Dirichlet series
| L(s) = 1 | − 16·3-s + 4·4-s + 8·7-s + 136·9-s − 64·12-s − 10·13-s + 6·16-s − 128·21-s − 16·23-s − 20·25-s − 816·27-s + 32·28-s − 4·29-s + 12·31-s + 544·36-s + 30·37-s + 160·39-s − 18·41-s − 32·43-s + 66·47-s − 96·48-s + 31·49-s − 40·52-s + 2·53-s − 36·59-s − 8·61-s + 1.08e3·63-s + ⋯ |
| L(s) = 1 | − 9.23·3-s + 2·4-s + 3.02·7-s + 45.3·9-s − 18.4·12-s − 2.77·13-s + 3/2·16-s − 27.9·21-s − 3.33·23-s − 4·25-s − 157.·27-s + 6.04·28-s − 0.742·29-s + 2.15·31-s + 90.6·36-s + 4.93·37-s + 25.6·39-s − 2.81·41-s − 4.87·43-s + 9.62·47-s − 13.8·48-s + 31/7·49-s − 5.54·52-s + 0.274·53-s − 4.68·59-s − 1.02·61-s + 137.·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.70418\times 10^{10}\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1467784049\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1467784049\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( ( 1 + T )^{16} \) | |
| 7 | \( 1 - 8 T + 33 T^{2} - 116 T^{3} + 46 p T^{4} - 570 T^{5} + 419 T^{6} + 1346 T^{7} - 6261 T^{8} + 1346 p T^{9} + 419 p^{2} T^{10} - 570 p^{3} T^{11} + 46 p^{5} T^{12} - 116 p^{5} T^{13} + 33 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 10 T + 23 T^{2} - 6 p T^{3} - 506 T^{4} - 1726 T^{5} - 4623 T^{6} + 17920 T^{7} + 163851 T^{8} + 17920 p T^{9} - 4623 p^{2} T^{10} - 1726 p^{3} T^{11} - 506 p^{4} T^{12} - 6 p^{6} T^{13} + 23 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 + 4 p T^{2} + 47 p T^{4} - 24 p T^{5} + 1888 T^{6} - 1938 T^{7} + 11808 T^{8} - 3792 p T^{9} + 67872 T^{10} - 121746 T^{11} + 76729 p T^{12} - 120954 p T^{13} + 2201808 T^{14} - 2694924 T^{15} + 11530099 T^{16} - 2694924 p T^{17} + 2201808 p^{2} T^{18} - 120954 p^{4} T^{19} + 76729 p^{5} T^{20} - 121746 p^{5} T^{21} + 67872 p^{6} T^{22} - 3792 p^{8} T^{23} + 11808 p^{8} T^{24} - 1938 p^{9} T^{25} + 1888 p^{10} T^{26} - 24 p^{12} T^{27} + 47 p^{13} T^{28} + 4 p^{15} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 - 108 T^{2} + 524 p T^{4} - 202678 T^{6} + 5279774 T^{8} - 108510450 T^{10} + 1826178193 T^{12} - 25727414170 T^{14} + 27905582807 p T^{16} - 25727414170 p^{2} T^{18} + 1826178193 p^{4} T^{20} - 108510450 p^{6} T^{22} + 5279774 p^{8} T^{24} - 202678 p^{10} T^{26} + 524 p^{13} T^{28} - 108 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 82 T^{2} - 4 T^{3} + 3384 T^{4} + 280 T^{5} - 92762 T^{6} - 10452 T^{7} + 1942613 T^{8} + 116028 T^{9} - 35540397 T^{10} + 4647570 T^{11} + 643529387 T^{12} - 174124440 T^{13} - 11865897791 T^{14} + 1612148132 T^{15} + 209658786411 T^{16} + 1612148132 p T^{17} - 11865897791 p^{2} T^{18} - 174124440 p^{3} T^{19} + 643529387 p^{4} T^{20} + 4647570 p^{5} T^{21} - 35540397 p^{6} T^{22} + 116028 p^{7} T^{23} + 1942613 p^{8} T^{24} - 10452 p^{9} T^{25} - 92762 p^{10} T^{26} + 280 p^{11} T^{27} + 3384 p^{12} T^{28} - 4 p^{13} T^{29} - 82 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 128 T^{2} + 8690 T^{4} - 415208 T^{6} + 15550811 T^{8} - 482959440 T^{10} + 12836439948 T^{12} - 296817904496 T^{14} + 6016663636925 T^{16} - 296817904496 p^{2} T^{18} + 12836439948 p^{4} T^{20} - 482959440 p^{6} T^{22} + 15550811 p^{8} T^{24} - 415208 