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Properties

Label 32-350e16-1.1-c2e16-0-0
Degree 3232
Conductor 5.071×10405.071\times 10^{40}
Sign 11
Analytic cond. 4.68219×10154.68219\times 10^{15}
Root an. cond. 3.088173.08817
Motivic weight 22
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  + 8·2-s − 2·3-s + 32·4-s − 16·6-s − 12·7-s + 96·8-s + 2·9-s + 40·11-s − 64·12-s − 16·13-s − 96·14-s + 264·16-s − 46·17-s + 16·18-s + 24·21-s + 320·22-s − 54·23-s − 192·24-s − 128·26-s + 28·27-s − 384·28-s − 208·31-s + 672·32-s − 80·33-s − 368·34-s + 64·36-s + 38·37-s + ⋯
L(s)  = 1  + 4·2-s − 2/3·3-s + 8·4-s − 8/3·6-s − 1.71·7-s + 12·8-s + 2/9·9-s + 3.63·11-s − 5.33·12-s − 1.23·13-s − 6.85·14-s + 33/2·16-s − 2.70·17-s + 8/9·18-s + 8/7·21-s + 14.5·22-s − 2.34·23-s − 8·24-s − 4.92·26-s + 1.03·27-s − 13.7·28-s − 6.70·31-s + 21·32-s − 2.42·33-s − 10.8·34-s + 16/9·36-s + 1.02·37-s + ⋯

Functional equation

Λ(s)=((216⋅532⋅716)s/2 ΓC(s)16 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((216⋅532⋅716)s/2 ΓC(s+1)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 216⋅532⋅7162^{16} \cdot 5^{32} \cdot 7^{16}
Sign: 11
Analytic conductor: 4.68219×10154.68219\times 10^{15}
Root analytic conductor: 3.088173.08817
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 216⋅532⋅716, ( :[1]16), 1)(32,\ 2^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.9260144603.926014460
L(12)L(\frac12) ≈\approx 3.9260144603.926014460
L(2)L(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−pT+pT2−p2T3+p2T4)4 ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{4}
5 1 1
7 1+12T+72T2+468T3−59T4−23880T5−172800T6−196440pT7−216060p2T8−196440p3T9−172800p4T10−23880p6T11−59p8T12+468p10T13+72p12T14+12p14T15+p16T16 1 + 12 T + 72 T^{2} + 468 T^{3} - 59 T^{4} - 23880 T^{5} - 172800 T^{6} - 196440 p T^{7} - 216060 p^{2} T^{8} - 196440 p^{3} T^{9} - 172800 p^{4} T^{10} - 23880 p^{6} T^{11} - 59 p^{8} T^{12} + 468 p^{10} T^{13} + 72 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16}
good3 1+2T+2T2−28T3−2T4+86T5+568T6+58p2T7−1541pT8+74p4T9+11452T10+106106T11+135058T12+1589180T13+3237770T14−5097334T15−9280004T16−5097334p2T17+3237770p4T18+1589180p6T19+135058p8T20+106106p10T21+11452p12T22+74p18T23−1541p17T24+58p20T25+568p20T26+86p22T27−2p24T28−28p26T29+2p28T30+2p30T31+p32T32 1 + 2 T + 2 T^{2} - 28 T^{3} - 2 T^{4} + 86 T^{5} + 568 T^{6} + 58 p^{2} T^{7} - 1541 p T^{8} + 74 p^{4} T^{9} + 11452 T^{10} + 106106 T^{11} + 135058 T^{12} + 1589180 T^{13} + 3237770 T^{14} - 5097334 T^{15} - 9280004 T^{16} - 5097334 p^{2} T^{17} + 3237770 p^{4} T^{18} + 1589180 p^{6} T^{19} + 135058 p^{8} T^{20} + 106106 p^{10} T^{21} + 11452 p^{12} T^{22} + 74 p^{18} T^{23} - 1541 p^{17} T^{24} + 58 p^{20} T^{25} + 568 p^{20} T^{26} + 86 p^{22} T^{27} - 2 p^{24} T^{28} - 28 p^{26} T^{29} + 2 p^{28} T^{30} + 2 p^{30} T^{31} + p^{32} T^{32}
