Properties

Degree 48
Conductor $ 2^{24} \cdot 7^{24} \cdot 431^{24} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·2-s + 7·3-s + 300·4-s + 8·5-s − 168·6-s − 24·7-s − 2.60e3·8-s − 2·9-s − 192·10-s + 15·11-s + 2.10e3·12-s − 7·13-s + 576·14-s + 56·15-s + 1.75e4·16-s − 5·17-s + 48·18-s + 6·19-s + 2.40e3·20-s − 168·21-s − 360·22-s + 3·23-s − 1.82e4·24-s − 22·25-s + 168·26-s − 128·27-s − 7.20e3·28-s + ⋯
L(s)  = 1  − 16.9·2-s + 4.04·3-s + 150·4-s + 3.57·5-s − 68.5·6-s − 9.07·7-s − 919.·8-s − 2/3·9-s − 60.7·10-s + 4.52·11-s + 606.·12-s − 1.94·13-s + 153.·14-s + 14.4·15-s + 4.38e3·16-s − 1.21·17-s + 11.3·18-s + 1.37·19-s + 536.·20-s − 36.6·21-s − 76.7·22-s + 0.625·23-s − 3.71e3·24-s − 4.39·25-s + 32.9·26-s − 24.6·27-s − 1.36e3·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 431^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24} \cdot 431^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(2^{24} \cdot 7^{24} \cdot 431^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6034} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(48,\ 2^{24} \cdot 7^{24} \cdot 431^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$
$L(1)$  $\approx$  $0.4934526781$
$L(\frac12)$  $\approx$  $0.4934526781$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \(F_p\) is a polynomial of degree 48. If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 47.
$p$$F_p$
bad2 \( ( 1 + T )^{24} \)
7 \( ( 1 + T )^{24} \)
431 \( ( 1 - T )^{24} \)
good3 \( 1 - 7 T + 17 p T^{2} - p^{5} T^{3} + 371 p T^{4} - 4222 T^{5} + 15241 T^{6} - 49387 T^{7} + 152549 T^{8} - 438883 T^{9} + 1208621 T^{10} - 3161227 T^{11} + 7949237 T^{12} - 6399158 p T^{13} + 4972093 p^{2} T^{14} - 100833106 T^{15} + 220034314 T^{16} - 465950248 T^{17} + 958114705 T^{18} - 638734448 p T^{19} + 3728834180 T^{20} - 7066579426 T^{21} + 13048015345 T^{22} - 23478432281 T^{23} + 41189959868 T^{24} - 23478432281 p T^{25} + 13048015345 p^{2} T^{26} - 7066579426 p^{3} T^{27} + 3728834180 p^{4} T^{28} - 638734448 p^{6} T^{29} + 958114705 p^{6} T^{30} - 465950248 p^{7} T^{31} + 220034314 p^{8} T^{32} - 100833106 p^{9} T^{33} + 4972093 p^{12} T^{34} - 6399158 p^{12} T^{35} + 7949237 p^{12} T^{36} - 3161227 p^{13} T^{37} + 1208621 p^{14} T^{38} - 438883 p^{15} T^{39} + 152549 p^{16} T^{40} - 49387 p^{17} T^{41} + 15241 p^{18} T^{42} - 4222 p^{19} T^{43} + 371 p^{21} T^{44} - p^{26} T^{45} + 17 p^{23} T^{46} - 7 p^{23} T^{47} + p^{24} T^{48} \)
5 \( 1 - 8 T + 86 T^{2} - 517 T^{3} + 3398 T^{4} - 16882 T^{5} + 86122 T^{6} - 371248 T^{7} + 1602484 T^{8} - 6167912 T^{9} + 23503216 T^{10} - 3290753 p^{2} T^{11} + 283543804 T^{12} - 913831604 T^{13} + 2892233394 T^{14} - 8656735662 T^{15} + 25412233104 T^{16} - 14213301636 p T^{17} + 194807375792 T^{18} - 102230724798 p T^{19} + 1314363592346 T^{20} - 3244787449189 T^{21} + 7849506549943 T^{22} - 18260915435173 T^{23} + 41627547461991 T^{24} - 18260915435173 p T^{25} + 7849506549943 p^{2} T^{26} - 3244787449189 p^{3} T^{27} + 1314363592346 p^{4} T^{28} - 102230724798 p^{6} T^{29} + 194807375792 p^{6} T^{30} - 14213301636 p^{8} T^{31} + 25412233104 p^{8} T^{32} - 8656735662 p^{9} T^{33} + 2892233394 p^{10} T^{34} - 913831604 p^{11} T^{35} + 283543804 p^{12} T^{36} - 3290753 p^{15} T^{37} + 23503216 p^{14} T^{38} - 6167912 p^{15} T^{39} + 1602484 p^{16} T^{40} - 371248 p^{17} T^{41} + 86122 p^{18} T^{42} - 16882 p^{19} T^{43} + 3398 p^{20} T^{44} - 517 p^{21} T^{45} + 86 p^{22} T^{46} - 8 p^{23} T^{47} + p^{24} T^{48} \)
