Dirichlet series
| L(s) = 1 | − 14·2-s − 14·3-s + 105·4-s − 3·5-s + 196·6-s + 4·7-s − 560·8-s + 105·9-s + 42·10-s + 5·11-s − 1.47e3·12-s + 14·13-s − 56·14-s + 42·15-s + 2.38e3·16-s − 7·17-s − 1.47e3·18-s + 31·19-s − 315·20-s − 56·21-s − 70·22-s + 14·23-s + 7.84e3·24-s − 25·25-s − 196·26-s − 560·27-s + 420·28-s + ⋯ |
| L(s) = 1 | − 9.89·2-s − 8.08·3-s + 52.5·4-s − 1.34·5-s + 80.0·6-s + 1.51·7-s − 197.·8-s + 35·9-s + 13.2·10-s + 1.50·11-s − 424.·12-s + 3.88·13-s − 14.9·14-s + 10.8·15-s + 595·16-s − 1.69·17-s − 346.·18-s + 7.11·19-s − 70.4·20-s − 12.2·21-s − 14.9·22-s + 2.91·23-s + 1.60e3·24-s − 5·25-s − 38.4·26-s − 107.·27-s + 79.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.99951\times 10^{25}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2818601024\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2818601024\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{14} \) |
| 3 | \( ( 1 + T )^{14} \) | |
| 13 | \( ( 1 - T )^{14} \) | |
| 103 | \( ( 1 + T )^{14} \) | |
| good | 5 | \( 1 + 3 T + 34 T^{2} + 87 T^{3} + 549 T^{4} + 1149 T^{5} + 5396 T^{6} + 8301 T^{7} + 34076 T^{8} + 25703 T^{9} + 25953 p T^{10} - 107288 T^{11} + 202568 T^{12} - 1634379 T^{13} - 162018 T^{14} - 1634379 p T^{15} + 202568 p^{2} T^{16} - 107288 p^{3} T^{17} + 25953 p^{5} T^{18} + 25703 p^{5} T^{19} + 34076 p^{6} T^{20} + 8301 p^{7} T^{21} + 5396 p^{8} T^{22} + 1149 p^{9} T^{23} + 549 p^{10} T^{24} + 87 p^{11} T^{25} + 34 p^{12} T^{26} + 3 p^{13} T^{27} + p^{14} T^{28} \) |
| 7 | \( 1 - 4 T + 31 T^{2} - 129 T^{3} + 592 T^{4} - 2141 T^{5} + 7785 T^{6} - 3555 p T^{7} + 77505 T^{8} - 216654 T^{9} + 616823 T^{10} - 1575650 T^{11} + 4236586 T^{12} - 10566813 T^{13} + 29009354 T^{14} - 10566813 p T^{15} + 4236586 p^{2} T^{16} - 1575650 p^{3} T^{17} + 616823 p^{4} T^{18} - 216654 p^{5} T^{19} + 77505 p^{6} T^{20} - 3555 p^{8} T^{21} + 7785 p^{8} T^{22} - 2141 p^{9} T^{23} + 592 p^{10} T^{24} - 129 p^{11} T^{25} + 31 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 - 5 T + 7 p T^{2} - 255 T^{3} + 2729 T^{4} - 6593 T^{5} + 65146 T^{6} - 112950 T^{7} + 1185032 T^{8} - 126028 p T^{9} + 17737693 T^{10} - 13301606 T^{11} + 20902498 p T^{12} - 117374261 T^{13} + 2659852280 T^{14} - 117374261 p T^{15} + 20902498 p^{3} T^{16} - 13301606 p^{3} T^{17} + 17737693 p^{4} T^{18} - 126028 p^{6} T^{19} + 1185032 p^{6} T^{20} - 112950 p^{7} T^{21} + 65146 p^{8} T^{22} - 6593 p^{9} T^{23} + 2729 p^{10} T^{24} - 255 p^{11} T^{25} + 7 p^{13} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 7 T + 123 T^{2} + 727 T^{3} + 7139 T^{4} + 36853 T^{5} + 264438 T^{6} + 71454 p T^{7} + 7123293 T^{8} + 29308688 T^{9} + 151804013 T^{10} + 566691726 T^{11} + 2783918143 T^{12} + 575359473 p T^{13} + 47887217988 T^{14} + 575359473 p^{2} T^{15} + 2783918143 p^{2} T^{16} + 566691726 p^{3} T^{17} + 151804013 p^{4} T^{18} + 29308688 p^{5} T^{19} + 7123293 p^{6} T^{20} + 71454 p^{8} T^{21} + 264438 p^{8} T^{22} + 36853 p^{9} T^{23} + 7139 p^{10} T^{24} + 727 p^{11} T^{25} + 123 p^{12} T^{26} + 7 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - 31 T + 642 T^{2} - 9741 