11 (1−20T+53T2+1868T3−19793T4+23808T5+812056T6+3236448T7−190136502T8+3236448p2T9+812056p4T10+23808p6T11−19793p8T12+1868p10T13+53p12T14−20p14T15+p16T16)2 ( 1 - 20 T + 53 T^{2} + 1868 T^{3} - 19793 T^{4} + 23808 T^{5} + 812056 T^{6} + 3236448 T^{7} - 190136502 T^{8} + 3236448 p^{2} T^{9} + 812056 p^{4} T^{10} + 23808 p^{6} T^{11} - 19793 p^{8} T^{12} + 1868 p^{10} T^{13} + 53 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2}
13 (1+8T+32T2+1112T3+33680T4+119592T5+497248T6+10550648T7+68709598T8+10550648p2T9+497248p4T10+119592p6T11+33680p8T12+1112p10T13+32p12T14+8p14T15+p16T16)2 ( 1 + 8 T + 32 T^{2} + 1112 T^{3} + 33680 T^{4} + 119592 T^{5} + 497248 T^{6} + 10550648 T^{7} + 68709598 T^{8} + 10550648 p^{2} T^{9} + 497248 p^{4} T^{10} + 119592 p^{6} T^{11} + 33680 p^{8} T^{12} + 1112 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2}
17 1+46T+1058T2+15228T3+49795T4−3577952T5−101322910T6−1912395662T7−35538003267T8−729089246500T9−13614614462848T10−149191614412260T11+396919406750154T12+3079380260285984pT13+1230645459219027588T14+18772373078273663580T15+ 1 + 46 T + 1058 T^{2} + 15228 T^{3} + 49795 T^{4} - 3577952 T^{5} - 101322910 T^{6} - 1912395662 T^{7} - 35538003267 T^{8} - 729089246500 T^{9} - 13614614462848 T^{10} - 149191614412260 T^{11} + 396919406750154 T^{12} + 3079380260285984 p T^{13} + 1230645459219027588 T^{14} + 18772373078273663580 T^{15} + 27 ⁣⋯ ⁣3027\!\cdots\!30T16+18772373078273663580p2T17+1230645459219027588p4T18+3079380260285984p7T19+396919406750154p8T20−149191614412260p10T21−13614614462848p12T22−729089246500p14T23−35538003267p16T24−1912395662p18T25−101322910p20T26−3577952p22T27+49795p24T28+15228p26T29+1058p28T30+46p30T31+p32T32 T^{16} + 18772373078273663580 p^{2} T^{17} + 1230645459219027588 p^{4} T^{18} + 3079380260285984 p^{7} T^{19} + 396919406750154 p^{8} T^{20} - 149191614412260 p^{10} T^{21} - 13614614462848 p^{12} T^{22} - 729089246500 p^{14} T^{23} - 35538003267 p^{16} T^{24} - 1912395662 p^{18} T^{25} - 101322910 p^{20} T^{26} - 3577952 p^{22} T^{27} + 49795 p^{24} T^{28} + 15228 p^{26} T^{29} + 1058 p^{28} T^{30} + 46 p^{30} T^{31} + p^{32} T^{32}