11 \( 1 - 15 T + 236 T^{2} - 2333 T^{3} + 22576 T^{4} - 173341 T^{5} + 1292391 T^{6} - 8293902 T^{7} + 51766668 T^{8} - 289082386 T^{9} + 1575541743 T^{10} - 7849537456 T^{11} + 38318910235 T^{12} - 173265277378 T^{13} + 771029425242 T^{14} - 3206340465249 T^{15} + 13189582342923 T^{16} - 51042522347753 T^{17} + 196594636122040 T^{18} - 716430229793788 T^{19} + 2616617182613221 T^{20} - 9088972618695500 T^{21} + 31854546957488804 T^{22} - 106606256301108123 T^{23} + 361542249176641409 T^{24} - 106606256301108123 p T^{25} + 31854546957488804 p^{2} T^{26} - 9088972618695500 p^{3} T^{27} + 2616617182613221 p^{4} T^{28} - 716430229793788 p^{5} T^{29} + 196594636122040 p^{6} T^{30} - 51042522347753 p^{7} T^{31} + 13189582342923 p^{8} T^{32} - 3206340465249 p^{9} T^{33} + 771029425242 p^{10} T^{34} - 173265277378 p^{11} T^{35} + 38318910235 p^{12} T^{36} - 7849537456 p^{13} T^{37} + 1575541743 p^{14} T^{38} - 289082386 p^{15} T^{39} + 51766668 p^{16} T^{40} - 8293902 p^{17} T^{41} + 1292391 p^{18} T^{42} - 173341 p^{19} T^{43} + 22576 p^{20} T^{44} - 2333 p^{21} T^{45} + 236 p^{22} T^{46} - 15 p^{23} T^{47} + p^{24} T^{48} \)
13 \( 1 + 7 T + 178 T^{2} + 1086 T^{3} + 15242 T^{4} + 83049 T^{5} + 845629 T^{6} + 4191160 T^{7} + 34476153 T^{8} + 12132157 p T^{9} + 1109872460 T^{10} + 4743327107 T^{11} + 29594433972 T^{12} + 119349675267 T^{13} + 676630733793 T^{14} + 2595464775351 T^{15} + 13608248856346 T^{16} + 49925048536458 T^{17} + 245139067925937 T^{18} + 862641857787015 T^{19} + 4000837921928294 T^{20} + 13510382821115724 T^{21} + 59519364006182043 T^{22} + 192627922090182559 T^{23} + 809157135971194064 T^{24} + 192627922090182559 p T^{25} + 59519364006182043 p^{2} T^{26} + 13510382821115724 p^{3} T^{27} + 4000837921928294 p^{4} T^{28} + 862641857787015 p^{5} T^{29} + 245139067925937 p^{6} T^{30} + 49925048536458 p^{7} T^{31} + 13608248856346 p^{8} T^{32} + 2595464775351 p^{9} T^{33} + 676630733793 p^{10} T^{34} + 119349675267 p^{11} T^{35} + 29594433972 p^{12} T^{36} + 4743327107 p^{13} T^{37} + 1109872460 p^{14} T^{38} + 12132157 p^{16} T^{39} + 34476153 p^{16} T^{40} + 4191160 p^{17} T^{41} + 845629 p^{18} T^{42} + 83049 p^{19} T^{43} + 15242 p^{20} T^{44} + 1086 p^{21} T^{45} + 178 p^{22} T^{46} + 7 p^{23} T^{47} + p^{24} T^{48} \)
17 \( 1 + 5 T + 196 T^{2} + 953 T^{3} + 20118 T^{4} + 93589 T^{5} + 1419040 T^{6} + 6298130 T^{7} + 76729682 T^{8} + 324941324 T^{9} + 3372365560 T^{10} + 13637181236 T^{11} + 124863255491 T^{12} + 482454968427 T^{13} + 3987263064857 T^{14} + 14724716978108 T^{15} + 111594559982232 T^{16} + 393806534543507 T^{17} + 2768244311764421 T^{18} + 9328346686446386 T^{19} + 61335708206015226 T^{20} + 197114828338807222 T^{21} + 1220024124684727891 T^{22} + 3732011522908004653 T^{23} + 21848310537592014194 T^{24} + 3732011522908004653 p T^{25} + 1220024124684727891 p^{2} T^{26} + 197114828338807222 p^{3} T^{27} + 61335708206015226 p^{4} T^{28} + 9328346686446386 p^{5} T^{29} + 2768244311764421 p^{6} T^{30} + 393806534543507 p^{7} T^{31} + 111594559982232 p^{8} T^{32} + 14724716978108 p^{9} T^{33} + 3987263064857 p^{10} T^{34} + 482454968427 p^{11} T^{35} + 124863255491 p^{12} T^{36} + 13637181236 p^{13} T^{37} + 3372365560 p^{14} T^{38} + 324941324 p^{15} T^{39} + 76729682 p^{16} T^{40} + 6298130 p^{17} T^{41} + 1419040 p^{18} T^{42} + 93589 p^{19} T^{43} + 20118 p^{20} T^{44} + 953 p^{21} T^{45} + 196 p^{22} T^{46} + 5 p^{23} T^{47} + p^{24} T^{48} \)
19 \( 1 - 6 T + 287 T^{2} - 1529 T^{3} + 39866 T^{4} - 191304 T^{5} + 3601388 T^{6} - 15755815 T^{7} + 239549063 T^{8} - 965468251 T^{9} + 12575279515 T^{10} - 47110261125 T^{11} + 544489671719 T^{12} - 1910191709140 T^{13} + 20033324264927 T^{14} - 66202356249648 T^{15} + 639411650080977 T^{16} - 1998751361378699 T^{17} + 17960147394452645 T^{18} - 53244294915978078 T^{19} + 448337804727115297 T^{20} - 1261959301377331236 T^{21} + 10010151200115666534 T^{22} - 26747509136279656793 T^{23} + \)\(20\!\cdots\!02\)\( T^{24} - 26747509136279656793 p T^{25} + 10010151200115666534 p^{2} T^{26} - 1261959301377331236 p^{3} T^{27} + 448337804727115297 p^{4} T^{28} - 53244294915978078 p^{5} T^{29} + 17960147394452645 p^{6} T^{30} - 1998751361378699 p^{7} T^{31} + 639411650080977 p^{8} T^{32} - 66202356249648 p^{9} T^{33} + 20033324264927 p^{10} T^{34} - 1910191709140 p^{11} T^{35} + 544489671719 p^{12} T^{36} - 47110261125 p^{13} T^{37} + 12575279515 p^{14} T^{38} - 965468251 p^{15} T^{39} + 239549063 p^{16} T^{40} - 15755815 p^{17} T^{41} + 3601388 p^{18} T^{42} - 191304 p^{19} T^{43} + 39866 p^{20} T^{44} - 1529 p^{21} T^{45} + 287 p^{22} T^{46} - 6 p^{23} T^{47} + p^{24} T^{48} \)