T^{3} + 121456 T^{4} - 1282111 T^{5} + 11854890 T^{6} - 97421684 T^{7} + 722292215 T^{8} - 4870544446 T^{9} + 30103406600 T^{10} - 171316923644 T^{11} + 901349486328 T^{12} - 4393822430629 T^{13} + 19877998451480 T^{14} - 4393822430629 p T^{15} + 901349486328 p^{2} T^{16} - 171316923644 p^{3} T^{17} + 30103406600 p^{4} T^{18} - 4870544446 p^{5} T^{19} + 722292215 p^{6} T^{20} - 97421684 p^{7} T^{21} + 11854890 p^{8} T^{22} - 1282111 p^{9} T^{23} + 121456 p^{10} T^{24} - 9741 p^{11} T^{25} + 642 p^{12} T^{26} - 31 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 - 14 T + 163 T^{2} - 1459 T^{3} + 12107 T^{4} - 88663 T^{5} + 613169 T^{6} - 3888147 T^{7} + 23623554 T^{8} - 135264391 T^{9} + 32574855 p T^{10} - 3960305747 T^{11} + 20420056474 T^{12} - 101542944321 T^{13} + 495218559334 T^{14} - 101542944321 p T^{15} + 20420056474 p^{2} T^{16} - 3960305747 p^{3} T^{17} + 32574855 p^{5} T^{18} - 135264391 p^{5} T^{19} + 23623554 p^{6} T^{20} - 3888147 p^{7} T^{21} + 613169 p^{8} T^{22} - 88663 p^{9} T^{23} + 12107 p^{10} T^{24} - 1459 p^{11} T^{25} + 163 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 - 9 T + 201 T^{2} - 1916 T^{3} + 22888 T^{4} - 202129 T^{5} + 1823927 T^{6} - 14392698 T^{7} + 109856612 T^{8} - 26664704 p T^{9} + 5232404216 T^{10} - 33076055245 T^{11} + 202467959958 T^{12} - 1157495988315 T^{13} + 6456011753450 T^{14} - 1157495988315 p T^{15} + 202467959958 p^{2} T^{16} - 33076055245 p^{3} T^{17} + 5232404216 p^{4} T^{18} - 26664704 p^{6} T^{19} + 109856612 p^{6} T^{20} - 14392698 p^{7} T^{21} + 1823927 p^{8} T^{22} - 202129 p^{9} T^{23} + 22888 p^{10} T^{24} - 1916 p^{11} T^{25} + 201 p^{12} T^{26} - 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 31 | \( 1 - 23 T + 12 p T^{2} - 4265 T^{3} + 41240 T^{4} - 326033 T^{5} + 2271256 T^{6} - 13473214 T^{7} + 73083297 T^{8} - 353741572 T^{9} + 1757621474 T^{10} - 8879255438 T^{11} + 52216752422 T^{12} - 301306059209 T^{13} + 1780869905636 T^{14} - 301306059209 p T^{15} + 52216752422 p^{2} T^{16} - 8879255438 p^{3} T^{17} + 1757621474 p^{4} T^{18} - 353741572 p^{5} T^{19} + 73083297 p^{6} T^{20} - 13473214 p^{7} T^{21} + 2271256 p^{8} T^{22} - 326033 p^{9} T^{23} + 41240 p^{10} T^{24} - 4265 p^{11} T^{25} + 12 p^{13} T^{26} - 23 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 - 12 T + 376 T^{2} - 3991 T^{3} + 69320 T^{4} - 655633 T^{5} + 8276402 T^{6} - 70179299 T^{7} + 713605277 T^{8} - 5446190073 T^{9} + 46951903950 T^{10} - 8728769392 p T^{11} + 2429171482682 T^{12} - 15038861584088 T^{13} + 100348643121968 T^{14} - 15038861584088 p T^{15} + 2429171482682 p^{2} T^{16} - 8728769392 p^{4} T^{17} + 46951903950 p^{4} T^{18} - 5446190073 p^{5} T^{19} + 713605277 p^{6} T^{20} - 70179299 p^{7} T^{21} + 8276402 p^{8} T^{22} - 655633 p^{9} T^{23} + 69320 p^{10} T^{24} - 3991 p^{11} T^{25} + 376 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 + 5 T + 232 T^{2} + 903 T^{3} + 27991 T^{4} + 86794 T^{5} + 2356178 T^{6} + 6064167 T^{7} + 157081775 T^{8} + 355737636 T^{9} + 8884986125 T^{10} + 18797724102 T^{11} + 439539202569 T^{12} + 21573633233 p T^{13} + 