19 1+962T2+285903T4−9599234T6−15336277051T8+2324255772132T10+984840646865914T12−821244130299946144T14− 1 + 962 T^{2} + 285903 T^{4} - 9599234 T^{6} - 15336277051 T^{8} + 2324255772132 T^{10} + 984840646865914 T^{12} - 821244130299946144 T^{14} - 53 ⁣⋯ ⁣8253\!\cdots\!82T16−821244130299946144p4T18+984840646865914p8T20+2324255772132p12T22−15336277051p16T24−9599234p20T26+285903p24T28+962p28T30+p32T32 T^{16} - 821244130299946144 p^{4} T^{18} + 984840646865914 p^{8} T^{20} + 2324255772132 p^{12} T^{22} - 15336277051 p^{16} T^{24} - 9599234 p^{20} T^{26} + 285903 p^{24} T^{28} + 962 p^{28} T^{30} + p^{32} T^{32}
23 1+54T+1458T2+41404T3+1531542T4+51187722T5+1388294360T6+34788210422T7+978568184441T8+28666106605246T9+745448729152316T10+17609421034026766T11+432143271417763898T12+11387865059627737708T13+ 1 + 54 T + 1458 T^{2} + 41404 T^{3} + 1531542 T^{4} + 51187722 T^{5} + 1388294360 T^{6} + 34788210422 T^{7} + 978568184441 T^{8} + 28666106605246 T^{9} + 745448729152316 T^{10} + 17609421034026766 T^{11} + 432143271417763898 T^{12} + 11387865059627737708 T^{13} + 28 ⁣⋯ ⁣3428\!\cdots\!34T14+ T^{14} + 63 ⁣⋯ ⁣3863\!\cdots\!38T15+ T^{15} + 13 ⁣⋯ ⁣3613\!\cdots\!36T16+ T^{16} + 63 ⁣⋯ ⁣3863\!\cdots\!38p2T17+ p^{2} T^{17} + 28 ⁣⋯ ⁣3428\!\cdots\!34p4T18+11387865059627737708p6T19+432143271417763898p8T20+17609421034026766p10T21+745448729152316p12T22+28666106605246p14T23+978568184441p16T24+34788210422p18T25+1388294360p20T26+51187722p22T27+1531542p24T28+41404p26T29+1458p28T30+54p30T31+p32T32 p^{4} T^{18} + 11387865059627737708 p^{6} T^{19} + 432143271417763898 p^{8} T^{20} + 17609421034026766 p^{10} T^{21} + 745448729152316 p^{12} T^{22} + 28666106605246 p^{14} T^{23} + 978568184441 p^{16} T^{24} + 34788210422 p^{18} T^{25} + 1388294360 p^{20} T^{26} + 51187722 p^{22} T^{27} + 1531542 p^{24} T^{28} + 41404 p^{26} T^{29} + 1458 p^{28} T^{30} + 54 p^{30} T^{31} + p^{32} T^{32}
29 (1−4830T2+11400705T4−16870918974T6+17005870320740T8−16870918974p4T10+11400705p8T12−4830p12T14+p16T16)2 ( 1 - 4830 T^{2} + 11400705 T^{4} - 16870918974 T^{6} + 17005870320740 T^{8} - 16870918974 p^{4} T^{10} + 11400705 p^{8} T^{12} - 4830 p^{12} T^{14} + p^{16} T^{16} )^{2}
31 (1+104T+3335T2+80216T3+6958945T4+299604368T5+5519548642T6+232310443200T7+11482603425294T8+232310443200p2T9+5519548642p4T10+299604368p6T11+6958945p8T12+80216p10T13+3335p12T14+104p14T15+p16T16)2 ( 1 + 104 T + 3335 T^{2} + 80216 T^{3} + 6958945 T^{4} + 299604368 T^{5} + 5519548642 T^{6} + 232310443200 T^{7} + 11482603425294 T^{8} + 232310443200 p^{2} T^{9} + 5519548642 p^{4} T^{10} + 299604368 p^{6} T^{11} + 6958945 p^{8} T^{12} + 80216 p^{10} T^{13} + 3335 p^{12} T^{14} + 104 p^{14} T^{15} + p^{16} T^{16} )^{2}
