Dirichlet series
| L(s) = 1 | + 5·4-s + 7-s − 4·8-s − 4·11-s + 5·13-s + 14·16-s + 2·17-s + 22·19-s − 4·23-s + 21·25-s + 5·28-s − 15·29-s + 3·31-s − 23·32-s + 4·37-s − 19·41-s + 11·43-s − 20·44-s − 5·47-s + 7·49-s + 25·52-s − 36·53-s − 4·56-s + 17·59-s + 44·61-s + 29·64-s − 52·67-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 0.377·7-s − 1.41·8-s − 1.20·11-s + 1.38·13-s + 7/2·16-s + 0.485·17-s + 5.04·19-s − 0.834·23-s + 21/5·25-s + 0.944·28-s − 2.78·29-s + 0.538·31-s − 4.06·32-s + 0.657·37-s − 2.96·41-s + 1.67·43-s − 3.01·44-s − 0.729·47-s + 49-s + 3.46·52-s − 4.94·53-s − 0.534·56-s + 2.21·59-s + 5.63·61-s + 29/8·64-s − 6.35·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 7^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11937\times 10^{13}\) |
| Root analytic conductor: | \(2.55729\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(39.67061042\) |
| \(L(\frac12)\) | \(\approx\) | \(39.67061042\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T - 6 T^{2} + 5 T^{3} + 4 p T^{4} + 81 T^{5} - 197 T^{6} - 67 p T^{7} + 2028 T^{8} - 67 p^{2} T^{9} - 197 p^{2} T^{10} + 81 p^{3} T^{11} + 4 p^{5} T^{12} + 5 p^{5} T^{13} - 6 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 6 T^{2} + 10 T^{3} + 19 p T^{4} - 720 T^{5} + 145 T^{6} - 4565 T^{7} + 64050 T^{8} - 4565 p T^{9} + 145 p^{2} T^{10} - 720 p^{3} T^{11} + 19 p^{5} T^{12} + 10 p^{5} T^{13} + 6 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - 5 T^{2} + p^{2} T^{3} + 11 T^{4} - 17 T^{5} + p T^{6} + 27 T^{7} - 61 T^{8} + 23 T^{9} + 137 T^{10} - 85 p T^{11} - 83 T^{12} + 79 p^{2} T^{13} - 211 T^{14} - 215 T^{15} + 719 T^{16} - 215 p T^{17} - 211 p^{2} T^{18} + 79 p^{5} T^{19} - 83 p^{4} T^{20} - 85 p^{6} T^{21} + 137 p^{6} T^{22} + 23 p^{7} T^{23} - 61 p^{8} T^{24} + 27 p^{9} T^{25} + p^{11} T^{26} - 17 p^{11} T^{27} + 11 p^{12} T^{28} + p^{15} T^{29} - 5 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( 1 - 21 T^{2} - 2 p T^{3} + 197 T^{4} + 163 T^{5} - 239 p T^{6} - 743 T^{7} + 6997 T^{8} - 33 p T^{9} - 10259 p T^{10} - 4408 T^{11} + 340012 T^{12} + 149669 T^{13} - 327916 p T^{14} - 505874 T^{15} + 7287304 T^{16} - 505874 p T^{17} - 327916 p^{3} T^{18} + 149669 p^{3} T^{19} + 340012 p^{4} T^{20} - 4408 p^{5} T^{21} - 10259 p^{7} T^{22} - 33 p^{8} T^{23} + 6997 p^{8} T^{24} - 743 p^{9} T^{25} - 239 p^{11} T^{26} + 163 p^{11} T^{27} + 197 p^{12} T^{28} - 2 p^{14} T^{29} - 21 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( ( 1 + 2 T + 47 T^{2} + 111 T^{3} + 1236 T^{4} + 2638 T^{5} + 21984 T^{6} + 41797 T^{7} + 279232 T^{8} + 41797 p T^{9} + 21984 p^{2} T^{10} + 2638 p^{3} T^{11} + 1236 p^{4} T^{12} + 111 p^{5} T^{13} + 47 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 2 T - 64 T^{2} + 156 T^{3} + 1833 T^{4} - 5492 T^{5} - 33046 T^{6} + 129314 T^{7} + 450298 T^{8} - 2431350 T^{9} - 4576644 T^{10} + 35905738 T^{11} + 9387682 T^{12} - 371638066 T^{13} + 950280703 T^{14} + 1942834138 T^{15} - 24004209966 T^{16} + 1942834138 p T^{17} + 950280703 p^{2} T^{18} - 371638066 p^{3} T^{19} + 9387682 p^{4} T^{20} + 35905738 p^{5} T^{21} - 4576644 p^{6} T^{22} - 2431350 