p^{10} T^{26} + 8690 p^{12} T^{28} - 128 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 16 T + 66 T^{2} - 140 T^{3} - 166 T^{4} + 16884 T^{5} + 96786 T^{6} + 191082 T^{7} + 640081 T^{8} + 6432372 T^{9} + 49803833 T^{10} + 329014498 T^{11} + 1370699533 T^{12} + 3673751986 T^{13} + 20094940149 T^{14} + 161647399878 T^{15} + 912460145467 T^{16} + 161647399878 p T^{17} + 20094940149 p^{2} T^{18} + 3673751986 p^{3} T^{19} + 1370699533 p^{4} T^{20} + 329014498 p^{5} T^{21} + 49803833 p^{6} T^{22} + 6432372 p^{7} T^{23} + 640081 p^{8} T^{24} + 191082 p^{9} T^{25} + 96786 p^{10} T^{26} + 16884 p^{11} T^{27} - 166 p^{12} T^{28} - 140 p^{13} T^{29} + 66 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 4 T - 86 T^{2} + 88 T^{3} + 5202 T^{4} - 20786 T^{5} - 127570 T^{6} + 1350404 T^{7} - 713179 T^{8} - 38405408 T^{9} + 185121661 T^{10} + 383989708 T^{11} - 6091484475 T^{12} + 10535407228 T^{13} + 3124876821 p T^{14} - 229396275918 T^{15} - 808236948365 T^{16} - 229396275918 p T^{17} + 3124876821 p^{3} T^{18} + 10535407228 p^{3} T^{19} - 6091484475 p^{4} T^{20} + 383989708 p^{5} T^{21} + 185121661 p^{6} T^{22} - 38405408 p^{7} T^{23} - 713179 p^{8} T^{24} + 1350404 p^{9} T^{25} - 127570 p^{10} T^{26} - 20786 p^{11} T^{27} + 5202 p^{12} T^{28} + 88 p^{13} T^{29} - 86 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 12 T + 178 T^{2} - 1560 T^{3} + 13526 T^{4} - 88560 T^{5} + 18310 p T^{6} - 2643174 T^{7} + 12175877 T^{8} - 22216488 T^{9} - 24227007 T^{10} + 1509848178 T^{11} - 9626729415 T^{12} + 72584640054 T^{13} - 309976990523 T^{14} + 1999645319814 T^{15} - 8070164087269 T^{16} + 1999645319814 p T^{17} - 309976990523 p^{2} T^{18} + 72584640054 p^{3} T^{19} - 9626729415 p^{4} T^{20} + 1509848178 p^{5} T^{21} - 24227007 p^{6} T^{22} - 22216488 p^{7} T^{23} + 12175877 p^{8} T^{24} - 2643174 p^{9} T^{25} + 18310 p^{11} T^{26} - 88560 p^{11} T^{27} + 13526 p^{12} T^{28} - 1560 p^{13} T^{29} + 178 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 30 T + 530 T^{2} - 6900 T^{3} + 71694 T^{4} - 615858 T^{5} + 4474894 T^{6} - 27618318 T^{7} + 142713869 T^{8} - 582401982 T^{9} + 1475912547 T^{10} + 2245257420 T^{11} - 58422138587 T^{12} + 448216608072 T^{13} - 2437496727085 T^{14} + 11364263152362 T^{15} - 59513910270921 T^{16} + 11364263152362 p T^{17} - 2437496727085 p^{2} T^{18} + 448216608072 p^{3} T^{19} - 58422138587 p^{4} T^{20} + 2245257420 p^{5} T^{21} + 1475912547 p^{6} T^{22} - 582401982 p^{7} T^{23} + 142713869 p^{8} T^{24} - 27618318 p^{9} T^{25} + 4474894 p^{10} T^{26} - 615858 p^{11} T^{27} + 71694 p^{12} T^{28} - 6900 p^{13} T^{29} + 530 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 18 T + 358 T^{2} + 4500 T^{3} + 54840 T^{4} + 546582 T^{5} + 5282002 T^{6} + 45829452 T^{7} + 391371517 T^{8} + 3136234194 T^{9} + 24733354269 T^{10} + 186013158912 T^{11} + 1365943595003 T^{12} + 234854728914 p T^{13} + 66175973009543 T^{14} + 10725583636596 p T^{15} + 2857961563758879 T^{16} + 10725583636596 p^{2} T^{17} + 66175973009543 p^{2} T^{18} + 234854728914 p^{4} T^{19} + 1365943595003 p^{4} T^{20} + 186013158912 p^{5} T^{21} + 24733354269 p^{6} T^{22} + 