37 1−38T+722T2−36204T3+921843T4−54419040T5+2057717682T6−7359428170T7+1748254042141T8+98996461623124T9−5173840906504960T10+1619684158242788pT11−5578617333398806678T12−6554354740117891808T13− 1 - 38 T + 722 T^{2} - 36204 T^{3} + 921843 T^{4} - 54419040 T^{5} + 2057717682 T^{6} - 7359428170 T^{7} + 1748254042141 T^{8} + 98996461623124 T^{9} - 5173840906504960 T^{10} + 1619684158242788 p T^{11} - 5578617333398806678 T^{12} - 6554354740117891808 T^{13} - 20 ⁣⋯ ⁣9620\!\cdots\!96T14+ T^{14} + 42 ⁣⋯ ⁣1642\!\cdots\!16T15− T^{15} - 83 ⁣⋯ ⁣5483\!\cdots\!54T16+ T^{16} + 42 ⁣⋯ ⁣1642\!\cdots\!16p2T17− p^{2} T^{17} - 20 ⁣⋯ ⁣9620\!\cdots\!96p4T18−6554354740117891808p6T19−5578617333398806678p8T20+1619684158242788p11T21−5173840906504960p12T22+98996461623124p14T23+1748254042141p16T24−7359428170p18T25+2057717682p20T26−54419040p22T27+921843p24T28−36204p26T29+722p28T30−38p30T31+p32T32 p^{4} T^{18} - 6554354740117891808 p^{6} T^{19} - 5578617333398806678 p^{8} T^{20} + 1619684158242788 p^{11} T^{21} - 5173840906504960 p^{12} T^{22} + 98996461623124 p^{14} T^{23} + 1748254042141 p^{16} T^{24} - 7359428170 p^{18} T^{25} + 2057717682 p^{20} T^{26} - 54419040 p^{22} T^{27} + 921843 p^{24} T^{28} - 36204 p^{26} T^{29} + 722 p^{28} T^{30} - 38 p^{30} T^{31} + p^{32} T^{32}
41 (1+18T+4237T2+27182T3+8225864T4+27182p2T5+4237p4T6+18p6T7+p8T8)4 ( 1 + 18 T + 4237 T^{2} + 27182 T^{3} + 8225864 T^{4} + 27182 p^{2} T^{5} + 4237 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{4}
43 (1+72T+2592T2+12288T3−872859T4+17006160T5+3562391520T6−141343265880T7−11372920448860T8−141343265880p2T9+3562391520p4T10+17006160p6T11−872859p8T12+12288p10T13+2592p12T14+72p14T15+p16T16)2 ( 1 + 72 T + 2592 T^{2} + 12288 T^{3} - 872859 T^{4} + 17006160 T^{5} + 3562391520 T^{6} - 141343265880 T^{7} - 11372920448860 T^{8} - 141343265880 p^{2} T^{9} + 3562391520 p^{4} T^{10} + 17006160 p^{6} T^{11} - 872859 p^{8} T^{12} + 12288 p^{10} T^{13} + 2592 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} )^{2}
47 1+46T+1058T2−161412T3−22698365T4−1010022752T5−9419259550T6+2715134508658T7+271485194118333T8+10484618609225180T9+53077017818847872T10−26010553538653727460T11− 1 + 46 T + 1058 T^{2} - 161412 T^{3} - 22698365 T^{4} - 1010022752 T^{5} - 9419259550 T^{6} + 2715134508658 T^{7} + 271485194118333 T^{8} + 10484618609225180 T^{9} + 53077017818847872 T^{10} - 26010553538653727460 T^{11} - 20 ⁣⋯ ⁣4620\!\cdots\!46T12− T^{12} - 72 ⁣⋯ ⁣1272\!\cdots\!12T13− T^{13} - 91 ⁣⋯ ⁣1291\!\cdots\!12T14+ T^{14} + 16 ⁣⋯ ⁣0016\!