p^{7} T^{23} + 450298 p^{8} T^{24} + 129314 p^{9} T^{25} - 33046 p^{10} T^{26} - 5492 p^{11} T^{27} + 1833 p^{12} T^{28} + 156 p^{13} T^{29} - 64 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 11 T + 140 T^{2} - 981 T^{3} + 7482 T^{4} - 40495 T^{5} + 238897 T^{6} - 1084061 T^{7} + 5342056 T^{8} - 1084061 p T^{9} + 238897 p^{2} T^{10} - 40495 p^{3} T^{11} + 7482 p^{4} T^{12} - 981 p^{5} T^{13} + 140 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 4 T - 118 T^{2} - 436 T^{3} + 7734 T^{4} + 26415 T^{5} - 344912 T^{6} - 1120862 T^{7} + 11400973 T^{8} + 1565586 p T^{9} - 293902113 T^{10} - 894901783 T^{11} + 6211644588 T^{12} + 16149910900 T^{13} - 118401154460 T^{14} - 140859369762 T^{15} + 2456835101046 T^{16} - 140859369762 p T^{17} - 118401154460 p^{2} T^{18} + 16149910900 p^{3} T^{19} + 6211644588 p^{4} T^{20} - 894901783 p^{5} T^{21} - 293902113 p^{6} T^{22} + 1565586 p^{8} T^{23} + 11400973 p^{8} T^{24} - 1120862 p^{9} T^{25} - 344912 p^{10} T^{26} + 26415 p^{11} T^{27} + 7734 p^{12} T^{28} - 436 p^{13} T^{29} - 118 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 15 T - 6 T^{2} - 815 T^{3} + 2345 T^{4} + 56626 T^{5} - 990 p T^{6} - 61073 p T^{7} + 2048957 T^{8} + 60143926 T^{9} + 115849537 T^{10} - 850143313 T^{11} - 6898778608 T^{12} + 21046980820 T^{13} + 452641755556 T^{14} + 138544395442 T^{15} - 12857101724960 T^{16} + 138544395442 p T^{17} + 452641755556 p^{2} T^{18} + 21046980820 p^{3} T^{19} - 6898778608 p^{4} T^{20} - 850143313 p^{5} T^{21} + 115849537 p^{6} T^{22} + 60143926 p^{7} T^{23} + 2048957 p^{8} T^{24} - 61073 p^{10} T^{25} - 990 p^{11} T^{26} + 56626 p^{11} T^{27} + 2345 p^{12} T^{28} - 815 p^{13} T^{29} - 6 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 3 T - 80 T^{2} + 753 T^{3} + 1092 T^{4} - 46457 T^{5} + 220332 T^{6} + 1046342 T^{7} - 14211319 T^{8} + 31888447 T^{9} + 369460130 T^{10} - 2912434243 T^{11} + 1186850305 T^{12} + 95815568717 T^{13} - 439310068799 T^{14} - 1217655157396 T^{15} + 18735868674340 T^{16} - 1217655157396 p T^{17} - 439310068799 p^{2} T^{18} + 95815568717 p^{3} T^{19} + 1186850305 p^{4} T^{20} - 2912434243 p^{5} T^{21} + 369460130 p^{6} T^{22} + 31888447 p^{7} T^{23} - 14211319 p^{8} T^{24} + 1046342 p^{9} T^{25} + 220332 p^{10} T^{26} - 46457 p^{11} T^{27} + 1092 p^{12} T^{28} + 753 p^{13} T^{29} - 80 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 4 T - 161 T^{2} + 698 T^{3} + 12653 T^{4} - 1527 p T^{5} - 646079 T^{6} + 2787577 T^{7} + 25616215 T^{8} - 84742349 T^{9} - 1011939655 T^{10} + 1209387882 T^{11} + 47747743954 T^{12} + 12036495365 T^{13} - 2303190372612 T^{14} - 505883343190 T^{15} + 94678266821288 T^{16} - 505883343190 p T^{17} - 2303190372612 p^{2} T^{18} + 12036495365 p^{3} T^{19} + 47747743954 p^{4} T^{20} + 1209387882 p^{5} T^{21} - 1011939655 p^{6} T^{22} - 84742349 p^{7} T^{23} + 25616215 p^{8} T^{24} + 2787577 p^{9} T^{25} - 646079 p^{10} T^{26} - 1527 p^{12} T^{27} + 12653 p^{12} T^{28} + 698 p^{13} T^{29} - 161 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 19 T + 55 T^{2} - 898 T^{3} - 5698 T^{4} - 16130 T^{5} - 337390 T^{6} - 2358667 