Dirichlet series
| L(s) = 1 | − 2·7-s − 12·11-s + 6·13-s − 18·17-s + 6·23-s + 16·25-s − 12·27-s + 6·29-s − 2·37-s − 6·41-s + 2·43-s + 36·47-s − 2·49-s − 36·53-s − 60·59-s + 28·67-s + 24·77-s − 32·79-s − 9·81-s − 24·89-s − 12·91-s + 6·97-s − 48·101-s − 42·103-s − 30·107-s + 4·109-s + 36·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 3.61·11-s + 1.66·13-s − 4.36·17-s + 1.25·23-s + 16/5·25-s − 2.30·27-s + 1.11·29-s − 0.328·37-s − 0.937·41-s + 0.304·43-s + 5.25·47-s − 2/7·49-s − 4.94·53-s − 7.81·59-s + 3.42·67-s + 2.73·77-s − 3.60·79-s − 81-s − 2.54·89-s − 1.25·91-s + 0.609·97-s − 4.77·101-s − 4.13·103-s − 2.90·107-s + 0.383·109-s + 3.38·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.10314\times 10^{14}\) |
| Root analytic conductor: | \(2.83706\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2804345033\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2804345033\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 4 p T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{4} T^{6} + 10 p^{2} T^{7} - 14 p^{2} T^{8} + 10 p^{3} T^{9} + p^{6} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} + 4 p^{6} T^{13} + p^{8} T^{16} \) | |
| 7 | \( 1 + 2 T + 6 T^{2} - 8 T^{3} + 23 T^{4} + 30 p T^{5} + 463 T^{6} + 164 p T^{7} - 1350 T^{8} + 164 p^{2} T^{9} + 463 p^{2} T^{10} + 30 p^{4} T^{11} + 23 p^{4} T^{12} - 8 p^{5} T^{13} + 6 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 16 T^{2} - 24 T^{3} + 123 T^{4} + 66 p T^{5} - 98 T^{6} - 2124 T^{7} - 4594 T^{8} - 102 T^{9} + 38946 T^{10} + 84396 T^{11} - 70916 T^{12} - 641196 T^{13} - 735151 T^{14} + 1444464 T^{15} + 6891966 T^{16} + 1444464 p T^{17} - 735151 p^{2} T^{18} - 641196 p^{3} T^{19} - 70916 p^{4} T^{20} + 84396 p^{5} T^{21} + 38946 p^{6} T^{22} - 102 p^{7} T^{23} - 4594 p^{8} T^{24} - 2124 p^{9} T^{25} - 98 p^{10} T^{26} + 66 p^{12} T^{27} + 123 p^{12} T^{28} - 24 p^{13} T^{29} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3735 T^{4} + 17808 T^{5} + 77432 T^{6} + 326142 T^{7} + 1351880 T^{8} + 5458080 T^{9} + 1935918 p T^{10} + 79655676 T^{11} + 287397262 T^{12} + 1023482544 T^{13} + 3603683689 T^{14} + 12488798178 T^{15} + 42242848866 T^{16} + 12488798178 p T^{17} + 3603683689 p^{2} T^{18} + 1023482544 p^{3} T^{19} + 287397262 p^{4} T^{20} + 79655676 p^{5} T^{21} + 1935918 p^{7} T^{22} + 5458080 p^{7} T^{23} + 1351880 p^{8} T^{24} + 326142 p^{9} T^{25} + 77432 p^{10} T^{26} + 17808 p^{11} T^{27} + 3735 p^{12} T^{28} + 696 p^{13} T^{29} + 106 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 57128796 p T^{15} - 178472592 p T^{16} + 57128796 p^{2} T^{17} - 179114935 p^{2} T^{18} + 72302586 p^{3} T^{19} - 19011605 p^{4} T^{20} + 6263526 p^{5} T^{21} - 1221396 p^{6} T^{22} + 214770 p^{7} T^{23} + 21641 p^{8} T^{24} - 12564 p^{9} T^{25} + 9181 p^{10} T^{26} - 2658 p^{11} T^{27} + 888 p^{12} T^{28} - 210 p^{13} T^{29} + 47 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 18 T + 95 T^{2} - 42 T^{3} - 849 T^{4} + 7584 T^{5} + 29152 T^{6} - 139356 T^{7} - 154873 T^{8} + 2564070 T^{9} - 13186653 T^{10} - 89037906 T^{11} + 222290986 T^{12} + 933503742 T^{13} - 5890959001 T^{14} - 449699490 p T^{15} + 98422426836 T^{16} - 449699490 p^{2} T^{17} - 5890959001 p^{2} T^{18} + 933503742 p^{3} T^{19} + 222290986 p^{4} T^{20} - 89037906 p^{5} T^{21} - 13186653 p^{6} T^{22} + 2564070 p^{7} T^{23} - 154873 p^{8} T^{24} - 139356 p^{9} T^{25} + 29152 