\cdots\!00T15+ T^{15} + 11 ⁣⋯ ⁣7011\!\cdots\!70T16+ T^{16} + 16 ⁣⋯ ⁣0016\!\cdots\!00p2T17− p^{2} T^{17} - 91 ⁣⋯ ⁣1291\!\cdots\!12p4T18− p^{4} T^{18} - 72 ⁣⋯ ⁣1272\!\cdots\!12p6T19− p^{6} T^{19} - 20 ⁣⋯ ⁣4620\!\cdots\!46p8T20−26010553538653727460p10T21+53077017818847872p12T22+10484618609225180p14T23+271485194118333p16T24+2715134508658p18T25−9419259550p20T26−1010022752p22T27−22698365p24T28−161412p26T29+1058p28T30+46p30T31+p32T32 p^{8} T^{20} - 26010553538653727460 p^{10} T^{21} + 53077017818847872 p^{12} T^{22} + 10484618609225180 p^{14} T^{23} + 271485194118333 p^{16} T^{24} + 2715134508658 p^{18} T^{25} - 9419259550 p^{20} T^{26} - 1010022752 p^{22} T^{27} - 22698365 p^{24} T^{28} - 161412 p^{26} T^{29} + 1058 p^{28} T^{30} + 46 p^{30} T^{31} + p^{32} T^{32}
53 1+30T+450T2−53260T3−13562801T4−264802000T5−422485750T6+1346619990690T7+94385910261585T8+3394198091700460T9+16961261878565400T10−11461088832223742300T11− 1 + 30 T + 450 T^{2} - 53260 T^{3} - 13562801 T^{4} - 264802000 T^{5} - 422485750 T^{6} + 1346619990690 T^{7} + 94385910261585 T^{8} + 3394198091700460 T^{9} + 16961261878565400 T^{10} - 11461088832223742300 T^{11} - 43 ⁣⋯ ⁣9443\!\cdots\!94T12− T^{12} - 38 ⁣⋯ ⁣8038\!\cdots\!80T13− T^{13} - 65 ⁣⋯ ⁣0065\!\cdots\!00T14+ T^{14} + 11 ⁣⋯ ⁣0011\!\cdots\!00T15+ T^{15} + 39 ⁣⋯ ⁣7439\!\cdots\!74T16+ T^{16} + 11 ⁣⋯ ⁣0011\!\cdots\!00p2T17− p^{2} T^{17} - 65 ⁣⋯ ⁣0065\!\cdots\!00p4T18− p^{4} T^{18} - 38 ⁣⋯ ⁣8038\!\cdots\!80p6T19− p^{6} T^{19} - 43 ⁣⋯ ⁣9443\!\cdots\!94p8T20−11461088832223742300p10T21+16961261878565400p12T22+3394198091700460p14T23+94385910261585p16T24+1346619990690p18T25−422485750p20T26−264802000p22T27−13562801p24T28−53260p26T29+450p28T30+30p30T31+p32T32 p^{8} T^{20} - 11461088832223742300 p^{10} T^{21} + 16961261878565400 p^{12} T^{22} + 3394198091700460 p^{14} T^{23} + 94385910261585 p^{16} T^{24} + 1346619990690 p^{18} T^{25} - 422485750 p^{20} T^{26} - 264802000 p^{22} T^{27} - 13562801 p^{24} T^{28} - 53260 p^{26} T^{29} + 450 p^{28} T^{30} + 30 p^{30} T^{31} + p^{32} T^{32}
59 1+6674T2−7729537T4−69884124210T6+388809824859333T8+851796830409747108T10− 1 + 6674 T^{2} - 7729537 T^{4} - 69884124210 T^{6} + 388809824859333 T^{8} + 851796830409747108 T^{10} - 72 ⁣⋯ ⁣9872\!\cdots\!98T12− T^{12} - 40 ⁣⋯ ⁣0840\!\cdots\!08T14+ T^{14} + 99 ⁣⋯ ⁣5099\!\cdots\!50T16− T^{16} - 40 ⁣⋯ ⁣0840\!\cdots\!08p4T18− p^{4} T^{18} - 72 ⁣⋯ ⁣9872\!\cdots\!98p8T20+851796830409747108p12T22+388809824859333p16T24−69884124210p20T26−7729537p24T28+6674p28T30+p32T32 p^{8} T^{20} + 851796830409747108 p^{12} T^{22} + 388809824859333 p^{16} T^{24} - 69884124210 p^{20} T^{26} - 7729537 p^{24} T^{28} + 6674 p^{28} T^{30} + p^{32} T^{32}