23 \( 1 - 3 T + 305 T^{2} - 1084 T^{3} + 46890 T^{4} - 186158 T^{5} + 4848949 T^{6} - 20620219 T^{7} + 378859924 T^{8} - 1673223450 T^{9} + 23789130049 T^{10} - 106625527997 T^{11} + 1246280683423 T^{12} - 5571119577880 T^{13} + 55830282324894 T^{14} - 245625897781400 T^{15} + 2175125824978434 T^{16} - 9321675607324154 T^{17} + 74578922206885242 T^{18} - 308775678645504277 T^{19} + 2269121551033675617 T^{20} - 9012502911349543888 T^{21} + 61601093218170916017 T^{22} - \)\(23\!\cdots\!00\)\( T^{23} + \)\(14\!\cdots\!66\)\( T^{24} - \)\(23\!\cdots\!00\)\( p T^{25} + 61601093218170916017 p^{2} T^{26} - 9012502911349543888 p^{3} T^{27} + 2269121551033675617 p^{4} T^{28} - 308775678645504277 p^{5} T^{29} + 74578922206885242 p^{6} T^{30} - 9321675607324154 p^{7} T^{31} + 2175125824978434 p^{8} T^{32} - 245625897781400 p^{9} T^{33} + 55830282324894 p^{10} T^{34} - 5571119577880 p^{11} T^{35} + 1246280683423 p^{12} T^{36} - 106625527997 p^{13} T^{37} + 23789130049 p^{14} T^{38} - 1673223450 p^{15} T^{39} + 378859924 p^{16} T^{40} - 20620219 p^{17} T^{41} + 4848949 p^{18} T^{42} - 186158 p^{19} T^{43} + 46890 p^{20} T^{44} - 1084 p^{21} T^{45} + 305 p^{22} T^{46} - 3 p^{23} T^{47} + p^{24} T^{48} \)
29 \( 1 - 5 T + 410 T^{2} - 2000 T^{3} + 82905 T^{4} - 395267 T^{5} + 11044276 T^{6} - 51477487 T^{7} + 1092168846 T^{8} - 4970853887 T^{9} + 85624104007 T^{10} - 379688721126 T^{11} + 5548127909971 T^{12} - 23901410423526 T^{13} + 305750065899245 T^{14} - 1275660755311768 T^{15} + 14629184555493513 T^{16} - 58927461302472889 T^{17} + 617066405308731044 T^{18} - 2392196392048637710 T^{19} + 23204401414054861066 T^{20} - 86289647282416458491 T^{21} + \)\(78\!\cdots\!62\)\( T^{22} - 96062691955944972944 p T^{23} + \)\(23\!\cdots\!28\)\( T^{24} - 96062691955944972944 p^{2} T^{25} + \)\(78\!\cdots\!62\)\( p^{2} T^{26} - 86289647282416458491 p^{3} T^{27} + 23204401414054861066 p^{4} T^{28} - 2392196392048637710 p^{5} T^{29} + 617066405308731044 p^{6} T^{30} - 58927461302472889 p^{7} T^{31} + 14629184555493513 p^{8} T^{32} - 1275660755311768 p^{9} T^{33} + 305750065899245 p^{10} T^{34} - 23901410423526 p^{11} T^{35} + 5548127909971 p^{12} T^{36} - 379688721126 p^{13} T^{37} + 85624104007 p^{14} T^{38} - 4970853887 p^{15} T^{39} + 1092168846 p^{16} T^{40} - 51477487 p^{17} T^{41} + 11044276 p^{18} T^{42} - 395267 p^{19} T^{43} + 82905 p^{20} T^{44} - 2000 p^{21} T^{45} + 410 p^{22} T^{46} - 5 p^{23} T^{47} + p^{24} T^{48} \)
31 \( 1 - 13 T + 506 T^{2} - 5468 T^{3} + 119641 T^{4} - 1111901 T^{5} + 577371 p T^{6} - 146449325 T^{7} + 1927088027 T^{8} - 14119240456 T^{9} + 160623722686 T^{10} - 1067245101057 T^{11} + 10869030053435 T^{12} - 66145521243671 T^{13} + 617748927584444 T^{14} - 3472276699269672 T^{15} + 30272728310298490 T^{16} - 158395811007190191 T^{17} + 1306748388430145380 T^{18} - 6415505871536368057 T^{19} + 50577175110078398144 T^{20} - \)\(23\!\cdots\!42\)\( T^{21} + \)\(17\!\cdots\!63\)\( T^{22} - \)\(78\!\cdots\!23\)\( T^{23} + \)\(57\!\cdots\!84\)\( T^{24} - \)\(78\!\cdots\!23\)\( p T^{25} + \)\(17\!\cdots\!63\)\( p^{2} T^{26} - \)\(23\!\cdots\!42\)\( p^{3} T^{27} + 50577175110078398144 p^{4} T^{28} - 6415505871536368057 p^{5} T^{29} + 1306748388430145380 p^{6} T^{30} - 158395811007190191 p^{7} T^{31} + 30272728310298490 p^{8} T^{32} - 3472276699269672 p^{9} T^{33} + 617748927584444 p^{10} T^{34} - 66145521243671 p^{11} T^{35} + 10869030053435 p^{12} T^{36} - 1067245101057 p^{13} T^{37} + 160623722686 p^{14} T^{38} - 14119240456 p^{15} T^{39} + 1927088027 p^{16} T^{40} - 146449325 p^{17} T^{41} + 577371 p^{19} T^{42} - 1111901 p^{19} T^{43} + 119641 p^{20} T^{44} - 5468 p^{21} T^{45} + 506 p^{22} T^{46} - 13 p^{23} T^{47} + p^{24} T^{48} \)