p^{10} T^{26} + 7584 p^{11} T^{27} - 849 p^{12} T^{28} - 42 p^{13} T^{29} + 95 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 80 T^{2} + 3483 T^{4} + 882 T^{5} + 107770 T^{6} + 26190 T^{7} + 2596658 T^{8} - 477594 T^{9} + 51326154 T^{10} - 58243698 T^{11} + 858124540 T^{12} - 2331197604 T^{13} + 13169269781 T^{14} - 60976736712 T^{15} + 222780924306 T^{16} - 60976736712 p T^{17} + 13169269781 p^{2} T^{18} - 2331197604 p^{3} T^{19} + 858124540 p^{4} T^{20} - 58243698 p^{5} T^{21} + 51326154 p^{6} T^{22} - 477594 p^{7} T^{23} + 2596658 p^{8} T^{24} + 26190 p^{9} T^{25} + 107770 p^{10} T^{26} + 882 p^{11} T^{27} + 3483 p^{12} T^{28} + 80 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 6 T + 130 T^{2} - 708 T^{3} + 8559 T^{4} - 45012 T^{5} + 399452 T^{6} - 2115900 T^{7} + 15122540 T^{8} - 81277422 T^{9} + 486352566 T^{10} - 2645846004 T^{11} + 13727139058 T^{12} - 75456395706 T^{13} + 352786829089 T^{14} - 1928667059838 T^{15} + 8415100540206 T^{16} - 1928667059838 p T^{17} + 352786829089 p^{2} T^{18} - 75456395706 p^{3} T^{19} + 13727139058 p^{4} T^{20} - 2645846004 p^{5} T^{21} + 486352566 p^{6} T^{22} - 81277422 p^{7} T^{23} + 15122540 p^{8} T^{24} - 2115900 p^{9} T^{25} + 399452 p^{10} T^{26} - 45012 p^{11} T^{27} + 8559 p^{12} T^{28} - 708 p^{13} T^{29} + 130 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 6 T + 196 T^{2} - 1104 T^{3} + 19989 T^{4} - 108858 T^{5} + 1435478 T^{6} - 7594740 T^{7} + 81322766 T^{8} - 416245488 T^{9} + 3844091772 T^{10} - 18872529852 T^{11} + 155981060614 T^{12} - 727660406082 T^{13} + 5509975202215 T^{14} - 24244157784798 T^{15} + 170557455776958 T^{16} - 24244157784798 p T^{17} + 5509975202215 p^{2} T^{18} - 727660406082 p^{3} T^{19} + 155981060614 p^{4} T^{20} - 18872529852 p^{5} T^{21} + 3844091772 p^{6} T^{22} - 416245488 p^{7} T^{23} + 81322766 p^{8} T^{24} - 7594740 p^{9} T^{25} + 1435478 p^{10} T^{26} - 108858 p^{11} T^{27} + 19989 p^{12} T^{28} - 1104 p^{13} T^{29} + 196 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 292 T^{2} + 41022 T^{4} - 3694910 T^{6} + 240356405 T^{8} - 12117919608 T^{10} + 500572972675 T^{12} - 17863857948586 T^{14} + 576631021225626 T^{16} - 17863857948586 p^{2} T^{18} + 500572972675 p^{4} T^{20} - 12117919608 p^{6} T^{22} + 240356405 p^{8} T^{24} - 3694910 p^{10} T^{26} + 41022 p^{12} T^{28} - 292 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 3343790405826 p T^{17} - 322717550496 p^{2} T^{18} + 116541321036 p^{3} T^{19} + 29966075382 p^{4} T^{20} + 801423570 p^{5} T^{21} - 417738792 p^{6} T^{22} - 50514234 p^{7} T^{23} - 5251023 p^{8} T^{24} - 747702 p^{9} T^{25} + 98460 p^{10} T^{26} + 48538 p^{11} T^{27} + 3854 p^{12} T^{28} - 656 p^{13} T^{29} - 116 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 6 T - 223 T^{2} - 1686 T^{3} + 25980 T^{4} + 231654 T^{5} - 1971341 T^{6} - 20408106 T^{7} + 109216031 T^{8} + 1268388768 T^{9} - 4836103872 T^{10} - 1400930916 p T^{11} + 192545389345 T^{12} + 1824668193534 T^{13} - 7677147470143 T^{14} - 27964729912410 T^{15} + 313424888729076 T^{16} - 27964729912410 p T^{17} - 7677147470143 p^{2} T^{18} + 1824668193534 p^{3} T^{19} + 192545389345 p^{4} T^{20} - 1400930916 p^{6} T^{21} - 4836103872 p^{6} T^{22} + 1268388768 p^{7} T^{23} + 109216031 p^{8} T^{24} - 20408106 p^{9} T^{25} - 1971341 p^{10} T^{26} + 231654 p^{11} T^{27} + 25980 p^{12} T^{28} - 1686 p^{13} T^{29} - 223 