Dirichlet series
| L(s) = 1 | − 2·7-s − 6·13-s − 18·17-s − 32·25-s − 6·29-s − 6·31-s − 2·37-s − 6·41-s + 2·43-s − 18·47-s + 7·49-s − 36·53-s + 30·59-s − 60·61-s − 14·67-s + 16·79-s − 24·89-s + 12·91-s − 6·97-s + 96·101-s − 30·107-s + 4·109-s − 36·113-s + 36·119-s + 68·121-s + 24·125-s + 127-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.66·13-s − 4.36·17-s − 6.39·25-s − 1.11·29-s − 1.07·31-s − 0.328·37-s − 0.937·41-s + 0.304·43-s − 2.62·47-s + 49-s − 4.94·53-s + 3.90·59-s − 7.68·61-s − 1.71·67-s + 1.80·79-s − 2.54·89-s + 1.25·91-s − 0.609·97-s + 9.55·101-s − 2.90·107-s + 0.383·109-s − 3.38·113-s + 3.30·119-s + 6.18·121-s + 2.14·125-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{48} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33580\times 10^{22}\) |
| Root analytic conductor: | \(4.91393\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5373612189\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5373612189\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 7 | \( 1 + 2 T - 3 T^{2} - 26 T^{3} - 67 T^{4} + 12 p T^{5} + 355 T^{6} - 40 T^{7} - 675 T^{8} - 40 p T^{9} + 355 p^{2} T^{10} + 12 p^{4} T^{11} - 67 p^{4} T^{12} - 26 p^{5} T^{13} - 3 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( ( 1 + 16 T^{2} - 12 T^{3} + 133 T^{4} - 138 T^{5} + 943 T^{6} - 696 T^{7} + 5539 T^{8} - 696 p T^{9} + 943 p^{2} T^{10} - 138 p^{3} T^{11} + 133 p^{4} T^{12} - 12 p^{5} T^{13} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( 1 - 68 T^{2} + 2622 T^{4} - 71710 T^{6} + 1545179 T^{8} - 27618072 T^{10} + 422760301 T^{12} - 5636332820 T^{14} + 66055843863 T^{16} - 5636332820 p^{2} T^{18} + 422760301 p^{4} T^{20} - 27618072 p^{6} T^{22} + 1545179 p^{8} T^{24} - 71710 p^{10} T^{26} + 2622 p^{12} T^{28} - 68 p^{14} T^{30} + p^{16} T^{32} \) | |
| 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 | \( 1 - 224 T^{2} + 24654 T^{4} - 1769830 T^{6} + 93146507 T^{8} - 3841107576 T^{10} + 129779286901 T^{12} - 3708814203716 T^{14} + 91488095720271 T^{16} - 3708814203716 p^{2} T^{18} + 129779286901 p^{4} T^{20} - 3841107576 p^{6} T^{22} + 93146507 p^{8} T^{24} - 1769830 p^{10} T^{26} + 24654 p^{12} T^{28} - 224 p^{14} T^{30} + p^{16} T^{32} \) | |
| 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 | \( 1 + 6 T + 164 T^{2} + 912 T^{3} + 14145 T^{4} + 82056 T^{5} + 882730 T^{6} + 5457276 T^{7} + 44537675 T^{8} + 289281588 T^{9} + 1950112812 T^{10} + 12838947000 T^{11} + 76837130221 T^{12} + 493000882878 T^{13} + 2756895706292 T^{14} + 16890347347524 T^{15} + 89911869890409 T^{16} + 16890347347524 p T^{17} + 2756895706292 p^{2} T^{18} + 493000882878 p^{3} T^{19} + 76837130221 p^{4} T^{20} + 12838947000 p^{5} T^{21} + 1950112812 p^{6} T^{22} + 289281588 p^{7} T^{23} + 44537675 p^{8} T^{24} + 5457276 p^{9} T^{25} + 882730 p^{10} T^{26} + 82056 p^{11} T^{27} + 14145 p^{12} T^{28} + 912 p^{13} T^{29} + 164 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 + 18 T - 55 T^{2} - 1722 T^{3} + 12957 T^{4} + 149790 T^{5} - 1485620 T^{6} - 10153800 T^{7} + 96840002 T^{8} + 357022848 T^{9} - 6832958010 T^{10} - 20400467106 T^{11} + 370230517390 T^{12} + 853239339180 T^{13} - 20043242784151 T^{14} - 7467517424682 T^{15} + 1132351959046185 T^{16} - 7467517424682 p T^{17} - 20043242784151 p^{2} T^{18} + 853239339180 p^{3} T^{19} + 370230517390 p^{4} T^{20} - 20400467106 p^{5} T^{21} - 6832958010 p^{6} T^{22} + 357022848 p^{7} T^{23} + 96840002 p^{8} T^{24} - 10153800 p^{9} T^{25} - 1485620 p^{10} T^{26} + 149790 p^{11} T^{27} + 12957 p^{12} T^{28} - 1722 p^{13} T^{29} - 55 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 - 30 T + 200 T^{2} + 1272 T^{3} + 1905 T^{4} - 286356 T^{5} - 330074 T^{6} + 21958908 T^{7} + 129465491 T^{8} - 1641971784 T^{9} - 13037705556 T^{10} + 82617584088 T^{11} + 950143418149 T^{12} - 1952014889118 T^{13} - 70581518213368 T^{14} + 49596128462484 T^{15} + 4103530612541721 T^{16} + 49596128462484 p T^{17} - 70581518213368 p^{2} T^{18} - 1952014889118 p^{3} T^{19} + 950143418149 p^{4} T^{20} + 82617584088 p^{5} T^{21} - 13037705556 p^{6} T^{22} - 1641971784 p^{7} T^{23} + 129465491 p^{8} T^{24} + 