Dirichlet series
| L(s) = 1 | + 8·11-s − 16·17-s + 8·19-s + 8·23-s + 12·25-s + 8·31-s − 8·37-s − 8·41-s + 8·43-s − 20·49-s − 24·53-s − 32·59-s − 48·61-s − 32·71-s − 40·73-s + 16·79-s − 48·83-s − 40·97-s − 8·103-s + 40·107-s + 24·109-s + 32·113-s + 64·121-s + 8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 3.88·17-s + 1.83·19-s + 1.66·23-s + 12/5·25-s + 1.43·31-s − 1.31·37-s − 1.24·41-s + 1.21·43-s − 2.85·49-s − 3.29·53-s − 4.16·59-s − 6.14·61-s − 3.79·71-s − 4.68·73-s + 1.80·79-s − 5.26·83-s − 4.06·97-s − 0.788·103-s + 3.86·107-s + 2.29·109-s + 3.01·113-s + 5.81·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{16} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.61063\times 10^{8}\) |
| Root analytic conductor: | \(1.80496\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{16} \cdot 17^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2714022210\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2714022210\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{8} )^{2} \) | |
| 17 | \( 1 + 16 T + 116 T^{2} + 576 T^{3} + 2678 T^{4} + 13776 T^{5} + 71076 T^{6} + 319456 T^{7} + 1315170 T^{8} + 319456 p T^{9} + 71076 p^{2} T^{10} + 13776 p^{3} T^{11} + 2678 p^{4} T^{12} + 576 p^{5} T^{13} + 116 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 12 T^{2} - 8 T^{3} + 72 T^{4} + 128 T^{5} - 268 T^{6} - 256 p T^{7} + 29 T^{8} + 9576 T^{9} + 8248 T^{10} - 49656 T^{11} - 84672 T^{12} + 36768 p T^{13} + 622528 T^{14} - 349416 T^{15} - 3525084 T^{16} - 349416 p T^{17} + 622528 p^{2} T^{18} + 36768 p^{4} T^{19} - 84672 p^{4} T^{20} - 49656 p^{5} T^{21} + 8248 p^{6} T^{22} + 9576 p^{7} T^{23} + 29 p^{8} T^{24} - 256 p^{10} T^{25} - 268 p^{10} T^{26} + 128 p^{11} T^{27} + 72 p^{12} T^{28} - 8 p^{13} T^{29} - 12 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( 1 + 20 T^{2} + 64 T^{3} + 200 T^{4} + 1408 T^{5} + 3676 T^{6} + 15520 T^{7} + 58704 T^{8} + 24544 p T^{9} + 618372 T^{10} + 1873920 T^{11} + 5693528 T^{12} + 16836032 T^{13} + 47744460 T^{14} + 134302848 T^{15} + 350582814 T^{16} + 134302848 p T^{17} + 47744460 p^{2} T^{18} + 16836032 p^{3} T^{19} + 5693528 p^{4} T^{20} + 1873920 p^{5} T^{21} + 618372 p^{6} T^{22} + 24544 p^{8} T^{23} + 58704 p^{8} T^{24} + 15520 p^{9} T^{25} + 3676 p^{10} T^{26} + 1408 p^{11} T^{27} + 200 p^{12} T^{28} + 64 p^{13} T^{29} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 8 T + 160 T^{3} - 256 T^{4} - 1432 T^{5} + 2736 T^{6} + 16344 T^{7} - 49855 T^{8} - 17096 p T^{9} + 1136192 T^{10} + 389880 T^{11} - 10654080 T^{12} - 3811536 T^{13} + 97612080 T^{14} + 59961128 T^{15} - 1202365152 T^{16} + 59961128 p T^{17} + 97612080 p^{2} T^{18} - 3811536 p^{3} T^{19} - 10654080 p^{4} T^{20} + 389880 p^{5} T^{21} + 1136192 p^{6} T^{22} - 17096 p^{8} T^{23} - 49855 p^{8} T^{24} + 16344 p^{9} T^{25} + 2736 p^{10} T^{26} - 1432 p^{11} T^{27} - 256 p^{12} T^{28} + 160 p^{13} T^{29} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 108 T^{2} + 5934 T^{4} - 220912 T^{6} + 6250705 T^{8} - 142697496 T^{10} + 2720139014 T^{12} - 44175794348 T^{14} + 617808556964 T^{16} - 44175794348 p^{2} T^{18} + 2720139014 p^{4} T^{20} - 142697496 p^{6} T^{22} + 6250705 p^{8} T^{24} - 220912 p^{10} T^{26} + 5934 