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 2 T - 209 T^{2} + 14 p T^{3} + 20774 T^{4} - 72052 T^{5} - 1457073 T^{6} + 4933350 T^{7} + 91147851 T^{8} - 235722522 T^{9} - 5466637218 T^{10} + 9144255228 T^{11} + 301177025103 T^{12} - 290829011802 T^{13} - 14773623661707 T^{14} + 4808085837138 T^{15} + 658882369500660 T^{16} + 4808085837138 p T^{17} - 14773623661707 p^{2} T^{18} - 290829011802 p^{3} T^{19} + 301177025103 p^{4} T^{20} + 9144255228 p^{5} T^{21} - 5466637218 p^{6} T^{22} - 235722522 p^{7} T^{23} + 91147851 p^{8} T^{24} + 4933350 p^{9} T^{25} - 1457073 p^{10} T^{26} - 72052 p^{11} T^{27} + 20774 p^{12} T^{28} + 14 p^{14} T^{29} - 209 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 18 T + 379 T^{2} - 4272 T^{3} + 53788 T^{4} - 467070 T^{5} + 4528654 T^{6} - 32587152 T^{7} + 257763508 T^{8} - 32587152 p T^{9} + 4528654 p^{2} T^{10} - 467070 p^{3} T^{11} + 53788 p^{4} T^{12} - 4272 p^{5} T^{13} + 379 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 36 T + 766 T^{2} + 12024 T^{3} + 153999 T^{4} + 1662408 T^{5} + 15312632 T^{6} + 121481640 T^{7} + 835010600 T^{8} + 5038052724 T^{9} + 28358007246 T^{10} + 182105031078 T^{11} + 1622318702806 T^{12} + 17289317216124 T^{13} + 177226593270661 T^{14} + 1605696832118286 T^{15} + 12594715934262750 T^{16} + 1605696832118286 p T^{17} + 177226593270661 p^{2} T^{18} + 17289317216124 p^{3} T^{19} + 1622318702806 p^{4} T^{20} + 182105031078 p^{5} T^{21} + 28358007246 p^{6} T^{22} + 5038052724 p^{7} T^{23} + 835010600 p^{8} T^{24} + 121481640 p^{9} T^{25} + 15312632 p^{10} T^{26} + 1662408 p^{11} T^{27} + 153999 p^{12} T^{28} + 12024 p^{13} T^{29} + 766 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 30 T + 700 T^{2} + 11136 T^{3} + 154015 T^{4} + 1732056 T^{5} + 17715805 T^{6} + 156334302 T^{7} + 1281973738 T^{8} + 156334302 p T^{9} + 17715805 p^{2} T^{10} + 1732056 p^{3} T^{11} + 154015 p^{4} T^{12} + 11136 p^{5} T^{13} + 700 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 472 T^{2} + 107634 T^{4} - 15742946 T^{6} + 1661711189 T^{8} - 136358974992 T^{10} + 9312516091879 T^{12} - 573351459745642 T^{14} + 34616863644604218 T^{16} - 573351459745642 p^{2} T^{18} + 9312516091879 p^{4} T^{20} - 136358974992 p^{6} T^{22} + 1661711189 p^{8} T^{24} - 15742946 p^{10} T^{26} + 107634 p^{12} T^{28} - 472 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 14 T + 435 T^{2} - 5596 T^{3} + 90428 T^{4} - 1008078 T^{5} + 11375926 T^{6} - 106594220 T^{7} + 933924096 T^{8} - 106594220 p T^{9} + 11375926 p^{2} T^{10} - 1008078 p^{3} T^{11} + 90428 p^{4} T^{12} - 5596 p^{5} T^{13} + 435 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 5475157404521894 p^{2} T^{18} + 63163988645884 p^{4} T^{20} - 640511863116 p^{6} T^{22} + 5566228364 p^{8} T^{24} - 38930632 p^{10} T^{26} + 199389 p^{12} T^{28} - 650 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 11427374402737884 p T^{17} + 1383495353685575 p^{2} T^{18} + 124668495857988 p^{3} T^{19} + 17706469884892 p^{4} T^{20} + 1064159524062 p^{5} T^{21} + 204710173188 p^{6} T^{22} + 6817588704 p^{7} T^{23} + 2027081504 p^{8} T^{24} + 29833866 p^{9} T^{25} + 16361620 p^{10} T^{26} + 70380 p^{11} T^{27} + 101253 p^{12} T^{28} + 434 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 16 T + 483 T^{2} + 6542 T^{3} + 113126 T^{4} + 1278384 T^{5} + 16380706 T^{6} + 152801626 T^{7} + 1573503228 T^{8} + 152801626 p