37 \( 1 - 2 T + 491 T^{2} - 1012 T^{3} + 118767 T^{4} - 257509 T^{5} + 18858017 T^{6} - 43389833 T^{7} + 2211197952 T^{8} - 5391999819 T^{9} + 204445412267 T^{10} - 523262701957 T^{11} + 15563810668784 T^{12} - 41121767271361 T^{13} + 1007213943826698 T^{14} - 2688462159660747 T^{15} + 56839844206309233 T^{16} - 149702845521671830 T^{17} + 2856033950757943815 T^{18} - 7270198928672531681 T^{19} + \)\(12\!\cdots\!41\)\( T^{20} - \)\(31\!\cdots\!06\)\( T^{21} + \)\(14\!\cdots\!75\)\( p T^{22} - \)\(12\!\cdots\!43\)\( T^{23} + \)\(20\!\cdots\!62\)\( T^{24} - \)\(12\!\cdots\!43\)\( p T^{25} + \)\(14\!\cdots\!75\)\( p^{3} T^{26} - \)\(31\!\cdots\!06\)\( p^{3} T^{27} + \)\(12\!\cdots\!41\)\( p^{4} T^{28} - 7270198928672531681 p^{5} T^{29} + 2856033950757943815 p^{6} T^{30} - 149702845521671830 p^{7} T^{31} + 56839844206309233 p^{8} T^{32} - 2688462159660747 p^{9} T^{33} + 1007213943826698 p^{10} T^{34} - 41121767271361 p^{11} T^{35} + 15563810668784 p^{12} T^{36} - 523262701957 p^{13} T^{37} + 204445412267 p^{14} T^{38} - 5391999819 p^{15} T^{39} + 2211197952 p^{16} T^{40} - 43389833 p^{17} T^{41} + 18858017 p^{18} T^{42} - 257509 p^{19} T^{43} + 118767 p^{20} T^{44} - 1012 p^{21} T^{45} + 491 p^{22} T^{46} - 2 p^{23} T^{47} + p^{24} T^{48} \)
41 \( 1 - 25 T + 21 p T^{2} - 15888 T^{3} + 329218 T^{4} - 4949575 T^{5} + 78432242 T^{6} - 1011502192 T^{7} + 13388373142 T^{8} - 152736335488 T^{9} + 1765122264001 T^{10} - 18170406524421 T^{11} + 188150310082698 T^{12} - 1771444864118803 T^{13} + 16708173063422946 T^{14} - 145233274825224226 T^{15} + 1261573573599584158 T^{16} - 10191274109410887537 T^{17} + 82147274286769154689 T^{18} - \)\(61\!\cdots\!09\)\( T^{19} + \)\(46\!\cdots\!76\)\( T^{20} - \)\(32\!\cdots\!08\)\( T^{21} + \)\(23\!\cdots\!25\)\( T^{22} - \)\(15\!\cdots\!00\)\( T^{23} + \)\(10\!\cdots\!30\)\( T^{24} - \)\(15\!\cdots\!00\)\( p T^{25} + \)\(23\!\cdots\!25\)\( p^{2} T^{26} - \)\(32\!\cdots\!08\)\( p^{3} T^{27} + \)\(46\!\cdots\!76\)\( p^{4} T^{28} - \)\(61\!\cdots\!09\)\( p^{5} T^{29} + 82147274286769154689 p^{6} T^{30} - 10191274109410887537 p^{7} T^{31} + 1261573573599584158 p^{8} T^{32} - 145233274825224226 p^{9} T^{33} + 16708173063422946 p^{10} T^{34} - 1771444864118803 p^{11} T^{35} + 188150310082698 p^{12} T^{36} - 18170406524421 p^{13} T^{37} + 1765122264001 p^{14} T^{38} - 152736335488 p^{15} T^{39} + 13388373142 p^{16} T^{40} - 1011502192 p^{17} T^{41} + 78432242 p^{18} T^{42} - 4949575 p^{19} T^{43} + 329218 p^{20} T^{44} - 15888 p^{21} T^{45} + 21 p^{23} T^{46} - 25 p^{23} T^{47} + p^{24} T^{48} \)
43 \( 1 + 15 T + 584 T^{2} + 7834 T^{3} + 169110 T^{4} + 2069576 T^{5} + 761448 p T^{6} + 369212873 T^{7} + 4784034435 T^{8} + 49978061719 T^{9} + 561971834828 T^{10} + 5461691657707 T^{11} + 55109814642767 T^{12} + 500300925467266 T^{13} + 4623084466079576 T^{14} + 39358126015375210 T^{15} + 337331499300421401 T^{16} + 2702504627388708253 T^{17} + 21660162821243673096 T^{18} + \)\(16\!\cdots\!90\)\( T^{19} + \)\(12\!\cdots\!55\)\( T^{20} + \)\(88\!\cdots\!77\)\( T^{21} + \)\(62\!\cdots\!76\)\( T^{22} + \)\(22\!\cdots\!80\)\( p^{2} T^{23} + \)\(28\!\cdots\!58\)\( T^{24} + \)\(22\!\cdots\!80\)\( p^{3} T^{25} + \)\(62\!\cdots\!76\)\( p^{2} T^{26} + \)\(88\!\cdots\!77\)\( p^{3} T^{27} + \)\(12\!\cdots\!55\)\( p^{4} T^{28} + \)\(16\!\cdots\!90\)\( p^{5} T^{29} + 21660162821243673096 p^{6} T^{30} + 2702504627388708253 p^{7} T^{31} + 337331499300421401 p^{8} T^{32} + 39358126015375210 p^{9} T^{33} + 4623084466079576 p^{10} T^{34} + 500300925467266 p^{11} T^{35} + 55109814642767 p^{12} T^{36} + 5461691657707 p^{13} T^{37} + 561971834828 p^{14} T^{38} + 49978061719 p^{15} T^{39} + 4784034435 p^{16} T^{40} + 369212873 p^{17} T^{41} + 761448 p^{19} T^{42} + 2069576 p^{19} T^{43} + 169110 p^{20} T^{44} + 7834 p^{21} T^{45} + 584 p^{22} T^{46} + 15 p^{23} T^{47} + p^{24} T^{48} \)