p^{12} T^{28} - 108 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 8 T + 32 T^{2} - 240 T^{3} + 2450 T^{4} - 10776 T^{5} + 36608 T^{6} - 235544 T^{7} + 1587905 T^{8} - 5461000 T^{9} + 17777088 T^{10} - 110339672 T^{11} + 757113090 T^{12} - 2708371664 T^{13} + 8825207840 T^{14} - 55352182392 T^{15} + 344756081028 T^{16} - 55352182392 p T^{17} + 8825207840 p^{2} T^{18} - 2708371664 p^{3} T^{19} + 757113090 p^{4} T^{20} - 110339672 p^{5} T^{21} + 17777088 p^{6} T^{22} - 5461000 p^{7} T^{23} + 1587905 p^{8} T^{24} - 235544 p^{9} T^{25} + 36608 p^{10} T^{26} - 10776 p^{11} T^{27} + 2450 p^{12} T^{28} - 240 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 8 T - 8 T^{2} + 416 T^{3} - 1760 T^{4} - 1400 T^{5} + 32672 T^{6} - 184536 T^{7} + 1221585 T^{8} - 2230792 T^{9} - 31595312 T^{10} + 186033896 T^{11} - 89861792 T^{12} - 514625408 T^{13} - 7747464 p^{2} T^{14} - 30808580184 T^{15} + 437519039264 T^{16} - 30808580184 p T^{17} - 7747464 p^{4} T^{18} - 514625408 p^{3} T^{19} - 89861792 p^{4} T^{20} + 186033896 p^{5} T^{21} - 31595312 p^{6} T^{22} - 2230792 p^{7} T^{23} + 1221585 p^{8} T^{24} - 184536 p^{9} T^{25} + 32672 p^{10} T^{26} - 1400 p^{11} T^{27} - 1760 p^{12} T^{28} + 416 p^{13} T^{29} - 8 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 20 T^{2} - 304 T^{3} + 200 T^{4} - 11504 T^{5} + 33964 T^{6} - 10400 p T^{7} + 2602560 T^{8} - 3320544 T^{9} + 90706804 T^{10} - 402757328 T^{11} + 1459627224 T^{12} - 14817672720 T^{13} + 70963800716 T^{14} - 376601357440 T^{15} + 1997123529278 T^{16} - 376601357440 p T^{17} + 70963800716 p^{2} T^{18} - 14817672720 p^{3} T^{19} + 1459627224 p^{4} T^{20} - 402757328 p^{5} T^{21} + 90706804 p^{6} T^{22} - 3320544 p^{7} T^{23} + 2602560 p^{8} T^{24} - 10400 p^{10} T^{25} + 33964 p^{10} T^{26} - 11504 p^{11} T^{27} + 200 p^{12} T^{28} - 304 p^{13} T^{29} + 20 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 8 T + 8 T^{2} + 152 T^{3} - 672 T^{4} - 3992 T^{5} + 9192 T^{6} + 247880 T^{7} - 2624988 T^{8} + 11751592 T^{9} + 6359544 T^{10} - 4871880 p T^{11} + 94988320 T^{12} + 3868834104 T^{13} + 5818413528 T^{14} - 370919655016 T^{15} + 2648960418374 T^{16} - 370919655016 p T^{17} + 5818413528 p^{2} T^{18} + 3868834104 p^{3} T^{19} + 94988320 p^{4} T^{20} - 4871880 p^{6} T^{21} + 6359544 p^{6} T^{22} + 11751592 p^{7} T^{23} - 2624988 p^{8} T^{24} + 247880 p^{9} T^{25} + 9192 p^{10} T^{26} - 3992 p^{11} T^{27} - 672 p^{12} T^{28} + 152 p^{13} T^{29} + 8 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 8 T - 32 T^{2} - 536 T^{3} - 704 T^{4} + 28600 T^{5} + 126288 T^{6} - 740904 T^{7} - 3678996 T^{8} + 31891880 T^{9} + 223174352 T^{10} - 1212632440 T^{11} - 10735241408 T^{12} + 53235595288 T^{13} + 629322731904 T^{14} - 452189134536 T^{15} - 23185807419674 T^{16} - 452189134536 p T^{17} + 629322731904 p^{2} T^{18} + 53235595288 p^{3} T^{19} - 10735241408 p^{4} T^{20} - 1212632440 p^{5} T^{21} + 223174352 p^{6} T^{22} + 31891880 p^{7} T^{23} - 3678996 p^{8} T^{24} - 740904 p^{9} T^{25} + 126288 p^{10} T^{26} + 28600 p^{11} T^{27} - 704 p^{12} T^{28} - 536 p^{13} T^{29} - 32 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 8 T + 28 T^{2} - 96 T^{3} - 1144 T^{4} - 12312 T^{5} - 42324 