T^{9} + 16380706 p^{2} T^{10} + 1278384 p^{3} T^{11} + 113126 p^{4} T^{12} + 6542 p^{5} T^{13} + 483 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 780079655467782 p T^{17} - 4030130568645907 p^{2} T^{18} + 25282823866380 p^{3} T^{19} + 41656592194789 p^{4} T^{20} - 393115428402 p^{5} T^{21} - 386467088226 p^{6} T^{22} + 4110229572 p^{7} T^{23} + 3170469341 p^{8} T^{24} - 29138634 p^{9} T^{25} - 22183883 p^{10} T^{26} + 132990 p^{11} T^{27} + 123774 p^{12} T^{28} - 312 p^{13} T^{29} - 487 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 + 24 T - 10 T^{2} - 3636 T^{3} + 1197 T^{4} + 543420 T^{5} + 1792912 T^{6} - 58468350 T^{7} - 544062388 T^{8} + 3510821196 T^{9} + 688519692 p T^{10} - 296095488138 T^{11} - 6613804791944 T^{12} + 16297764313308 T^{13} + 597103648531739 T^{14} + 218011613471172 T^{15} - 39896652829458150 T^{16} + 218011613471172 p T^{17} + 597103648531739 p^{2} T^{18} + 16297764313308 p^{3} T^{19} - 6613804791944 p^{4} T^{20} - 296095488138 p^{5} T^{21} + 688519692 p^{7} T^{22} + 3510821196 p^{7} T^{23} - 544062388 p^{8} T^{24} - 58468350 p^{9} T^{25} + 1792912 p^{10} T^{26} + 543420 p^{11} T^{27} + 1197 p^{12} T^{28} - 3636 p^{13} T^{29} - 10 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 22669145285986746 p T^{17} + 2133943680044999 p^{2} T^{18} - 194363495836338 p^{3} T^{19} + 21002714173786 p^{4} T^{20} - 1605167799666 p^{5} T^{21} + 171203907675 p^{6} T^{22} - 14418445794 p^{7} T^{23} + 1233764039 p^{8} T^{24} - 109039644 p^{9} T^{25} + 8922340 p^{10} T^{26} - 571872 p^{11} T^{27} + 68379 p^{12} T^{28} - 2298 p^{13} T^{29} + 395 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.70800901348579261685388367750, −2.45632328609385521811229435489, −2.37715503789728528147049668330, −2.36367660008575014289293756107, −2.25304715276910005108236408472, −2.21530251244089000801214741148, −2.04940535050340770480579677888, −1.99308742354117623182517138779, −1.97717368222187136662104086433, −1.92058563110458559456097388204, −1.89248315857836021008729477786, −1.81506765614139829971059940854, −1.45876863514054403179112047587, −1.41540202436882191732809345093, −1.36357429780631348776080165327, −1.31862393606902695681812425467, −1.31323499469751763751269370200, −1.17420731286212451814425753596, −1.02425777230571826959045152477, −0.66009213726643097970309970315, −0.59609788735989439180387685783, −0.58786449694970707345100190086, −0.35412072304373766868994969863, −0.19069735290478271632564933445, −0.081349855707083990640500630889, 0.081349855707083990640500630889, 0.19069735290478271632564933445, 0.35412072304373766868994969863, 0.58786449694970707345100190086, 0.59609788735989439180387685783, 0.66009213726643097970309970315, 1.02425777230571826959045152477, 1.17420731286212451814425753596, 1.31323499469751763751269370200, 1.31862393606902695681812425467, 1.36357429780631348776080165327, 1.41540202436882191732809345093, 1.45876863514054403179112047587, 1.81506765614139829971059940854, 1.89248315857836021008729477786, 1.92058563110458559456097388204, 1.97717368222187136662104086433, 1.99308742354117623182517138779, 2.04940535050340770480579677888, 2.21530251244089000801214741148, 2.25304715276910005108236408472, 2.36367660008575014289293756107, 2.37715503789728528147049668330, 2.45632328609385521811229435489, 2.70800901348579261685388367750
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.