Dirichlet series
| L(s) = 1 | + 4·4-s + 4·5-s + 4·7-s + 12·11-s − 8·13-s + 6·16-s + 16·17-s + 4·19-s + 16·20-s − 4·23-s + 6·25-s + 16·28-s + 48·29-s + 16·31-s + 16·35-s + 20·37-s − 12·41-s + 4·43-s + 48·44-s − 64·47-s + 38·49-s − 32·52-s + 32·53-s + 48·55-s − 4·59-s + 4·61-s − 32·65-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.78·5-s + 1.51·7-s + 3.61·11-s − 2.21·13-s + 3/2·16-s + 3.88·17-s + 0.917·19-s + 3.57·20-s − 0.834·23-s + 6/5·25-s + 3.02·28-s + 8.91·29-s + 2.87·31-s + 2.70·35-s + 3.28·37-s − 1.87·41-s + 0.609·43-s + 7.23·44-s − 9.33·47-s + 38/7·49-s − 4.43·52-s + 4.39·53-s + 6.47·55-s − 0.520·59-s + 0.512·61-s − 3.96·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.82469\times 10^{7}\) |
| Root analytic conductor: | \(1.76469\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(132.5555519\) |
| \(L(\frac12)\) | \(\approx\) | \(132.5555519\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( ( 1 - T^{4} + T^{8} )^{2} \) | |
| 5 | \( 1 - 4 T + 2 p T^{2} - 16 T^{3} + 33 T^{4} - 64 T^{5} + 2 p^{2} T^{6} - 124 T^{7} + 116 T^{8} - 124 p T^{9} + 2 p^{4} T^{10} - 64 p^{3} T^{11} + 33 p^{4} T^{12} - 16 p^{5} T^{13} + 2 p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 8 T + 44 T^{2} + 128 T^{3} + 274 T^{4} - 280 T^{5} - 3008 T^{6} - 19608 T^{7} - 59973 T^{8} - 19608 p T^{9} - 3008 p^{2} T^{10} - 280 p^{3} T^{11} + 274 p^{4} T^{12} + 128 p^{5} T^{13} + 44 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 - 4 T - 22 T^{2} + 8 p T^{3} + 415 T^{4} - 416 T^{5} - 4586 T^{6} + 156 T^{7} + 31397 T^{8} + 11944 T^{9} - 144044 T^{10} + 104504 T^{11} + 830266 T^{12} - 2275424 T^{13} - 10166848 T^{14} + 8934456 T^{15} + 96042762 T^{16} + 8934456 p T^{17} - 10166848 p^{2} T^{18} - 2275424 p^{3} T^{19} + 830266 p^{4} T^{20} + 104504 p^{5} T^{21} - 144044 p^{6} T^{22} + 11944 p^{7} T^{23} + 31397 p^{8} T^{24} + 156 p^{9} T^{25} - 4586 p^{10} T^{26} - 416 p^{11} T^{27} + 415 p^{12} T^{28} + 8 p^{14} T^{29} - 22 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) |
| 11 | \( 1 - 12 T + 78 T^{2} - 376 T^{3} + 1371 T^{4} - 4584 T^{5} + 17282 T^{6} - 73652 T^{7} + 328937 T^{8} - 1306176 T^{9} + 4725668 T^{10} - 16480096 T^{11} + 56598750 T^{12} - 203440184 T^{13} + 724087160 T^{14} - 2516654320 T^{15} + 8585626930 T^{16} - 2516654320 p T^{17} + 724087160 p^{2} T^{18} - 203440184 p^{3} T^{19} + 56598750 p^{4} T^{20} - 16480096 p^{5} T^{21} + 4725668 p^{6} T^{22} - 1306176 p^{7} T^{23} + 328937 p^{8} T^{24} - 73652 p^{9} T^{25} + 17282 p^{10} T^{26} - 4584 p^{11} T^{27} + 1371 p^{12} T^{28} - 376 p^{13} T^{29} + 78 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 16 T + 194 T^{2} - 1488 T^{3} + 559 p T^{4} - 43176 T^{5} + 166694 T^{6} - 420584 T^{7} + 1249221 T^{8} - 5101288 T^{9} + 49686636 T^{10} - 317536376 T^{11} + 1721846106 T^{12} - 6168742856 T^{13} + 16968240520 T^{14} - 19631588904 T^{15} + 21613586442 T^{16} - 19631588904 p T^{17} + 16968240520 p^{2} T^{18} - 6168742856 p^{3} T^{19} + 1721846106 p^{4} T^{20} - 317536376 p^{5} T^{21} + 49686636 p^{6} T^{22} - 5101288 p^{7} T^{23} + 1249221 p^{8} T^{24} - 420584 p^{9} T^{25} + 166694 p^{10} T^{26} - 43176 p^{11} T^{27} + 559 p^{13} T^{28} - 1488 p^{13} T^{29} + 194 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 4 T + 26 T^{2} - 144 T^{3} - 29 T^{4} - 48 p T^{5} - 2618 T^{6} + 23908 T^{7} + 107577 T^{8} - 385024 T^{9} + 3735852 T^{10} - 16280608 T^{11} + 69539054 T^{12} - 70967144 T^{13} - 49174248 T^{14} + 665386720 T^{15} - 27736806046 T^{16} + 665386720 p T^{17} - 49174248 p^{2} T^{18} - 70967144 p^{3} T^{19} + 69539054 p^{4} T^{20} - 16280608 p^{5} T^{21} + 3735852 p^{6} T^{22} - 385024 p^{7} T^{23} + 107577 p^{8} T^{24} + 23908 