3136234194 p^{7} T^{23} + 391371517 p^{8} T^{24} + 45829452 p^{9} T^{25} + 5282002 p^{10} T^{26} + 546582 p^{11} T^{27} + 54840 p^{12} T^{28} + 4500 p^{13} T^{29} + 358 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 32 T + 386 T^{2} + 1956 T^{3} + 3028 T^{4} + 21752 T^{5} + 145614 T^{6} - 5040252 T^{7} - 76956731 T^{8} - 431456972 T^{9} - 1420003353 T^{10} - 10453792098 T^{11} - 69895637053 T^{12} + 208576009948 T^{13} + 5960716600089 T^{14} + 38060480693916 T^{15} + 189113657800291 T^{16} + 38060480693916 p T^{17} + 5960716600089 p^{2} T^{18} + 208576009948 p^{3} T^{19} - 69895637053 p^{4} T^{20} - 10453792098 p^{5} T^{21} - 1420003353 p^{6} T^{22} - 431456972 p^{7} T^{23} - 76956731 p^{8} T^{24} - 5040252 p^{9} T^{25} + 145614 p^{10} T^{26} + 21752 p^{11} T^{27} + 3028 p^{12} T^{28} + 1956 p^{13} T^{29} + 386 p^{14} T^{30} + 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 66 T + 2301 T^{2} - 56034 T^{3} + 1061827 T^{4} - 16584684 T^{5} + 221550956 T^{6} - 2600902890 T^{7} + 27436895546 T^{8} - 265294801890 T^{9} + 2393966242239 T^{10} - 20471188232046 T^{11} + 167711622861328 T^{12} - 28160940748968 p T^{13} + 10066116607722236 T^{14} - 73587368388097878 T^{15} + 515611185246520285 T^{16} - 73587368388097878 p T^{17} + 10066116607722236 p^{2} T^{18} - 28160940748968 p^{4} T^{19} + 167711622861328 p^{4} T^{20} - 20471188232046 p^{5} T^{21} + 2393966242239 p^{6} T^{22} - 265294801890 p^{7} T^{23} + 27436895546 p^{8} T^{24} - 2600902890 p^{9} T^{25} + 221550956 p^{10} T^{26} - 16584684 p^{11} T^{27} + 1061827 p^{12} T^{28} - 56034 p^{13} T^{29} + 2301 p^{14} T^{30} - 66 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 2 T - 195 T^{2} - 1318 T^{3} + 22409 T^{4} + 269300 T^{5} - 557738 T^{6} - 27538680 T^{7} - 96170256 T^{8} + 1346417800 T^{9} + 12401375043 T^{10} - 17177770406 T^{11} - 633298455796 T^{12} - 1787982426838 T^{13} + 14498252101676 T^{14} + 62417595792192 T^{15} - 108566606387647 T^{16} + 62417595792192 p T^{17} + 14498252101676 p^{2} T^{18} - 1787982426838 p^{3} T^{19} - 633298455796 p^{4} T^{20} - 17177770406 p^{5} T^{21} + 12401375043 p^{6} T^{22} + 1346417800 p^{7} T^{23} - 96170256 p^{8} T^{24} - 27538680 p^{9} T^{25} - 557738 p^{10} T^{26} + 269300 p^{11} T^{27} + 22409 p^{12} T^{28} - 1318 p^{13} T^{29} - 195 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 36 T + 838 T^{2} + 14616 T^{3} + 213245 T^{4} + 2685852 T^{5} + 30015338 T^{6} + 299798364 T^{7} + 2684139012 T^{8} + 21300298320 T^{9} + 145626266164 T^{10} + 792521466936 T^{11} + 2415357989795 T^{12} - 13731479051352 T^{13} - 5705015711544 p T^{14} - 3837260333121780 T^{15} - 32976566032834097 T^{16} - 3837260333121780 p T^{17} - 5705015711544 p^{3} T^{18} - 13731479051352 p^{3} T^{19} + 2415357989795 p^{4} T^{20} + 792521466936 p^{5} T^{21} + 145626266164 p^{6} T^{22} + 21300298320 p^{7} T^{23} + 2684139012 p^{8} T^{24} + 299798364 p^{9} T^{25} + 30015338 p^{10} T^{26} + 2685852 p^{11} T^{27} + 213245 p^{12} T^{28} + 14616 p^{13} T^{29} + 838 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 4 T + 332 T^{2} + 508 T^{3} + 46868 T^{4} - 38844 T^{5} + 3974100 