47 \( 1 - 35 T + 1174 T^{2} - 25902 T^{3} + 533688 T^{4} - 8947063 T^{5} + 141127760 T^{6} - 1947372766 T^{7} + 25513637069 T^{8} - 302627509031 T^{9} + 3434473776051 T^{10} - 35971677944828 T^{11} + 362787085688574 T^{12} - 3417314951500364 T^{13} + 31176517773706026 T^{14} - 267872764206229066 T^{15} + 2242775978234119450 T^{16} - 17806909672984508105 T^{17} + \)\(13\!\cdots\!14\)\( T^{18} - \)\(10\!\cdots\!22\)\( T^{19} + \)\(76\!\cdots\!26\)\( T^{20} - \)\(54\!\cdots\!89\)\( T^{21} + \)\(38\!\cdots\!19\)\( T^{22} - \)\(26\!\cdots\!79\)\( T^{23} + \)\(18\!\cdots\!16\)\( T^{24} - \)\(26\!\cdots\!79\)\( p T^{25} + \)\(38\!\cdots\!19\)\( p^{2} T^{26} - \)\(54\!\cdots\!89\)\( p^{3} T^{27} + \)\(76\!\cdots\!26\)\( p^{4} T^{28} - \)\(10\!\cdots\!22\)\( p^{5} T^{29} + \)\(13\!\cdots\!14\)\( p^{6} T^{30} - 17806909672984508105 p^{7} T^{31} + 2242775978234119450 p^{8} T^{32} - 267872764206229066 p^{9} T^{33} + 31176517773706026 p^{10} T^{34} - 3417314951500364 p^{11} T^{35} + 362787085688574 p^{12} T^{36} - 35971677944828 p^{13} T^{37} + 3434473776051 p^{14} T^{38} - 302627509031 p^{15} T^{39} + 25513637069 p^{16} T^{40} - 1947372766 p^{17} T^{41} + 141127760 p^{18} T^{42} - 8947063 p^{19} T^{43} + 533688 p^{20} T^{44} - 25902 p^{21} T^{45} + 1174 p^{22} T^{46} - 35 p^{23} T^{47} + p^{24} T^{48} \)
53 \( 1 - 2 T + 624 T^{2} - 493 T^{3} + 192184 T^{4} + 75325 T^{5} + 39340369 T^{6} + 60080942 T^{7} + 6072915611 T^{8} + 15733451762 T^{9} + 758289954057 T^{10} + 2698449928394 T^{11} + 79967266151419 T^{12} + 352647041373912 T^{13} + 7326849934531288 T^{14} + 37539209306694723 T^{15} + 594749671751158866 T^{16} + 3375475136740726636 T^{17} + 43369719151216128761 T^{18} + \)\(26\!\cdots\!48\)\( T^{19} + \)\(28\!\cdots\!59\)\( T^{20} + \)\(33\!\cdots\!78\)\( p T^{21} + \)\(17\!\cdots\!29\)\( T^{22} + \)\(10\!\cdots\!27\)\( T^{23} + \)\(95\!\cdots\!60\)\( T^{24} + \)\(10\!\cdots\!27\)\( p T^{25} + \)\(17\!\cdots\!29\)\( p^{2} T^{26} + \)\(33\!\cdots\!78\)\( p^{4} T^{27} + \)\(28\!\cdots\!59\)\( p^{4} T^{28} + \)\(26\!\cdots\!48\)\( p^{5} T^{29} + 43369719151216128761 p^{6} T^{30} + 3375475136740726636 p^{7} T^{31} + 594749671751158866 p^{8} T^{32} + 37539209306694723 p^{9} T^{33} + 7326849934531288 p^{10} T^{34} + 352647041373912 p^{11} T^{35} + 79967266151419 p^{12} T^{36} + 2698449928394 p^{13} T^{37} + 758289954057 p^{14} T^{38} + 15733451762 p^{15} T^{39} + 6072915611 p^{16} T^{40} + 60080942 p^{17} T^{41} + 39340369 p^{18} T^{42} + 75325 p^{19} T^{43} + 192184 p^{20} T^{44} - 493 p^{21} T^{45} + 624 p^{22} T^{46} - 2 p^{23} T^{47} + p^{24} T^{48} \)
59 \( 1 - 35 T + 1499 T^{2} - 36514 T^{3} + 938104 T^{4} - 18007424 T^{5} + 350958449 T^{6} - 5648860937 T^{7} + 91167004314 T^{8} - 1276761689078 T^{9} + 17856070264089 T^{10} - 222964259734491 T^{11} + 2778062036365215 T^{12} - 31465117388285622 T^{13} + 355793966571612492 T^{14} - 3701177923627146710 T^{15} + 38475231180912288714 T^{16} - \)\(37\!\cdots\!40\)\( T^{17} + \)\(35\!\cdots\!48\)\( T^{18} - \)\(32\!\cdots\!97\)\( T^{19} + \)\(28\!\cdots\!63\)\( T^{20} - \)\(24\!\cdots\!64\)\( T^{21} + \)\(20\!\cdots\!15\)\( T^{22} - \)\(16\!\cdots\!54\)\( T^{23} + \)\(12\!\cdots\!54\)\( T^{24} - \)\(16\!\cdots\!54\)\( p T^{25} + \)\(20\!\cdots\!15\)\( p^{2} T^{26} - \)\(24\!\cdots\!64\)\( p^{3} T^{27} + \)\(28\!\cdots\!63\)\( p^{4} T^{28} - \)\(32\!\cdots\!97\)\( p^{5} T^{29} + \)\(35\!\cdots\!48\)\( p^{6} T^{30} - \)\(37\!\cdots\!40\)\( p^{7} T^{31} + 38475231180912288714 p^{8} T^{32} - 3701177923627146710 p^{9} T^{33} + 355793966571612492 p^{10} T^{34} - 31465117388285622 p^{11} T^{35} + 2778062036365215 p^{12} T^{36} - 222964259734491 p^{13} T^{37} + 17856070264089 p^{14} T^{38} - 1276761689078 p^{15} T^{39} + 91167004314 p^{16} T^{40} - 5648860937 p^{17} T^{41} + 350958449 p^{18} T^{42} - 18007424 p^{19} T^{43} + 938104 p^{20} T^{44} - 36514 p^{21} T^{45} + 1499 p^{22} T^{46} - 35 p^{23} T^{47} + p^{24} T^{48} \)