21958908 p^{9} T^{25} - 330074 p^{10} T^{26} - 286356 p^{11} T^{27} + 1905 p^{12} T^{28} + 1272 p^{13} T^{29} + 200 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 60 T + 2036 T^{2} + 50160 T^{3} + 994527 T^{4} + 16748022 T^{5} + 247695826 T^{6} + 3290908266 T^{7} + 39930949007 T^{8} + 448092194298 T^{9} + 4697101745298 T^{10} + 46372941826116 T^{11} + 434191490396749 T^{12} + 3878280141400200 T^{13} + 33210009451501436 T^{14} + 273658668372668376 T^{15} + 2175265354887641205 T^{16} + 273658668372668376 p T^{17} + 33210009451501436 p^{2} T^{18} + 3878280141400200 p^{3} T^{19} + 434191490396749 p^{4} T^{20} + 46372941826116 p^{5} T^{21} + 4697101745298 p^{6} T^{22} + 448092194298 p^{7} T^{23} + 39930949007 p^{8} T^{24} + 3290908266 p^{9} T^{25} + 247695826 p^{10} T^{26} + 16748022 p^{11} T^{27} + 994527 p^{12} T^{28} + 50160 p^{13} T^{29} + 2036 p^{14} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 14 T - 239 T^{2} - 5102 T^{3} + 20453 T^{4} + 910354 T^{5} + 617796 T^{6} - 105135588 T^{7} - 361795050 T^{8} + 8998911156 T^{9} + 52479815982 T^{10} - 604126066254 T^{11} - 5379501807402 T^{12} + 30548371355232 T^{13} + 457126004374545 T^{14} - 769012940073186 T^{15} - 33126987823382439 T^{16} - 769012940073186 p T^{17} + 457126004374545 p^{2} T^{18} + 30548371355232 p^{3} T^{19} - 5379501807402 p^{4} T^{20} - 604126066254 p^{5} T^{21} + 52479815982 p^{6} T^{22} + 8998911156 p^{7} T^{23} - 361795050 p^{8} T^{24} - 105135588 p^{9} T^{25} + 617796 p^{10} T^{26} + 910354 p^{11} T^{27} + 20453 p^{12} T^{28} - 5102 p^{13} T^{29} - 239 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 - 16 T - 227 T^{2} + 5356 T^{3} + 15491 T^{4} - 818138 T^{5} + 465174 T^{6} + 79955730 T^{7} - 161428446 T^{8} - 5129891178 T^{9} + 6184544046 T^{10} + 147661653486 T^{11} + 2275304241738 T^{12} + 3934261822074 T^{13} - 453211437102513 T^{14} - 289666157971230 T^{15} + 45106559543464353 T^{16} - 289666157971230 p T^{17} - 453211437102513 p^{2} T^{18} + 3934261822074 p^{3} T^{19} + 2275304241738 p^{4} T^{20} + 147661653486 p^{5} T^{21} + 6184544046 p^{6} T^{22} - 5129891178 p^{7} T^{23} - 161428446 p^{8} T^{24} + 79955730 p^{9} T^{25} + 465174 p^{10} T^{26} - 818138 p^{11} T^{27} + 15491 p^{12} T^{28} + 5356 p^{13} T^{29} - 227 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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
−2.16840875211571459220351682006, −1.96902140745980186696422430377, −1.93890809641360118813679430837, −1.87964523765859066159825576390, −1.87122382204427054643519954717, −1.72112976633752104296356705712, −1.71325500021392584590369726135, −1.63071965086958680822565853328, −1.62647025121189820663948539312, −1.61983239069538729371967457931, −1.50869749297038142397201177383, −1.44901777096490411229679263919, −1.37348183703325662578305153386, −1.23973032863149171800665479240, −1.20659247729142842555226612095, −0.969702666371568298605471133780, −0.820549032405487601572779391780, −0.71518999230132278158759151769, −0.51944176790316498217490433291, −0.47822527598616712658658178342, −0.34678950649650720744455183822, −0.33666112902928117538698842204, −0.22785483891183799421711728330, −0.15912393604904100537367906386, −0.14182473951111429681928647259, 0.14182473951111429681928647259, 0.15912393604904100537367906386, 0.22785483891183799421711728330, 0.33666112902928117538698842204, 0.34678950649650720744455183822, 0.47822527598616712658658178342, 0.51944176790316498217490433291, 0.71518999230132278158759151769, 0.820549032405487601572779391780, 0.969702666371568298605471133780, 1.20659247729142842555226612095, 1.23973032863149171800665479240, 1.37348183703325662578305153386, 1.44901777096490411229679263919, 1.50869749297038142397201177383, 1.61983239069538729371967457931, 1.62647025121189820663948539312, 1.63071965086958680822565853328, 1.71325500021392584590369726135, 1.72112976633752104296356705712, 1.87122382204427054643519954717, 1.87964523765859066159825576390, 1.93890809641360118813679430837, 1.96902140745980186696422430377, 2.16840875211571459220351682006
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.