T^{6} - 10198436 T^{7} + 259069430 T^{8} - 10198436 p T^{9} + 3974100 p^{2} T^{10} - 38844 p^{3} T^{11} + 46868 p^{4} T^{12} + 508 p^{5} T^{13} + 332 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 374 T^{2} + 83627 T^{4} - 13802854 T^{6} + 1824872063 T^{8} - 201472927114 T^{10} + 19036310914896 T^{12} - 1559874560238336 T^{14} + 111682489688488927 T^{16} - 1559874560238336 p^{2} T^{18} + 19036310914896 p^{4} T^{20} - 201472927114 p^{6} T^{22} + 1824872063 p^{8} T^{24} - 13802854 p^{10} T^{26} + 83627 p^{12} T^{28} - 374 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 + 30 T + 496 T^{2} + 5880 T^{3} + 47763 T^{4} + 239550 T^{5} - 184432 T^{6} - 17087508 T^{7} - 154821892 T^{8} - 456732348 T^{9} + 8514713412 T^{10} + 158385211890 T^{11} + 1465807912525 T^{12} + 7253307274650 T^{13} - 20262310103312 T^{14} - 798123905876142 T^{15} - 8825867569357317 T^{16} - 798123905876142 p T^{17} - 20262310103312 p^{2} T^{18} + 7253307274650 p^{3} T^{19} + 1465807912525 p^{4} T^{20} + 158385211890 p^{5} T^{21} + 8514713412 p^{6} T^{22} - 456732348 p^{7} T^{23} - 154821892 p^{8} T^{24} - 17087508 p^{9} T^{25} - 184432 p^{10} T^{26} + 239550 p^{11} T^{27} + 47763 p^{12} T^{28} + 5880 p^{13} T^{29} + 496 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 18 T + 301 T^{2} + 3474 T^{3} + 27698 T^{4} + 113148 T^{5} - 236309 T^{6} - 20620026 T^{7} - 249891430 T^{8} - 2525884308 T^{9} - 21352907538 T^{10} - 134607558612 T^{11} - 277508772423 T^{12} + 33901633560 p T^{13} + 104687240819980 T^{14} + 1238446915840422 T^{15} + 11497606771536233 T^{16} + 1238446915840422 p T^{17} + 104687240819980 p^{2} T^{18} + 33901633560 p^{4} T^{19} - 277508772423 p^{4} T^{20} - 134607558612 p^{5} T^{21} - 21352907538 p^{6} T^{22} - 2525884308 p^{7} T^{23} - 249891430 p^{8} T^{24} - 20620026 p^{9} T^{25} - 236309 p^{10} T^{26} + 113148 p^{11} T^{27} + 27698 p^{12} T^{28} + 3474 p^{13} T^{29} + 301 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 24 T - 116 T^{2} - 6316 T^{3} + 8751 T^{4} + 1024592 T^{5} - 1057936 T^{6} - 117334194 T^{7} + 354271940 T^{8} + 10928923888 T^{9} - 72443598888 T^{10} - 846862897898 T^{11} + 9962166960773 T^{12} + 50168591523642 T^{13} - 1051373374276888 T^{14} - 1522192926505076 T^{15} + 90217363743814155 T^{16} - 1522192926505076 p T^{17} - 1051373374276888 p^{2} T^{18} + 50168591523642 p^{3} T^{19} + 9962166960773 p^{4} T^{20} - 846862897898 p^{5} T^{21} - 72443598888 p^{6} T^{22} + 10928923888 p^{7} T^{23} + 354271940 p^{8} T^{24} - 117334194 p^{9} T^{25} - 1057936 p^{10} T^{26} + 1024592 p^{11} T^{27} + 8751 p^{12} T^{28} - 6316 p^{13} T^{29} - 116 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 484 T^{2} + 142268 T^{4} - 29580086 T^{6} + 4879595558 T^{8} - 662523125234 T^{10} + 76909497524985 T^{12} - 7735213826859018 T^{14} + 684662977430406493 T^{16} - 7735213826859018 p^{2} T^{18} + 76909497524985 p^{4} T^{20} - 662523125234 p^{6} T^{22} + 4879595558 p^{8} T^{24} - 29580086 p^{10} T^{26} + 142268 p^{12} T^{28} - 484 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 + 42 T + 1223 T^{2} + 26670 T^{3} + 484115 T^{4} + 7501362 T^{5} + 102284584 