T^{7} + 9439631 T^{8} + 161858061 T^{9} + 581239977 T^{10} + 3187530408 T^{11} + 37046289522 T^{12} + 51167015014 T^{13} - 1547466225765 T^{14} - 9408233383915 T^{15} - 34873463295586 T^{16} - 9408233383915 p T^{17} - 1547466225765 p^{2} T^{18} + 51167015014 p^{3} T^{19} + 37046289522 p^{4} T^{20} + 3187530408 p^{5} T^{21} + 581239977 p^{6} T^{22} + 161858061 p^{7} T^{23} + 9439631 p^{8} T^{24} - 2358667 p^{9} T^{25} - 337390 p^{10} T^{26} - 16130 p^{11} T^{27} - 5698 p^{12} T^{28} - 898 p^{13} T^{29} + 55 p^{14} T^{30} + 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 11 T - 135 T^{2} + 2100 T^{3} + 6137 T^{4} - 176179 T^{5} - 40953 T^{6} + 8402878 T^{7} + 3448032 T^{8} - 271175414 T^{9} - 1428475026 T^{10} + 9515158664 T^{11} + 124800942246 T^{12} - 411307083359 T^{13} - 6208904645339 T^{14} + 198702541525 p T^{15} + 260553692815250 T^{16} + 198702541525 p^{2} T^{17} - 6208904645339 p^{2} T^{18} - 411307083359 p^{3} T^{19} + 124800942246 p^{4} T^{20} + 9515158664 p^{5} T^{21} - 1428475026 p^{6} T^{22} - 271175414 p^{7} T^{23} + 3448032 p^{8} T^{24} + 8402878 p^{9} T^{25} - 40953 p^{10} T^{26} - 176179 p^{11} T^{27} + 6137 p^{12} T^{28} + 2100 p^{13} T^{29} - 135 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 5 T - 168 T^{2} - 161 T^{3} + 16599 T^{4} - 36967 T^{5} - 885940 T^{6} + 5198908 T^{7} + 21412749 T^{8} - 271776345 T^{9} + 346611318 T^{10} + 4913373621 T^{11} - 20631932251 T^{12} + 180472147091 T^{13} - 1523045035175 T^{14} - 7154378798768 T^{15} + 147263530000230 T^{16} - 7154378798768 p T^{17} - 1523045035175 p^{2} T^{18} + 180472147091 p^{3} T^{19} - 20631932251 p^{4} T^{20} + 4913373621 p^{5} T^{21} + 346611318 p^{6} T^{22} - 271776345 p^{7} T^{23} + 21412749 p^{8} T^{24} + 5198908 p^{9} T^{25} - 885940 p^{10} T^{26} - 36967 p^{11} T^{27} + 16599 p^{12} T^{28} - 161 p^{13} T^{29} - 168 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 36 T + 508 T^{2} + 3932 T^{3} + 29444 T^{4} + 280817 T^{5} + 1836996 T^{6} + 8495258 T^{7} + 63588909 T^{8} + 353688724 T^{9} + 1151048267 T^{10} + 22337541825 T^{11} + 242039231570 T^{12} + 1659435705048 T^{13} + 17509974066650 T^{14} + 155428900722104 T^{15} + 1072759047046030 T^{16} + 155428900722104 p T^{17} + 17509974066650 p^{2} T^{18} + 1659435705048 p^{3} T^{19} + 242039231570 p^{4} T^{20} + 22337541825 p^{5} T^{21} + 1151048267 p^{6} T^{22} + 353688724 p^{7} T^{23} + 63588909 p^{8} T^{24} + 8495258 p^{9} T^{25} + 1836996 p^{10} T^{26} + 280817 p^{11} T^{27} + 29444 p^{12} T^{28} + 3932 p^{13} T^{29} + 508 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 17 T - 117 T^{2} + 1678 T^{3} + 31523 T^{4} - 155083 T^{5} - 3653861 T^{6} - 1779514 T^{7} + 328712872 T^{8} + 1147364866 T^{9} - 17281370072 T^{10} - 132663891622 T^{11} + 533757239924 T^{12} + 7300564633963 T^{13} + 7414801168177 T^{14} - 192564944881873 T^{15} - 1192310545014150 T^{16} - 192564944881873 p T^{17} + 7414801168177 p^{2} T^{18} + 7300564633963 p^{3} T^{19} + 533757239924 p^{4} T^{20} - 132663891622 p^{5} T^{21} - 17281370072 p^{6} T^{22} + 1147364866 p^{7} T^{23} + 328712872 p^{8} T^{24} - 1779514 p^{9} T^{25} - 