T^{6} - 234376 T^{7} - 2032499 T^{8} - 17947192 T^{9} - 49661896 T^{10} - 581365720 T^{11} - 1328372992 T^{12} + 1385524688 T^{13} + 145215391376 T^{14} + 2143000153208 T^{15} + 22744964436660 T^{16} + 2143000153208 p T^{17} + 145215391376 p^{2} T^{18} + 1385524688 p^{3} T^{19} - 1328372992 p^{4} T^{20} - 581365720 p^{5} T^{21} - 49661896 p^{6} T^{22} - 17947192 p^{7} T^{23} - 2032499 p^{8} T^{24} - 234376 p^{9} T^{25} - 42324 p^{10} T^{26} - 12312 p^{11} T^{27} - 1144 p^{12} T^{28} - 96 p^{13} T^{29} + 28 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 8 T + 32 T^{2} - 48 T^{3} - 3006 T^{4} + 18088 T^{5} - 47360 T^{6} - 383480 T^{7} + 1480001 T^{8} + 40054104 T^{9} - 328162112 T^{10} + 2239204552 T^{11} + 3617123458 T^{12} - 113774449968 T^{13} + 568630457632 T^{14} - 1334961024376 T^{15} - 8633731471004 T^{16} - 1334961024376 p T^{17} + 568630457632 p^{2} T^{18} - 113774449968 p^{3} T^{19} + 3617123458 p^{4} T^{20} + 2239204552 p^{5} T^{21} - 328162112 p^{6} T^{22} + 40054104 p^{7} T^{23} + 1480001 p^{8} T^{24} - 383480 p^{9} T^{25} - 47360 p^{10} T^{26} + 18088 p^{11} T^{27} - 3006 p^{12} T^{28} - 48 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 400 T^{2} + 84584 T^{4} - 12212976 T^{6} + 1335722556 T^{8} - 116642835920 T^{10} + 8387181034328 T^{12} - 505521339926960 T^{14} + 25788166193361158 T^{16} - 505521339926960 p^{2} T^{18} + 8387181034328 p^{4} T^{20} - 116642835920 p^{6} T^{22} + 1335722556 p^{8} T^{24} - 12212976 p^{10} T^{26} + 84584 p^{12} T^{28} - 400 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 24 T + 288 T^{2} + 2632 T^{3} + 17840 T^{4} + 87256 T^{5} + 419936 T^{6} + 2531336 T^{7} + 28574684 T^{8} + 373168504 T^{9} + 3704157984 T^{10} + 31755943528 T^{11} + 225824213584 T^{12} + 1274186201784 T^{13} + 6808043719008 T^{14} + 36227374837672 T^{15} + 213084153666822 T^{16} + 36227374837672 p T^{17} + 6808043719008 p^{2} T^{18} + 1274186201784 p^{3} T^{19} + 225824213584 p^{4} T^{20} + 31755943528 p^{5} T^{21} + 3704157984 p^{6} T^{22} + 373168504 p^{7} T^{23} + 28574684 p^{8} T^{24} + 2531336 p^{9} T^{25} + 419936 p^{10} T^{26} + 87256 p^{11} T^{27} + 17840 p^{12} T^{28} + 2632 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 32 T + 512 T^{2} + 7056 T^{3} + 110100 T^{4} + 1523408 T^{5} + 292736 p T^{6} + 190600128 T^{7} + 2184104168 T^{8} + 22742834240 T^{9} + 213186049664 T^{10} + 2013959811088 T^{11} + 19033305660668 T^{12} + 164042572435024 T^{13} + 1331257053692160 T^{14} + 11015737447779872 T^{15} + 88573117387346190 T^{16} + 11015737447779872 p T^{17} + 1331257053692160 p^{2} T^{18} + 164042572435024 p^{3} T^{19} + 19033305660668 p^{4} T^{20} + 2013959811088 p^{5} T^{21} + 213186049664 p^{6} T^{22} + 22742834240 p^{7} T^{23} + 2184104168 p^{8} T^{24} + 190600128 p^{9} T^{25} + 292736 p^{11} T^{26} + 1523408 p^{11} T^{27} + 110100 p^{12} T^{28} + 7056 p^{13} T^{29} + 512 p^{14} T^{30} + 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 48 T + 1080 T^{2} + 14432 T^{3} + 114720 T^{4} + 293728 T^{5} - 6145576 T^{6} - 114631344 T^{7} - 1159130452 T^{8} - 7416578896 T^{9} - 14557334680 T^{10} + 310229752608 T^{11} + 4354637138528 T^{12} + 28725077974816 T^{13} + 65600976808328 T^{14} - 747059554714032 