19178916375602 T^{14} + 21573633233 p^{2} T^{15} + 439539202569 p^{2} T^{16} + 18797724102 p^{3} T^{17} + 8884986125 p^{4} T^{18} + 355737636 p^{5} T^{19} + 157081775 p^{6} T^{20} + 6064167 p^{7} T^{21} + 2356178 p^{8} T^{22} + 86794 p^{9} T^{23} + 27991 p^{10} T^{24} + 903 p^{11} T^{25} + 232 p^{12} T^{26} + 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 + 2 T + 364 T^{2} + 680 T^{3} + 66871 T^{4} + 109964 T^{5} + 8178187 T^{6} + 11596709 T^{7} + 17263629 p T^{8} + 907283845 T^{9} + 52880438636 T^{10} + 56375649615 T^{11} + 3047263202649 T^{12} + 67205234587 p T^{13} + 144238454924186 T^{14} + 67205234587 p^{2} T^{15} + 3047263202649 p^{2} T^{16} + 56375649615 p^{3} T^{17} + 52880438636 p^{4} T^{18} + 907283845 p^{5} T^{19} + 17263629 p^{7} T^{20} + 11596709 p^{7} T^{21} + 8178187 p^{8} T^{22} + 109964 p^{9} T^{23} + 66871 p^{10} T^{24} + 680 p^{11} T^{25} + 364 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 + 11 T + 397 T^{2} + 4309 T^{3} + 82643 T^{4} + 827019 T^{5} + 11547735 T^{6} + 104704066 T^{7} + 1188810483 T^{8} + 9771070409 T^{9} + 95017480563 T^{10} + 708893639374 T^{11} + 6077161764777 T^{12} + 41137288674466 T^{13} + 315981100890954 T^{14} + 41137288674466 p T^{15} + 6077161764777 p^{2} T^{16} + 708893639374 p^{3} T^{17} + 95017480563 p^{4} T^{18} + 9771070409 p^{5} T^{19} + 1188810483 p^{6} T^{20} + 104704066 p^{7} T^{21} + 11547735 p^{8} T^{22} + 827019 p^{9} T^{23} + 82643 p^{10} T^{24} + 4309 p^{11} T^{25} + 397 p^{12} T^{26} + 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 - 6 T + 412 T^{2} - 33 p T^{3} + 83252 T^{4} - 247756 T^{5} + 11201644 T^{6} - 22303687 T^{7} + 1133664124 T^{8} - 1409415786 T^{9} + 92045778336 T^{10} - 67796356263 T^{11} + 6221456621063 T^{12} - 2954360400877 T^{13} + 356868631966448 T^{14} - 2954360400877 p T^{15} + 6221456621063 p^{2} T^{16} - 67796356263 p^{3} T^{17} + 92045778336 p^{4} T^{18} - 1409415786 p^{5} T^{19} + 1133664124 p^{6} T^{20} - 22303687 p^{7} T^{21} + 11201644 p^{8} T^{22} - 247756 p^{9} T^{23} + 83252 p^{10} T^{24} - 33 p^{12} T^{25} + 412 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 4 T + 435 T^{2} + 547 T^{3} + 1490 p T^{4} - 121965 T^{5} + 11584809 T^{6} - 43984765 T^{7} + 1192198149 T^{8} - 6391994138 T^{9} + 105987857061 T^{10} - 600490468346 T^{11} + 8219291912012 T^{12} - 42863794863453 T^{13} + 534923792078606 T^{14} - 42863794863453 p T^{15} + 8219291912012 p^{2} T^{16} - 600490468346 p^{3} T^{17} + 105987857061 p^{4} T^{18} - 6391994138 p^{5} T^{19} + 1192198149 p^{6} T^{20} - 43984765 p^{7} T^{21} + 11584809 p^{8} T^{22} - 121965 p^{9} T^{23} + 1490 p^{11} T^{24} + 547 p^{11} T^{25} + 435 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 12 T + 482 T^{2} - 4638 T^{3} + 106787 T^{4} - 828032 T^{5} + 14480297 T^{6} - 87816605 T^{7} + 1344767383 T^{8} - 5842521801 T^{9} + 91879935358 T^{10} - 230541494581 T^{11} + 5135016712645 T^{12} - 4901001983979 T^{13} + 289194053480718 T^{14} - 4901001983979 p T^{15} + 5135016712645 p^{2} T^{16} - 230541494581 p^{3} T^{17} + 91879935358 p^{4} T^{18} - 5842521801 p^{5} T^{19} + 