{21} T^{11} + 23 ⁣⋯ ⁣7723\!\cdots\!77p28T12−11058578082007940412p35T13−22406789187435p42T14+175012p49T15+p56T16 p^{28} T^{12} - 11058578082007940412 p^{35} T^{13} - 22406789187435 p^{42} T^{14} + 175012 p^{49} T^{15} + p^{56} T^{16}
73 (1−7137890T+48989573892505T2− ( 1 - 7137890 T + 48989573892505 T^{2} - 21 ⁣⋯ ⁣7821\!\cdots\!78T3+ T^{3} + 81 ⁣⋯ ⁣1281\!\cdots\!12T4− T^{4} - 21 ⁣⋯ ⁣7821\!\cdots\!78p7T5+48989573892505p14T6−7137890p21T7+p28T8)2 p^{7} T^{5} + 48989573892505 p^{14} T^{6} - 7137890 p^{21} T^{7} + p^{28} T^{8} )^{2}
79 (1+7053952T+797675870260pT2+ ( 1 + 7053952 T + 797675870260 p T^{2} + 36 ⁣⋯ ⁣4036\!\cdots\!40T3+ T^{3} + 17 ⁣⋯ ⁣6617\!\cdots\!66T4+ T^{4} + 36 ⁣⋯ ⁣4036\!\cdots\!40p7T5+797675870260p15T6+7053952p21T7+p28T8)2 p^{7} T^{5} + 797675870260 p^{15} T^{6} + 7053952 p^{21} T^{7} + p^{28} T^{8} )^{2}
83 (1−657288T+707565190612pT2+83704316584548117240T3+ ( 1 - 657288 T + 707565190612 p T^{2} + 83704316584548117240 T^{3} + 16 ⁣⋯ ⁣2616\!\cdots\!26T4+83704316584548117240p7T5+707565190612p15T6−657288p21T7+p28T8)2 T^{4} + 83704316584548117240 p^{7} T^{5} + 707565190612 p^{15} T^{6} - 657288 p^{21} T^{7} + p^{28} T^{8} )^{2}
89 1+11452234T+45324518002715T2+ 1 + 11452234 T + 45324518002715 T^{2} + 11 ⁣⋯ ⁣0611\!\cdots\!06T3+ T^{3} + 99 ⁣⋯ ⁣4599\!\cdots\!45T4+ T^{4} + 24 ⁣⋯ ⁣6024\!\cdots\!60T5+ T^{5} + 52 ⁣⋯ ⁣7452\!\cdots\!74T6+ T^{6} + 37 ⁣⋯ ⁣0837\!\cdots\!08T7+ T^{7} + 66 ⁣⋯ ⁣2266\!\cdots\!22T8+ T^{8} + 37 ⁣⋯ ⁣0837\!\cdots\!08p7T9+ p^{7} T^{9} + 52 ⁣⋯ ⁣7452\!\cdots\!74p14T10+ p^{14} T^{10} + 24 ⁣⋯ ⁣6024\!\cdots\!60p21T11+ p^{21} T^{11} + 99 ⁣⋯ ⁣4599\!\cdots\!45p28T12+ p^{28} T^{12} + 11 ⁣⋯ ⁣0611\!\cdots\!06p35T13+45324518002715p42T14+11452234p49T15+p56T16 p^{35} T^{13} + 45324518002715 p^{42} T^{14} + 11452234 p^{49} T^{15} + p^{56} T^{16}
97 1+428002T−262783180278813T2− 1 + 428002 T - 262783180278813 T^{2} - 24 ⁣⋯ ⁣5024\!\cdots\!50T3+ T^{3} + 39 ⁣⋯ ⁣7339\!\cdots\!73T4+ T^{4} + 35 ⁣⋯ ⁣3235\!\cdots\!32T5− T^{5} - 42 ⁣⋯ ⁣2642\!\cdots\!26T6− T^{6} - 13 ⁣⋯ ⁣8013\!\cdots\!80T7+ T^{7} + 37 ⁣⋯ ⁣6637\!\cdots\!66T8− T^{8} - 13 ⁣⋯ ⁣8013\!\cdots\!80p7T9− p^{7} T^{9} - 42 ⁣⋯ ⁣2642\!\cdots\!26p14T10+ p^{14} T^{10} + 35 ⁣⋯ ⁣3235\!\cdots\!32p21T11+ p^{21} T^{11} + 39 ⁣⋯ ⁣7339\!\cdots\!73p28T12− p^{28} T^{12} - 24 ⁣⋯ ⁣5024\!\cdots\!50p35T13−262783180278813p42T14+428002p49T15+p56T16 p^{35} T^{13} - 262783180278813 p^{42} T^{14} + 428002 p^{49} T^{15} + p^{56} T^{16}
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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

−6.81168357888632453050945433734, −6.41202421035417755654813853444, −6.34734425840558225407562277420, −6.01761739580206786286193250518, −5.78443546173782474486460611623, −5.61392653010655724868642819762, −5.53261499213347158230359298570, −5.46484886820953479772836985539, −4.79274759649410613750469996864, −4.68103046049712956632384351648, −4.67975183157922891483601731651, −4.52559608014640163369828050410, −4.36270168228763664302675118250, −3.83752993980241522668129601175, −3.55158826630784322412726203703, −3.39338269765668062466063672318, −2.78085853788222646183678354315, −2.71018029187727279853891285228, −2.38291324254866173764632379237, −2.20771799778827641035146128298, −2.10824556057425073623084577750, −1.74495065676387056838552167449, −0.77624088404093377737204112699, −0.56010130648833225088231556862, −0.28637075526289471709435495819, 0.28637075526289471709435495819, 0.56010130648833225088231556862, 0.77624088404093377737204112699, 1.74495065676387056838552167449, 2.10824556057425073623084577750, 2.20771799778827641035146128298, 2.38291324254866173764632379237, 2.71018029187727279853891285228, 2.78085853788222646183678354315, 3.39338269765668062466063672318, 3.55158826630784322412726203703, 3.83752993980241522668129601175, 4.36270168228763664302675118250, 4.52559608014640163369828050410, 4.67975183157922891483601731651, 4.68103046049712956632384351648, 4.79274759649410613750469996864, 5.46484886820953479772836985539, 5.53261499213347158230359298570, 5.61392653010655724868642819762, 5.78443546173782474486460611623, 6.01761739580206786286193250518, 6.34734425840558225407562277420, 6.41202421035417755654813853444, 6.81168357888632453050945433734

Graph of the ZZ-function along the critical line

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