T^{6} + 1234036176 T^{7} + 13272648956 T^{8} + 125811854232 T^{9} + 1025419523107 T^{10} + 6607490860506 T^{11} + 22957660774650 T^{12} - 190819882694682 T^{13} - 5259490183409106 T^{14} - 72261362209903842 T^{15} - 758184563465183243 T^{16} - 72261362209903842 p T^{17} - 5259490183409106 p^{2} T^{18} - 190819882694682 p^{3} T^{19} + 22957660774650 p^{4} T^{20} + 6607490860506 p^{5} T^{21} + 1025419523107 p^{6} T^{22} + 125811854232 p^{7} T^{23} + 13272648956 p^{8} T^{24} + 1234036176 p^{9} T^{25} + 102284584 p^{10} T^{26} + 7501362 p^{11} T^{27} + 484115 p^{12} T^{28} + 26670 p^{13} T^{29} + 1223 p^{14} T^{30} + 42 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 6 T + 503 T^{2} + 2946 T^{3} + 127209 T^{4} + 657234 T^{5} + 21377574 T^{6} + 92067138 T^{7} + 2753435634 T^{8} + 9901707102 T^{9} + 312594305607 T^{10} + 1072349902938 T^{11} + 35099100528878 T^{12} + 134138470329528 T^{13} + 3917180676344122 T^{14} + 16065358655112540 T^{15} + 403313590778631681 T^{16} + 16065358655112540 p T^{17} + 3917180676344122 p^{2} T^{18} + 134138470329528 p^{3} T^{19} + 35099100528878 p^{4} T^{20} + 1072349902938 p^{5} T^{21} + 312594305607 p^{6} T^{22} + 9901707102 p^{7} T^{23} + 2753435634 p^{8} T^{24} + 92067138 p^{9} T^{25} + 21377574 p^{10} T^{26} + 657234 p^{11} T^{27} + 127209 p^{12} T^{28} + 2946 p^{13} T^{29} + 503 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.83403908325169652170767103082, −2.66880367279473077278535098296, −2.63588716763148293006983507237, −2.59503668622980124093250033322, −2.49702489626518820096562935514, −2.34672975396414332119022067909, −2.23597753137304716862048231817, −2.13730714193709009237375914592, −2.10900590845158351613828712453, −1.95728948487042987304263140602, −1.91444189738648243390873449488, −1.82715330830193313971236708004, −1.76920158681622420554762709363, −1.57877091058893325277929142266, −1.54904658840062278290561864307, −1.33549587749925029921506640631, −1.31417530672618486817241386344, −1.19672517269895242111465137050, −1.18961953355634180809476784921, −1.15862143308814748377867151411, −0.65134728135642011864161717466, −0.46611362898529670140560049562, −0.40136022021364167954458068201, −0.33177607405403785101908541174, −0.22969981611222044836233206231, 0.22969981611222044836233206231, 0.33177607405403785101908541174, 0.40136022021364167954458068201, 0.46611362898529670140560049562, 0.65134728135642011864161717466, 1.15862143308814748377867151411, 1.18961953355634180809476784921, 1.19672517269895242111465137050, 1.31417530672618486817241386344, 1.33549587749925029921506640631, 1.54904658840062278290561864307, 1.57877091058893325277929142266, 1.76920158681622420554762709363, 1.82715330830193313971236708004, 1.91444189738648243390873449488, 1.95728948487042987304263140602, 2.10900590845158351613828712453, 2.13730714193709009237375914592, 2.23597753137304716862048231817, 2.34672975396414332119022067909, 2.49702489626518820096562935514, 2.59503668622980124093250033322, 2.63588716763148293006983507237, 2.66880367279473077278535098296, 2.83403908325169652170767103082
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.