61 (1+60T−8926T2−393256T3+57653141T4+1511546176T5−266254549814T6−2710779938676T7+1000035050441884T8−2710779938676p2T9−266254549814p4T10+1511546176p6T11+57653141p8T12−393256p10T13−8926p12T14+60p14T15+p16T16)2 ( 1 + 60 T - 8926 T^{2} - 393256 T^{3} + 57653141 T^{4} + 1511546176 T^{5} - 266254549814 T^{6} - 2710779938676 T^{7} + 1000035050441884 T^{8} - 2710779938676 p^{2} T^{9} - 266254549814 p^{4} T^{10} + 1511546176 p^{6} T^{11} + 57653141 p^{8} T^{12} - 393256 p^{10} T^{13} - 8926 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} )^{2}
67 1−74T+2738T2+587900T3−50988730T4+3339904986T5+65267378776T6−10012501185306T7+918173402551977T8+2374991498059262T9+329325818085928764T10−1284143109513809266T11+ 1 - 74 T + 2738 T^{2} + 587900 T^{3} - 50988730 T^{4} + 3339904986 T^{5} + 65267378776 T^{6} - 10012501185306 T^{7} + 918173402551977 T^{8} + 2374991498059262 T^{9} + 329325818085928764 T^{10} - 1284143109513809266 T^{11} + 25 ⁣⋯ ⁣9025\!\cdots\!90T12− T^{12} - 13 ⁣⋯ ⁣4413\!\cdots\!44T13+ T^{13} + 44 ⁣⋯ ⁣5444\!\cdots\!54T14+ T^{14} + 11 ⁣⋯ ⁣7811\!\cdots\!78T15− T^{15} - 84 ⁣⋯ ⁣1284\!\cdots\!12T16+ T^{16} + 11 ⁣⋯ ⁣7811\!\cdots\!78p2T17+ p^{2} T^{17} + 44 ⁣⋯ ⁣5444\!\cdots\!54p4T18− p^{4} T^{18} - 13 ⁣⋯ ⁣4413\!\cdots\!44p6T19+ p^{6} T^{19} + 25 ⁣⋯ ⁣9025\!\cdots\!90p8T20−1284143109513809266p10T21+329325818085928764p12T22+2374991498059262p14T23+918173402551977p16T24−10012501185306p18T25+65267378776p20T26+3339904986p22T27−50988730p24T28+587900p26T29+2738p28T30−74p30T31+p32T32 p^{8} T^{20} - 1284143109513809266 p^{10} T^{21} + 329325818085928764 p^{12} T^{22} + 2374991498059262 p^{14} T^{23} + 918173402551977 p^{16} T^{24} - 10012501185306 p^{18} T^{25} + 65267378776 p^{20} T^{26} + 3339904986 p^{22} T^{27} - 50988730 p^{24} T^{28} + 587900 p^{26} T^{29} + 2738 p^{28} T^{30} - 74 p^{30} T^{31} + p^{32} T^{32}
71 (1−4T+10238T2+37292T3+54945234T4+37292p2T5+10238p4T6−4p6T7+p8T8)4 ( 1 - 4 T + 10238 T^{2} + 37292 T^{3} + 54945234 T^{4} + 37292 p^{2} T^{5} + 10238 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{4}