1344767383 p^{6} T^{20} - 87816605 p^{7} T^{21} + 14480297 p^{8} T^{22} - 828032 p^{9} T^{23} + 106787 p^{10} T^{24} - 4638 p^{11} T^{25} + 482 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 24 T + 728 T^{2} - 11067 T^{3} + 196045 T^{4} - 2144049 T^{5} + 28074428 T^{6} - 224629985 T^{7} + 2399433248 T^{8} - 12472021562 T^{9} + 120741266923 T^{10} - 110918142082 T^{11} + 2706198176098 T^{12} + 37496687932729 T^{13} + 246324937482 T^{14} + 37496687932729 p T^{15} + 2706198176098 p^{2} T^{16} - 110918142082 p^{3} T^{17} + 120741266923 p^{4} T^{18} - 12472021562 p^{5} T^{19} + 2399433248 p^{6} T^{20} - 224629985 p^{7} T^{21} + 28074428 p^{8} T^{22} - 2144049 p^{9} T^{23} + 196045 p^{10} T^{24} - 11067 p^{11} T^{25} + 728 p^{12} T^{26} - 24 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 - 20 T + 625 T^{2} - 8542 T^{3} + 149291 T^{4} - 1431774 T^{5} + 17687299 T^{6} - 106070439 T^{7} + 1019982579 T^{8} - 522497538 T^{9} + 10266084344 T^{10} + 578764158604 T^{11} - 2595505604495 T^{12} + 61931023687227 T^{13} - 244091435636824 T^{14} + 61931023687227 p T^{15} - 2595505604495 p^{2} T^{16} + 578764158604 p^{3} T^{17} + 10266084344 p^{4} T^{18} - 522497538 p^{5} T^{19} + 1019982579 p^{6} T^{20} - 106070439 p^{7} T^{21} + 17687299 p^{8} T^{22} - 1431774 p^{9} T^{23} + 149291 p^{10} T^{24} - 8542 p^{11} T^{25} + 625 p^{12} T^{26} - 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( 1 - 2 T + 443 T^{2} - 1375 T^{3} + 93120 T^{4} - 417438 T^{5} + 12418477 T^{6} - 75514184 T^{7} + 1185969174 T^{8} - 9498474967 T^{9} + 87804401592 T^{10} - 927760695467 T^{11} + 5594072409883 T^{12} - 76465537821639 T^{13} + 372417748110468 T^{14} - 76465537821639 p T^{15} + 5594072409883 p^{2} T^{16} - 927760695467 p^{3} T^{17} + 87804401592 p^{4} T^{18} - 9498474967 p^{5} T^{19} + 1185969174 p^{6} T^{20} - 75514184 p^{7} T^{21} + 12418477 p^{8} T^{22} - 417438 p^{9} T^{23} + 93120 p^{10} T^{24} - 1375 p^{11} T^{25} + 443 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 - 59 T + 2407 T^{2} - 71445 T^{3} + 1755844 T^{4} - 36460999 T^{5} + 665505041 T^{6} - 10797401048 T^{7} + 158556521206 T^{8} - 2121765780844 T^{9} + 26133042315189 T^{10} - 297453653405510 T^{11} + 3146959336917445 T^{12} - 31001951260452789 T^{13} + 285206931233742566 T^{14} - 31001951260452789 p T^{15} + 3146959336917445 p^{2} T^{16} - 297453653405510 p^{3} T^{17} + 26133042315189 p^{4} T^{18} - 2121765780844 p^{5} T^{19} + 158556521206 p^{6} T^{20} - 10797401048 p^{7} T^{21} + 665505041 p^{8} T^{22} - 36460999 p^{9} T^{23} + 1755844 p^{10} T^{24} - 71445 p^{11} T^{25} + 2407 p^{12} T^{26} - 59 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - T + 578 T^{2} - 451 T^{3} + 171663 T^{4} - 702 p T^{5} + 34522926 T^{6} + 7365824 T^{7} + 5258965667 T^{8} + 3990734318 T^{9} + 646035334504 T^{10} + 762906296233 T^{11} + 66697995009421 T^{12} + 91565920089015 T^{13} + 5936164767077296 T^{14} + 91565920089015 p T^{15} + 66697995009421 p^{2} T^{16} + 762906296233 p^{3} T^{17} + 646035334504 p^{4} T^{18} + 3990734318 p^{5} T^{19} + 5258965667 p^{6} T^{20} + 7365824 p^{7} T^{21} + 