T^{15} - 9824487658931994 T^{16} - 747059554714032 p T^{17} + 65600976808328 p^{2} T^{18} + 28725077974816 p^{3} T^{19} + 4354637138528 p^{4} T^{20} + 310229752608 p^{5} T^{21} - 14557334680 p^{6} T^{22} - 7416578896 p^{7} T^{23} - 1159130452 p^{8} T^{24} - 114631344 p^{9} T^{25} - 6145576 p^{10} T^{26} + 293728 p^{11} T^{27} + 114720 p^{12} T^{28} + 14432 p^{13} T^{29} + 1080 p^{14} T^{30} + 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 236 T^{2} + 976 T^{3} + 31712 T^{4} + 174928 T^{5} + 3258404 T^{6} + 18338752 T^{7} + 244778526 T^{8} + 18338752 p T^{9} + 3258404 p^{2} T^{10} + 174928 p^{3} T^{11} + 31712 p^{4} T^{12} + 976 p^{5} T^{13} + 236 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 32 T + 576 T^{2} + 8272 T^{3} + 102912 T^{4} + 1124016 T^{5} + 11077824 T^{6} + 106469472 T^{7} + 1074410052 T^{8} + 11048322784 T^{9} + 112988051776 T^{10} + 1135163106928 T^{11} + 10793398601216 T^{12} + 95402473867664 T^{13} + 796853531131840 T^{14} + 92690682449760 p T^{15} + 55188886227832326 T^{16} + 92690682449760 p^{2} T^{17} + 796853531131840 p^{2} T^{18} + 95402473867664 p^{3} T^{19} + 10793398601216 p^{4} T^{20} + 1135163106928 p^{5} T^{21} + 112988051776 p^{6} T^{22} + 11048322784 p^{7} T^{23} + 1074410052 p^{8} T^{24} + 106469472 p^{9} T^{25} + 11077824 p^{10} T^{26} + 1124016 p^{11} T^{27} + 102912 p^{12} T^{28} + 8272 p^{13} T^{29} + 576 p^{14} T^{30} + 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 40 T + 748 T^{2} + 8680 T^{3} + 70152 T^{4} + 420504 T^{5} + 1586020 T^{6} - 12433416 T^{7} - 463461744 T^{8} - 7133433016 T^{9} - 71339455812 T^{10} - 496952630488 T^{11} - 2190940117480 T^{12} + 1809451099608 T^{13} + 197197687946292 T^{14} + 3053545893986328 T^{15} + 31068550757888670 T^{16} + 3053545893986328 p T^{17} + 197197687946292 p^{2} T^{18} + 1809451099608 p^{3} T^{19} - 2190940117480 p^{4} T^{20} - 496952630488 p^{5} T^{21} - 71339455812 p^{6} T^{22} - 7133433016 p^{7} T^{23} - 463461744 p^{8} T^{24} - 12433416 p^{9} T^{25} + 1586020 p^{10} T^{26} + 420504 p^{11} T^{27} + 70152 p^{12} T^{28} + 8680 p^{13} T^{29} + 748 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 16 T + 244 T^{2} - 3056 T^{3} + 32584 T^{4} - 223184 T^{5} + 1877692 T^{6} - 5737616 T^{7} + 28936144 T^{8} - 37302928 T^{9} + 4118420324 T^{10} - 48388405968 T^{11} + 1213057172184 T^{12} - 9656421875440 T^{13} + 124211418889516 T^{14} - 958213726155792 T^{15} + 9984485295094686 T^{16} - 958213726155792 p T^{17} + 124211418889516 p^{2} T^{18} - 9656421875440 p^{3} T^{19} + 1213057172184 p^{4} T^{20} - 48388405968 p^{5} T^{21} + 4118420324 p^{6} T^{22} - 37302928 p^{7} T^{23} + 28936144 p^{8} T^{24} - 5737616 p^{9} T^{25} + 1877692 p^{10} T^{26} - 223184 p^{11} T^{27} + 32584 p^{12} T^{28} - 3056 p^{13} T^{29} + 244 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 48 T + 1152 T^{2} + 21424 T^{3} + 360856 T^{4} + 5339696 T^{5} + 70093184 T^{6} + 864249904 T^{7} + 9998587004 T^{8} + 106800729136 T^{9} + 1086383393920 T^{10} + 10670425964976 T^{11} + 99431823746856 T^{12} + 900702738138032 T^{13} + 8178086186704256 T^{14} + 73275277141191600 T^{15} + 657228499657031110 T^{16} + 73275277141191600 p T^{17} + 8178086186704256 p^{2} T^{18} + 