73 1−54T+1458T2−376300T3−9032525T4−1427883664T5+161075984306T6−3449490767706T7+2006002204707837T8−28704897132082748T9+2397705340658794944T10− 1 - 54 T + 1458 T^{2} - 376300 T^{3} - 9032525 T^{4} - 1427883664 T^{5} + 161075984306 T^{6} - 3449490767706 T^{7} + 2006002204707837 T^{8} - 28704897132082748 T^{9} + 2397705340658794944 T^{10} - 54 ⁣⋯ ⁣9654\!\cdots\!96T11− T^{11} - 52 ⁣⋯ ⁣5052\!\cdots\!50T12− T^{12} - 41 ⁣⋯ ⁣8441\!\cdots\!84T13+ T^{13} + 17 ⁣⋯ ⁣0417\!\cdots\!04T14− T^{14} - 42 ⁣⋯ ⁣7242\!\cdots\!72T15+ T^{15} + 19 ⁣⋯ ⁣5819\!\cdots\!58T16− T^{16} - 42 ⁣⋯ ⁣7242\!\cdots\!72p2T17+ p^{2} T^{17} + 17 ⁣⋯ ⁣0417\!\cdots\!04p4T18− p^{4} T^{18} - 41 ⁣⋯ ⁣8441\!\cdots\!84p6T19− p^{6} T^{19} - 52 ⁣⋯ ⁣5052\!\cdots\!50p8T20− p^{8} T^{20} - 54 ⁣⋯ ⁣9654\!\cdots\!96p10T21+2397705340658794944p12T22−28704897132082748p14T23+2006002204707837p16T24−3449490767706p18T25+161075984306p20T26−1427883664p22T27−9032525p24T28−376300p26T29+1458p28T30−54p30T31+p32T32 p^{10} T^{21} + 2397705340658794944 p^{12} T^{22} - 28704897132082748 p^{14} T^{23} + 2006002204707837 p^{16} T^{24} - 3449490767706 p^{18} T^{25} + 161075984306 p^{20} T^{26} - 1427883664 p^{22} T^{27} - 9032525 p^{24} T^{28} - 376300 p^{26} T^{29} + 1458 p^{28} T^{30} - 54 p^{30} T^{31} + p^{32} T^{32}
79 1+42538T2+982751207T4+15945124184102T6+201431875816036549T8+ 1 + 42538 T^{2} + 982751207 T^{4} + 15945124184102 T^{6} + 201431875816036549 T^{8} + 20 ⁣⋯ ⁣9220\!\cdots\!92T10+ T^{10} + 18 ⁣⋯ ⁣3418\!\cdots\!34T12+ T^{12} + 14 ⁣⋯ ⁣2414\!\cdots\!24T14+ T^{14} + 93 ⁣⋯ ⁣9093\!\cdots\!90T16+ T^{16} + 14 ⁣⋯ ⁣2414\!\cdots\!24p4T18+ p^{4} T^{18} + 18 ⁣⋯ ⁣3418\!\cdots\!34p8T20+ p^{8} T^{20} + 20 ⁣⋯ ⁣9220\!\cdots\!92p12T22+201431875816036549p16T24+15945124184102p20T26+982751207p24T28+42538p28T30+p32T32 p^{12} T^{22} + 201431875816036549 p^{16} T^{24} + 15945124184102 p^{20} T^{26} + 982751207 p^{24} T^{28} + 42538 p^{28} T^{30} + p^{32} T^{32}
83 (1+32T+512T2−252712T3−42882475T4+1371597568T5+97778626848T6+7242029223592T7+506959325886468T8+7242029223592p2T9+97778626848p4T10+1371597568p6T11−42882475p8T12−252712p10T13+512p12T14+32p14T15+p16T16)2 ( 1 + 32 T + 512 T^{2} - 252712 T^{3} - 42882475 T^{4} + 1371597568 T^{5} + 97778626848 T^{6} + 7242029223592 T^{7} + 506959325886468 T^{8} + 7242029223592 p^{2} T^{9} + 97778626848 p^{4} T^{10} + 1371597568 p^{6} T^{11} - 42882475 p^{8} T^{12} - 252712 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} )^{2}
89 1+32552T2+612475958T4+7153003757936T6+52449920499191449T8+ 1 + 32552 T^{2} + 612475958 T^{4} + 7153003757936 T^{6} + 52449920499191449 T^{8} + 12 ⁣⋯ ⁣9212\!\cdots\!92T10− T^{10} - 24 ⁣⋯ ⁣6624\!\cdots\!66T12− T^{12} - 42 ⁣⋯ ⁣6442\!\cdots\!64T14− T^{14} - 40 ⁣⋯ ⁣1240\!\cdots\!12T16− T^{16} - 42 ⁣⋯ ⁣6442\!\cdots\!64p4T18− p^{4} T^{18} - 24 ⁣⋯ ⁣6624\!\cdots\!66p8T20+ p^{8} T^{20} + 12 ⁣⋯ ⁣9212\!\cdots\!92p12T22+52449920499191449p16T24+7153003757936p20T26+612475958p24T28+32552p28T30+p32T32 p^{12} T^{22} + 52449920499191449 p^{16} T^{24} + 7153003757936 p^{20} T^{26} + 612475958 p^{24} T^{28} + 32552 p^{28} T^{30} + p^{32} T^{32}