61 \( 1 + 7 T + 796 T^{2} + 6384 T^{3} + 326728 T^{4} + 2867271 T^{5} + 91359943 T^{6} + 849983213 T^{7} + 19432248452 T^{8} + 187182708308 T^{9} + 3331926227602 T^{10} + 32618194575876 T^{11} + 476916135411979 T^{12} + 4674149669674084 T^{13} + 58291677589141069 T^{14} + 564854371778035523 T^{15} + 6178093844046151776 T^{16} + 58560053642586863263 T^{17} + \)\(57\!\cdots\!03\)\( T^{18} + \)\(52\!\cdots\!10\)\( T^{19} + \)\(47\!\cdots\!74\)\( T^{20} + \)\(41\!\cdots\!31\)\( T^{21} + \)\(34\!\cdots\!78\)\( T^{22} + \)\(28\!\cdots\!46\)\( T^{23} + \)\(22\!\cdots\!30\)\( T^{24} + \)\(28\!\cdots\!46\)\( p T^{25} + \)\(34\!\cdots\!78\)\( p^{2} T^{26} + \)\(41\!\cdots\!31\)\( p^{3} T^{27} + \)\(47\!\cdots\!74\)\( p^{4} T^{28} + \)\(52\!\cdots\!10\)\( p^{5} T^{29} + \)\(57\!\cdots\!03\)\( p^{6} T^{30} + 58560053642586863263 p^{7} T^{31} + 6178093844046151776 p^{8} T^{32} + 564854371778035523 p^{9} T^{33} + 58291677589141069 p^{10} T^{34} + 4674149669674084 p^{11} T^{35} + 476916135411979 p^{12} T^{36} + 32618194575876 p^{13} T^{37} + 3331926227602 p^{14} T^{38} + 187182708308 p^{15} T^{39} + 19432248452 p^{16} T^{40} + 849983213 p^{17} T^{41} + 91359943 p^{18} T^{42} + 2867271 p^{19} T^{43} + 326728 p^{20} T^{44} + 6384 p^{21} T^{45} + 796 p^{22} T^{46} + 7 p^{23} T^{47} + p^{24} T^{48} \)
67 \( 1 - 10 T + 847 T^{2} - 8317 T^{3} + 351179 T^{4} - 3348780 T^{5} + 94545444 T^{6} - 865947494 T^{7} + 18495590049 T^{8} - 160851656944 T^{9} + 2788800998846 T^{10} - 22740196420111 T^{11} + 335616711585827 T^{12} - 2528223131386142 T^{13} + 32954855867774207 T^{14} - 225120135719880201 T^{15} + 2680985354324953432 T^{16} - 16198601507753907929 T^{17} + \)\(18\!\cdots\!86\)\( T^{18} - \)\(94\!\cdots\!33\)\( T^{19} + \)\(10\!\cdots\!10\)\( T^{20} - \)\(46\!\cdots\!35\)\( T^{21} + \)\(60\!\cdots\!50\)\( T^{22} - \)\(22\!\cdots\!48\)\( T^{23} + \)\(36\!\cdots\!76\)\( T^{24} - \)\(22\!\cdots\!48\)\( p T^{25} + \)\(60\!\cdots\!50\)\( p^{2} T^{26} - \)\(46\!\cdots\!35\)\( p^{3} T^{27} + \)\(10\!\cdots\!10\)\( p^{4} T^{28} - \)\(94\!\cdots\!33\)\( p^{5} T^{29} + \)\(18\!\cdots\!86\)\( p^{6} T^{30} - 16198601507753907929 p^{7} T^{31} + 2680985354324953432 p^{8} T^{32} - 225120135719880201 p^{9} T^{33} + 32954855867774207 p^{10} T^{34} - 2528223131386142 p^{11} T^{35} + 335616711585827 p^{12} T^{36} - 22740196420111 p^{13} T^{37} + 2788800998846 p^{14} T^{38} - 160851656944 p^{15} T^{39} + 18495590049 p^{16} T^{40} - 865947494 p^{17} T^{41} + 94545444 p^{18} T^{42} - 3348780 p^{19} T^{43} + 351179 p^{20} T^{44} - 8317 p^{21} T^{45} + 847 p^{22} T^{46} - 10 p^{23} T^{47} + p^{24} T^{48} \)
71 \( 1 - 58 T + 2614 T^{2} - 86448 T^{3} + 2463696 T^{4} - 60384335 T^{5} + 1334159343 T^{6} - 26681629993 T^{7} + 492810404136 T^{8} - 8440526422165 T^{9} + 135479964016856 T^{10} - 2044485716605035 T^{11} + 29190368874443930 T^{12} - 395254723405321750 T^{13} + 5096509340691741298 T^{14} - 62687271237276662184 T^{15} + \)\(73\!\cdots\!11\)\( T^{16} - \)\(83\!\cdots\!66\)\( T^{17} + \)\(89\!\cdots\!66\)\( T^{18} - \)\(93\!\cdots\!67\)\( T^{19} + \)\(93\!\cdots\!85\)\( T^{20} - \)\(89\!\cdots\!42\)\( T^{21} + \)\(82\!\cdots\!98\)\( T^{22} - \)\(73\!\cdots\!25\)\( T^{23} + \)\(63\!\cdots\!88\)\( T^{24} - \)\(73\!\cdots\!25\)\( p T^{25} + \)\(82\!\cdots\!98\)\( p^{2} T^{26} - \)\(89\!\cdots\!42\)\( p^{3} T^{27} + \)\(93\!\cdots\!85\)\( p^{4} T^{28} - \)\(93\!\cdots\!67\)\( p^{5} T^{29} + \)\(89\!\cdots\!66\)\( p^{6} T^{30} - \)\(83\!\cdots\!66\)\( p^{7} T^{31} + \)\(73\!\cdots\!11\)\( p^{8} T^{32} - 62687271237276662184 p^{9} T^{33} + 5096509340691741298 p^{10} T^{34} - 395254723405321750 p^{11} T^{35} + 29190368874443930 p^{12} T^{36} - 2044485716605035 p^{13} T^{37} + 135479964016856 p^{14} T^{38} - 8440526422165 p^{15} T^{39} + 492810404136 p^{16} T^{40} - 26681629993 p^{17} T^{41} + 1334159343 p^{18} T^{42} - 60384335 p^{19} T^{43} + 2463696 p^{20} T^{44} - 86448 p^{21} T^{45} + 2614 p^{22} T^{46} - 58 p^{23} T^{47} + p^{24} T^{48} \)