900702738138032 p^{3} T^{19} + 99431823746856 p^{4} T^{20} + 10670425964976 p^{5} T^{21} + 1086383393920 p^{6} T^{22} + 106800729136 p^{7} T^{23} + 9998587004 p^{8} T^{24} + 864249904 p^{9} T^{25} + 70093184 p^{10} T^{26} + 5339696 p^{11} T^{27} + 360856 p^{12} T^{28} + 21424 p^{13} T^{29} + 1152 p^{14} T^{30} + 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 896 T^{2} + 395332 T^{4} - 114593088 T^{6} + 24550488968 T^{8} - 4139968731072 T^{10} + 570019370003020 T^{12} - 65447372604414336 T^{14} + 6334013156204728718 T^{16} - 65447372604414336 p^{2} T^{18} + 570019370003020 p^{4} T^{20} - 4139968731072 p^{6} T^{22} + 24550488968 p^{8} T^{24} - 114593088 p^{10} T^{26} + 395332 p^{12} T^{28} - 896 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 40 T + 428 T^{2} - 3032 T^{3} - 74488 T^{4} + 204232 T^{5} + 9581268 T^{6} - 17705336 T^{7} - 1105307088 T^{8} + 4563320440 T^{9} + 173501516172 T^{10} - 283656949064 T^{11} - 20846739710696 T^{12} + 2828884134232 T^{13} + 2121633010495540 T^{14} + 2379683583448280 T^{15} - 166247640075511330 T^{16} + 2379683583448280 p T^{17} + 2121633010495540 p^{2} T^{18} + 2828884134232 p^{3} T^{19} - 20846739710696 p^{4} T^{20} - 283656949064 p^{5} T^{21} + 173501516172 p^{6} T^{22} + 4563320440 p^{7} T^{23} - 1105307088 p^{8} T^{24} - 17705336 p^{9} T^{25} + 9581268 p^{10} T^{26} + 204232 p^{11} T^{27} - 74488 p^{12} T^{28} - 3032 p^{13} T^{29} + 428 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.11707326202576637544517842514, −3.09955620857960024884494148621, −3.06747024927656244847667188088, −3.02337375567703204028464444317, −2.89708750736276991274530080136, −2.87162136922617149845721191375, −2.70841805580405019798952780282, −2.58839673477010223157254614796, −2.36659186381873564481833690902, −2.13762545588162023489381366413, −2.06473547565380975677943236733, −2.00895399211302510087879511084, −1.92982897764986029940694187708, −1.87766364272804359388755197878, −1.73896908438363585908600174920, −1.66982574259377548604599095102, −1.51010555654218843281338079306, −1.37047026824774404494423953420, −1.30585673712410378759385065236, −1.20133531873565416401679627030, −1.18862013902925942258270849708, −1.07211120378201935978681924278, −0.55253424396642926064936555704, −0.34274436243799145208824363735, −0.07199047777914608200748366516, 0.07199047777914608200748366516, 0.34274436243799145208824363735, 0.55253424396642926064936555704, 1.07211120378201935978681924278, 1.18862013902925942258270849708, 1.20133531873565416401679627030, 1.30585673712410378759385065236, 1.37047026824774404494423953420, 1.51010555654218843281338079306, 1.66982574259377548604599095102, 1.73896908438363585908600174920, 1.87766364272804359388755197878, 1.92982897764986029940694187708, 2.00895399211302510087879511084, 2.06473547565380975677943236733, 2.13762545588162023489381366413, 2.36659186381873564481833690902, 2.58839673477010223157254614796, 2.70841805580405019798952780282, 2.87162136922617149845721191375, 2.89708750736276991274530080136, 3.02337375567703204028464444317, 3.06747024927656244847667188088, 3.09955620857960024884494148621, 3.11707326202576637544517842514
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.