97 (1+68T+2312T2−258100T3−260409124T4−11295624444T5−132728762504T6+78472536305132T7+30681693645016390T8+78472536305132p2T9−132728762504p4T10−11295624444p6T11−260409124p8T12−258100p10T13+2312p12T14+68p14T15+p16T16)2 ( 1 + 68 T + 2312 T^{2} - 258100 T^{3} - 260409124 T^{4} - 11295624444 T^{5} - 132728762504 T^{6} + 78472536305132 T^{7} + 30681693645016390 T^{8} + 78472536305132 p^{2} T^{9} - 132728762504 p^{4} T^{10} - 11295624444 p^{6} T^{11} - 260409124 p^{8} T^{12} - 258100 p^{10} T^{13} + 2312 p^{12} T^{14} + 68 p^{14} T^{15} + p^{16} T^{16} )^{2}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.01307807732938402045314332677, −2.99653379307592492214321392124, −2.97114079215687035321242641359, −2.86088903464508881917130250408, −2.72258284216068091004220047778, −2.60659435539921888578166445346, −2.28630148068930637446954861760, −2.22521932024942430720084467704, −2.10576688455264223440229481494, −1.97446962325651689153871585845, −1.90158932566270721751815637562, −1.86891463511483541839670015134, −1.86520926195608365142348070494, −1.78966940463849846455120417345, −1.73156284992252938926146584433, −1.55572566885583637016576085585, −1.51625050112658521325803460433, −1.36525816149981641074589550976, −1.11757500767264879902756466459, −0.927076026138612437754517464481, −0.67146317879544329990780575248, −0.57433651463646520564403538208, −0.39671176946970811038469733563, −0.22078123416638756730545541016, −0.06446521100032377221072541871, 0.06446521100032377221072541871, 0.22078123416638756730545541016, 0.39671176946970811038469733563, 0.57433651463646520564403538208, 0.67146317879544329990780575248, 0.927076026138612437754517464481, 1.11757500767264879902756466459, 1.36525816149981641074589550976, 1.51625050112658521325803460433, 1.55572566885583637016576085585, 1.73156284992252938926146584433, 1.78966940463849846455120417345, 1.86520926195608365142348070494, 1.86891463511483541839670015134, 1.90158932566270721751815637562, 1.97446962325651689153871585845, 2.10576688455264223440229481494, 2.22521932024942430720084467704, 2.28630148068930637446954861760, 2.60659435539921888578166445346, 2.72258284216068091004220047778, 2.86088903464508881917130250408, 2.97114079215687035321242641359, 2.99653379307592492214321392124, 3.01307807732938402045314332677

Graph of the ZZ-function along the critical line

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