3653861 p^{10} T^{26} - 155083 p^{11} T^{27} + 31523 p^{12} T^{28} + 1678 p^{13} T^{29} - 117 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 22 T + 579 T^{2} - 8159 T^{3} + 124286 T^{4} - 1305086 T^{5} + 14599980 T^{6} - 122294457 T^{7} + 1091766164 T^{8} - 122294457 p T^{9} + 14599980 p^{2} T^{10} - 1305086 p^{3} T^{11} + 124286 p^{4} T^{12} - 8159 p^{5} T^{13} + 579 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 26 T + 509 T^{2} + 6689 T^{3} + 76718 T^{4} + 701954 T^{5} + 6084092 T^{6} + 47009719 T^{7} + 390863832 T^{8} + 47009719 p T^{9} + 6084092 p^{2} T^{10} + 701954 p^{3} T^{11} + 76718 p^{4} T^{12} + 6689 p^{5} T^{13} + 509 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 9 T - 386 T^{2} - 3589 T^{3} + 81235 T^{4} + 746618 T^{5} - 12406552 T^{6} - 106145741 T^{7} + 1537530601 T^{8} + 11338510824 T^{9} - 161456593527 T^{10} - 918377967383 T^{11} + 14822377342250 T^{12} + 52736724869388 T^{13} - 1218705028862170 T^{14} - 1440399689396096 T^{15} + 90831121962753112 T^{16} - 1440399689396096 p T^{17} - 1218705028862170 p^{2} T^{18} + 52736724869388 p^{3} T^{19} + 14822377342250 p^{4} T^{20} - 918377967383 p^{5} T^{21} - 161456593527 p^{6} T^{22} + 11338510824 p^{7} T^{23} + 1537530601 p^{8} T^{24} - 106145741 p^{9} T^{25} - 12406552 p^{10} T^{26} + 746618 p^{11} T^{27} + 81235 p^{12} T^{28} - 3589 p^{13} T^{29} - 386 p^{14} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 6 T - 165 T^{2} + 1806 T^{3} + 26272 T^{4} - 359064 T^{5} + 1422661 T^{6} + 45649262 T^{7} - 424402259 T^{8} + 411144950 T^{9} + 51687211764 T^{10} - 358428082760 T^{11} - 635781335617 T^{12} + 45102010283078 T^{13} - 217262895796719 T^{14} - 1212829928109410 T^{15} + 31435042908856916 T^{16} - 1212829928109410 p T^{17} - 217262895796719 p^{2} T^{18} + 45102010283078 p^{3} T^{19} - 635781335617 p^{4} T^{20} - 358428082760 p^{5} T^{21} + 51687211764 p^{6} T^{22} + 411144950 p^{7} T^{23} - 424402259 p^{8} T^{24} + 45649262 p^{9} T^{25} + 1422661 p^{10} T^{26} - 359064 p^{11} T^{27} + 26272 p^{12} T^{28} + 1806 p^{13} T^{29} - 165 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 16 T - 231 T^{2} + 4320 T^{3} + 34115 T^{4} - 635696 T^{5} - 4129200 T^{6} + 64433872 T^{7} + 434574435 T^{8} - 5166633824 T^{9} - 32859288299 T^{10} + 304450300496 T^{11} + 1757236296274 T^{12} - 12357156907472 T^{13} - 63637790409375 T^{14} + 230917397502624 T^{15} + 3042650590829296 T^{16} + 230917397502624 p T^{17} - 63637790409375 p^{2} T^{18} - 12357156907472 p^{3} T^{19} + 1757236296274 p^{4} T^{20} + 304450300496 p^{5} T^{21} - 32859288299 p^{6} T^{22} - 5166633824 p^{7} T^{23} + 434574435 p^{8} T^{24} + 64433872 p^{9} T^{25} - 4129200 p^{10} T^{26} - 635696 p^{11} T^{27} + 34115 p^{12} T^{28} + 4320 p^{13} T^{29} - 231 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 18 T + 468 T^{2} + 5152 T^{3} + 82570 T^{4} + 680729 T^{5} + 9262051 T^{6} + 65493403 T^{7} + 834546596 T^{8} + 65493403 p T^{9} + 9262051 p^{2} T^{10} + 680729 p^{3} T^{11} + 82570 p^{4} T^{12} + 5152 p^{5} T^{13} + 468 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 20 T - 230 T^{2} - 6910 