73 \( 1 - 9 T + 716 T^{2} - 7415 T^{3} + 262777 T^{4} - 2944072 T^{5} + 66247763 T^{6} - 762011687 T^{7} + 12852939167 T^{8} - 146461143921 T^{9} + 2033676325700 T^{10} - 22511570954773 T^{11} + 272588929851293 T^{12} - 2903780739134311 T^{13} + 31895758614160945 T^{14} - 325556072902761764 T^{15} + 3338993147904997320 T^{16} - 32599976582023540819 T^{17} + \)\(31\!\cdots\!91\)\( T^{18} - \)\(29\!\cdots\!58\)\( T^{19} + \)\(28\!\cdots\!34\)\( T^{20} - \)\(34\!\cdots\!15\)\( p T^{21} + \)\(22\!\cdots\!05\)\( T^{22} - \)\(19\!\cdots\!12\)\( T^{23} + \)\(17\!\cdots\!04\)\( T^{24} - \)\(19\!\cdots\!12\)\( p T^{25} + \)\(22\!\cdots\!05\)\( p^{2} T^{26} - \)\(34\!\cdots\!15\)\( p^{4} T^{27} + \)\(28\!\cdots\!34\)\( p^{4} T^{28} - \)\(29\!\cdots\!58\)\( p^{5} T^{29} + \)\(31\!\cdots\!91\)\( p^{6} T^{30} - 32599976582023540819 p^{7} T^{31} + 3338993147904997320 p^{8} T^{32} - 325556072902761764 p^{9} T^{33} + 31895758614160945 p^{10} T^{34} - 2903780739134311 p^{11} T^{35} + 272588929851293 p^{12} T^{36} - 22511570954773 p^{13} T^{37} + 2033676325700 p^{14} T^{38} - 146461143921 p^{15} T^{39} + 12852939167 p^{16} T^{40} - 762011687 p^{17} T^{41} + 66247763 p^{18} T^{42} - 2944072 p^{19} T^{43} + 262777 p^{20} T^{44} - 7415 p^{21} T^{45} + 716 p^{22} T^{46} - 9 p^{23} T^{47} + p^{24} T^{48} \)
79 \( 1 - 31 T + 1449 T^{2} - 36989 T^{3} + 1023239 T^{4} - 21984222 T^{5} + 466434597 T^{6} - 8649048001 T^{7} + 154280880620 T^{8} - 2524314340574 T^{9} + 39532034589256 T^{10} - 580896743779752 T^{11} + 8181373713537081 T^{12} - 109437759610976819 T^{13} + 1407187641420226917 T^{14} - 17310623302243406434 T^{15} + 2598575629767551637 p T^{16} - \)\(23\!\cdots\!24\)\( T^{17} + \)\(25\!\cdots\!75\)\( T^{18} - \)\(27\!\cdots\!72\)\( T^{19} + \)\(28\!\cdots\!96\)\( T^{20} - \)\(27\!\cdots\!50\)\( T^{21} + \)\(26\!\cdots\!66\)\( T^{22} - \)\(25\!\cdots\!12\)\( T^{23} + \)\(22\!\cdots\!20\)\( T^{24} - \)\(25\!\cdots\!12\)\( p T^{25} + \)\(26\!\cdots\!66\)\( p^{2} T^{26} - \)\(27\!\cdots\!50\)\( p^{3} T^{27} + \)\(28\!\cdots\!96\)\( p^{4} T^{28} - \)\(27\!\cdots\!72\)\( p^{5} T^{29} + \)\(25\!\cdots\!75\)\( p^{6} T^{30} - \)\(23\!\cdots\!24\)\( p^{7} T^{31} + 2598575629767551637 p^{9} T^{32} - 17310623302243406434 p^{9} T^{33} + 1407187641420226917 p^{10} T^{34} - 109437759610976819 p^{11} T^{35} + 8181373713537081 p^{12} T^{36} - 580896743779752 p^{13} T^{37} + 39532034589256 p^{14} T^{38} - 2524314340574 p^{15} T^{39} + 154280880620 p^{16} T^{40} - 8649048001 p^{17} T^{41} + 466434597 p^{18} T^{42} - 21984222 p^{19} T^{43} + 1023239 p^{20} T^{44} - 36989 p^{21} T^{45} + 1449 p^{22} T^{46} - 31 p^{23} T^{47} + p^{24} T^{48} \)
83 \( 1 + T + 1002 T^{2} + 535 T^{3} + 508461 T^{4} + 65737 T^{5} + 174003389 T^{6} - 37154900 T^{7} + 45122687080 T^{8} - 22274012001 T^{9} + 9445471685217 T^{10} - 6725278890574 T^{11} + 1659966752719149 T^{12} - 1448099495339411 T^{13} + 251453618199547640 T^{14} - 246628593237975121 T^{15} + 33443471250692967936 T^{16} - 34943342986473173630 T^{17} + \)\(39\!\cdots\!98\)\( T^{18} - \)\(42\!\cdots\!50\)\( T^{19} + \)\(42\!\cdots\!30\)\( T^{20} - \)\(44\!\cdots\!95\)\( T^{21} + \)\(40\!\cdots\!06\)\( T^{22} - \)\(42\!\cdots\!81\)\( T^{23} + \)\(35\!\cdots\!34\)\( T^{24} - \)\(42\!\cdots\!81\)\( p T^{25} + \)\(40\!\cdots\!06\)\( p^{2} T^{26} - \)\(44\!\cdots\!95\)\( p^{3} T^{27} + \)\(42\!\cdots\!30\)\( p^{4} T^{28} - \)\(42\!\cdots\!50\)\( p^{5} T^{29} + \)\(39\!\cdots\!98\)\( p^{6} T^{30} - 34943342986473173630 p^{7} T^{31} + 33443471250692967936 p^{8} T^{32} - 246628593237975121 p^{9} T^{33} + 251453618199547640 p^{10} T^{34} - 1448099495339411 p^{11} T^{35} + 1659966752719149 p^{12} T^{36} - 6725278890574 p^{13} T^{37} + 9445471685217 p^{14} T^{38} - 22274012001 p^{15} T^{39} + 45122687080 p^{16} T^{40} - 37154900 p^{17} T^{41} + 174003389 p^{18} T^{42} + 65737 p^{19} T^{43} + 508461 p^{20} T^{44} + 535 p^{21} T^{45} + 1002 p^{22} T^{46} + p^{23} T^{47} + p^{24} T^{48} \)