34522926 p^{8} T^{22} - 702 p^{10} T^{23} + 171663 p^{10} T^{24} - 451 p^{11} T^{25} + 578 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 - 6 T + 314 T^{2} - 1552 T^{3} + 53306 T^{4} - 187401 T^{5} + 6529520 T^{6} - 13363945 T^{7} + 697212845 T^{8} - 486604536 T^{9} + 64791377804 T^{10} + 53496142742 T^{11} + 5464351022744 T^{12} + 12805961052930 T^{13} + 458759171432148 T^{14} + 12805961052930 p T^{15} + 5464351022744 p^{2} T^{16} + 53496142742 p^{3} T^{17} + 64791377804 p^{4} T^{18} - 486604536 p^{5} T^{19} + 697212845 p^{6} T^{20} - 13363945 p^{7} T^{21} + 6529520 p^{8} T^{22} - 187401 p^{9} T^{23} + 53306 p^{10} T^{24} - 1552 p^{11} T^{25} + 314 p^{12} T^{26} - 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 + 6 T + 539 T^{2} + 4360 T^{3} + 176400 T^{4} + 1470473 T^{5} + 41966887 T^{6} + 343070551 T^{7} + 7783476451 T^{8} + 61109403984 T^{9} + 1174953427935 T^{10} + 8706554053390 T^{11} + 147557628968604 T^{12} + 1018584151007508 T^{13} + 15576640897061022 T^{14} + 1018584151007508 p T^{15} + 147557628968604 p^{2} T^{16} + 8706554053390 p^{3} T^{17} + 1174953427935 p^{4} T^{18} + 61109403984 p^{5} T^{19} + 7783476451 p^{6} T^{20} + 343070551 p^{7} T^{21} + 41966887 p^{8} T^{22} + 1470473 p^{9} T^{23} + 176400 p^{10} T^{24} + 4360 p^{11} T^{25} + 539 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.69069598273965836967808354039, −1.62662402491856799271559236727, −1.53885317331267813843548413787, −1.53495714481227442292984935248, −1.47406441777318208127510455585, −1.41253399391008934625001735483, −1.39999677892357748464225810833, −1.39911162262814804442677282062, −1.37754656900372574842551980265, −1.31633656835623084207108412431, −1.27939950253271391610590256979, −0.931947859253951152177642075376, −0.918449103972021885604456540543, −0.903478823951766187924656229891, −0.811643889722881371981280178321, −0.70820018753726197319537706233, −0.62445002108060695948292282719, −0.59713687676140533316405372415, −0.57679898831678751243497204913, −0.53278370470825405966474709112, −0.52975759548789676351445315703, −0.50590575184155791352023525741, −0.49925470795214414458870578708, −0.29849432704311060972946178695, −0.22739236479326189977802727441, 0.22739236479326189977802727441, 0.29849432704311060972946178695, 0.49925470795214414458870578708, 0.50590575184155791352023525741, 0.52975759548789676351445315703, 0.53278370470825405966474709112, 0.57679898831678751243497204913, 0.59713687676140533316405372415, 0.62445002108060695948292282719, 0.70820018753726197319537706233, 0.811643889722881371981280178321, 0.903478823951766187924656229891, 0.918449103972021885604456540543, 0.931947859253951152177642075376, 1.27939950253271391610590256979, 1.31633656835623084207108412431, 1.37754656900372574842551980265, 1.39911162262814804442677282062, 1.39999677892357748464225810833, 1.41253399391008934625001735483, 1.47406441777318208127510455585, 1.53495714481227442292984935248, 1.53885317331267813843548413787, 1.62662402491856799271559236727, 1.69069598273965836967808354039
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.