T^{3} + 23173 T^{4} + 1212909 T^{5} - 1800159 T^{6} - 144338618 T^{7} + 324101485 T^{8} + 13502896607 T^{9} - 69738425178 T^{10} - 1068854739785 T^{11} + 10722335634037 T^{12} + 66540867336460 T^{13} - 1270995603836848 T^{14} - 24065235744079 p T^{15} + 1388943676654486 p T^{16} - 24065235744079 p^{2} T^{17} - 1270995603836848 p^{2} T^{18} + 66540867336460 p^{3} T^{19} + 10722335634037 p^{4} T^{20} - 1068854739785 p^{5} T^{21} - 69738425178 p^{6} T^{22} + 13502896607 p^{7} T^{23} + 324101485 p^{8} T^{24} - 144338618 p^{9} T^{25} - 1800159 p^{10} T^{26} + 1212909 p^{11} T^{27} + 23173 p^{12} T^{28} - 6910 p^{13} T^{29} - 230 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 7 T - 310 T^{2} + 2485 T^{3} + 42400 T^{4} - 379400 T^{5} - 3621348 T^{6} + 31983102 T^{7} + 207352950 T^{8} - 1459259835 T^{9} + 6146393957 T^{10} - 52607230120 T^{11} - 3573659402933 T^{12} + 20216002617763 T^{13} + 477664863434550 T^{14} - 1221840745845096 T^{15} - 46591681297798206 T^{16} - 1221840745845096 p T^{17} + 477664863434550 p^{2} T^{18} + 20216002617763 p^{3} T^{19} - 3573659402933 p^{4} T^{20} - 52607230120 p^{5} T^{21} + 6146393957 p^{6} T^{22} - 1459259835 p^{7} T^{23} + 207352950 p^{8} T^{24} + 31983102 p^{9} T^{25} - 3621348 p^{10} T^{26} - 379400 p^{11} T^{27} + 42400 p^{12} T^{28} + 2485 p^{13} T^{29} - 310 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.82133644036110197068371648257, −2.59079724979655038206422614439, −2.45176276020761611991307544419, −2.42205530628917753960023504827, −2.37735001493055377379015418227, −2.36530249722327871942503934275, −2.35241562899336740396753506959, −2.12388897658819517734965265929, −2.09622343197798029410947677713, −1.96505401548148924666114132088, −1.80064115270935418277348550727, −1.56481994745208551397946564599, −1.53557730294854673516762020124, −1.48699418714608180642156120355, −1.47480998365317350508900396581, −1.43600376873028875432278563480, −1.42942637659312802826033175704, −1.22653396605298153167751040270, −1.18727514358592665360195438869, −1.07768804512887670531943938517, −0.70913159138250785947221616637, −0.66457634493936205488864598850, −0.53846120171215439535674616875, −0.34576097113995388115049028433, −0.28988813346988428727932095573, 0.28988813346988428727932095573, 0.34576097113995388115049028433, 0.53846120171215439535674616875, 0.66457634493936205488864598850, 0.70913159138250785947221616637, 1.07768804512887670531943938517, 1.18727514358592665360195438869, 1.22653396605298153167751040270, 1.42942637659312802826033175704, 1.43600376873028875432278563480, 1.47480998365317350508900396581, 1.48699418714608180642156120355, 1.53557730294854673516762020124, 1.56481994745208551397946564599, 1.80064115270935418277348550727, 1.96505401548148924666114132088, 2.09622343197798029410947677713, 2.12388897658819517734965265929, 2.35241562899336740396753506959, 2.36530249722327871942503934275, 2.37735001493055377379015418227, 2.42205530628917753960023504827, 2.45176276020761611991307544419, 2.59079724979655038206422614439, 2.82133644036110197068371648257
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.