89 \( 1 - 45 T + 2000 T^{2} - 58283 T^{3} + 1592195 T^{4} - 35385877 T^{5} + 737921282 T^{6} - 13433372421 T^{7} + 230860714937 T^{8} - 3571743031435 T^{9} + 52423637469959 T^{10} - 702360481211805 T^{11} + 8947153326685256 T^{12} - 104323142362853702 T^{13} + 1153927643937484991 T^{14} - 11553926088854732159 T^{15} + \)\(10\!\cdots\!39\)\( T^{16} - \)\(87\!\cdots\!04\)\( T^{17} + \)\(61\!\cdots\!94\)\( T^{18} - \)\(26\!\cdots\!21\)\( T^{19} - \)\(56\!\cdots\!57\)\( T^{20} + \)\(38\!\cdots\!95\)\( T^{21} - \)\(58\!\cdots\!41\)\( T^{22} + \)\(73\!\cdots\!25\)\( T^{23} - \)\(72\!\cdots\!12\)\( T^{24} + \)\(73\!\cdots\!25\)\( p T^{25} - \)\(58\!\cdots\!41\)\( p^{2} T^{26} + \)\(38\!\cdots\!95\)\( p^{3} T^{27} - \)\(56\!\cdots\!57\)\( p^{4} T^{28} - \)\(26\!\cdots\!21\)\( p^{5} T^{29} + \)\(61\!\cdots\!94\)\( p^{6} T^{30} - \)\(87\!\cdots\!04\)\( p^{7} T^{31} + \)\(10\!\cdots\!39\)\( p^{8} T^{32} - 11553926088854732159 p^{9} T^{33} + 1153927643937484991 p^{10} T^{34} - 104323142362853702 p^{11} T^{35} + 8947153326685256 p^{12} T^{36} - 702360481211805 p^{13} T^{37} + 52423637469959 p^{14} T^{38} - 3571743031435 p^{15} T^{39} + 230860714937 p^{16} T^{40} - 13433372421 p^{17} T^{41} + 737921282 p^{18} T^{42} - 35385877 p^{19} T^{43} + 1592195 p^{20} T^{44} - 58283 p^{21} T^{45} + 2000 p^{22} T^{46} - 45 p^{23} T^{47} + p^{24} T^{48} \)
97 \( 1 + 9 T + 1138 T^{2} + 9132 T^{3} + 6518 p T^{4} + 4607644 T^{5} + 230556935 T^{6} + 1558007421 T^{7} + 62656424720 T^{8} + 400906262453 T^{9} + 13654999027862 T^{10} + 84138417870400 T^{11} + 2503593036950029 T^{12} + 14994717742628511 T^{13} + 399108193981778238 T^{14} + 2326364318589261974 T^{15} + 56622914995519057231 T^{16} + \)\(31\!\cdots\!54\)\( T^{17} + \)\(72\!\cdots\!67\)\( T^{18} + \)\(39\!\cdots\!02\)\( T^{19} + \)\(85\!\cdots\!59\)\( T^{20} + \)\(44\!\cdots\!23\)\( T^{21} + \)\(92\!\cdots\!12\)\( T^{22} + \)\(46\!\cdots\!89\)\( T^{23} + \)\(93\!\cdots\!40\)\( T^{24} + \)\(46\!\cdots\!89\)\( p T^{25} + \)\(92\!\cdots\!12\)\( p^{2} T^{26} + \)\(44\!\cdots\!23\)\( p^{3} T^{27} + \)\(85\!\cdots\!59\)\( p^{4} T^{28} + \)\(39\!\cdots\!02\)\( p^{5} T^{29} + \)\(72\!\cdots\!67\)\( p^{6} T^{30} + \)\(31\!\cdots\!54\)\( p^{7} T^{31} + 56622914995519057231 p^{8} T^{32} + 2326364318589261974 p^{9} T^{33} + 399108193981778238 p^{10} T^{34} + 14994717742628511 p^{11} T^{35} + 2503593036950029 p^{12} T^{36} + 84138417870400 p^{13} T^{37} + 13654999027862 p^{14} T^{38} + 400906262453 p^{15} T^{39} + 62656424720 p^{16} T^{40} + 1558007421 p^{17} T^{41} + 230556935 p^{18} T^{42} + 4607644 p^{19} T^{43} + 6518 p^{21} T^{44} + 9132 p^{21} T^{45} + 1138 p^{22} T^{46} + 9 p^{23} T^{47} + p^{24} T^{48} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.21432741836490434637083578161, −1.21161353638808184750685067757, −1.13959688450968355724241334381, −1.04343208229475818274465379360, −0.951335473960372593942065499980, −0.943225675075969223661102077749, −0.938636392979836618123654551160, −0.852565163474016829081619831074, −0.824166769725427444514092361864, −0.76433237562782662723588425727, −0.64015562112288881359343420960, −0.60072733068810213797099636593, −0.58436541222449823467147406233, −0.54185854538306070155732138915, −0.52105020623183248076912944279, −0.51152000154552941138218756977, −0.50653418709760305676460211404, −0.45877292100159491106272449041, −0.44961642214952977618446974730, −0.44342655580290455017389707574, −0.42354721186780244481109080566, −0.31686712646607045448801382303, −0.29895669106892863316162302198, −0.23172483262533671477432803943, −0.22015366979512233102814908325, 0.22015366979512233102814908325, 0.23172483262533671477432803943, 0.29895669106892863316162302198, 0.31686712646607045448801382303, 0.42354721186780244481109080566, 0.44342655580290455017389707574, 0.44961642214952977618446974730, 0.45877292100159491106272449041, 0.50653418709760305676460211404, 0.51152000154552941138218756977, 0.52105020623183248076912944279, 0.54185854538306070155732138915, 0.58436541222449823467147406233, 0.60072733068810213797099636593, 0.64015562112288881359343420960, 0.76433237562782662723588425727, 0.824166769725427444514092361864, 0.852565163474016829081619831074, 0.938636392979836618123654551160, 0.943225675075969223661102077749, 0.951335473960372593942065499980, 1.04343208229475818274465379360, 1.13959688450968355724241334381, 1.21161353638808184750685067757, 1.21432741836490434637083578161

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

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