p^{9} T^{25} - 2618 p^{10} T^{26} - 48 p^{12} T^{27} - 29 p^{12} T^{28} - 144 p^{13} T^{29} + 26 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 4 T + 44 T^{2} + 40 T^{3} - 148 T^{4} - 5356 T^{5} - 28800 T^{6} - 126796 T^{7} - 31782 T^{8} + 1371520 T^{9} + 19588412 T^{10} + 57381844 T^{11} + 343704688 T^{12} - 437378340 T^{13} - 2406162292 T^{14} - 45331104864 T^{15} - 184647762061 T^{16} - 45331104864 p T^{17} - 2406162292 p^{2} T^{18} - 437378340 p^{3} T^{19} + 343704688 p^{4} T^{20} + 57381844 p^{5} T^{21} + 19588412 p^{6} T^{22} + 1371520 p^{7} T^{23} - 31782 p^{8} T^{24} - 126796 p^{9} T^{25} - 28800 p^{10} T^{26} - 5356 p^{11} T^{27} - 148 p^{12} T^{28} + 40 p^{13} T^{29} + 44 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 48 T + 1262 T^{2} - 23712 T^{3} + 353527 T^{4} - 4427016 T^{5} + 48278890 T^{6} - 470232960 T^{7} + 4166908373 T^{8} - 34060663128 T^{9} + 259461909764 T^{10} - 1855747133208 T^{11} + 12529078658282 T^{12} - 80156785221072 T^{13} + 487269163598776 T^{14} - 2819815265593704 T^{15} + 15551871314055082 T^{16} - 2819815265593704 p T^{17} + 487269163598776 p^{2} T^{18} - 80156785221072 p^{3} T^{19} + 12529078658282 p^{4} T^{20} - 1855747133208 p^{5} T^{21} + 259461909764 p^{6} T^{22} - 34060663128 p^{7} T^{23} + 4166908373 p^{8} T^{24} - 470232960 p^{9} T^{25} + 48278890 p^{10} T^{26} - 4427016 p^{11} T^{27} + 353527 p^{12} T^{28} - 23712 p^{13} T^{29} + 1262 p^{14} T^{30} - 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 16 T + 128 T^{2} - 688 T^{3} + 3936 T^{4} - 36656 T^{5} + 319360 T^{6} - 2203280 T^{7} + 12963260 T^{8} - 77022032 T^{9} + 503742080 T^{10} - 3321679216 T^{11} + 20803773600 T^{12} - 121069665520 T^{13} + 715309440 p^{2} T^{14} - 126978655536 p T^{15} + 22211322219142 T^{16} - 126978655536 p^{2} T^{17} + 715309440 p^{4} T^{18} - 121069665520 p^{3} T^{19} + 20803773600 p^{4} T^{20} - 3321679216 p^{5} T^{21} + 503742080 p^{6} T^{22} - 77022032 p^{7} T^{23} + 12963260 p^{8} T^{24} - 2203280 p^{9} T^{25} + 319360 p^{10} T^{26} - 36656 p^{11} T^{27} + 3936 p^{12} T^{28} - 688 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 20 T - 24 T^{2} + 2208 T^{3} + 7438 T^{4} - 247444 T^{5} - 661408 T^{6} + 445708 p T^{7} + 80127489 T^{8} - 960270652 T^{9} - 6005983632 T^{10} + 39000103804 T^{11} + 386098066990 T^{12} - 1217230760088 T^{13} - 18982168718456 T^{14} + 16262787956276 T^{15} + 785925244643428 T^{16} + 16262787956276 p T^{17} - 18982168718456 p^{2} T^{18} - 1217230760088 p^{3} T^{19} + 386098066990 p^{4} T^{20} + 39000103804 p^{5} T^{21} - 6005983632 p^{6} T^{22} - 960270652 p^{7} T^{23} + 80127489 p^{8} T^{24} + 445708 p^{10} T^{25} - 661408 p^{10} T^{26} - 247444 p^{11} T^{27} + 7438 p^{12} T^{28} + 2208 p^{13} T^{29} - 24 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 12 T + 90 T^{2} + 1120 T^{3} + 11147 T^{4} + 86488 T^{5} + 717302 T^{6} + 5959284 T^{7} + 47282985 T^{8} + 356599864 T^{9} + 2583269332 T^{10} + 18922882632 T^{11} + 137017529838 T^{12} + 935337415968 T^{13} + 6057195240824 T^{14} + 40725971476552 T^{15} + 273033384795618 T^{16} + 40725971476552 p T^{17} + 6057195240824 p^{2} T^{18} + 935337415968 p^{3} T^{19} + 137017529838 p^{4} T^{20} + 18922882632 p^{5} T^{21} + 2583269332 p^{6} T^{22} + 356599864 p^{7} T^{23} + 47282985 p^{8} T^{24} + 5959284 p^{9} T^{25} + 717302 p^{10} T^{26} + 86488 p^{11} T^{27} + 11147 p^{12} T^{28} + 1120 p^{13} T^{29} + 90 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 4 T - 28 T^{2} + 128 T^{3} - 244 T^{4} + 14676 T^{5} - 992 p T^{6} - 192012 T^{7} + 3201786 T^{8} - 15498616 T^{9} - 100178956 T^{10} + 582115324 T^{11} + 3387912176 T^{12} + 11264846372 T^{13} - 194373541052 T^{14} - 953132684712 T^{15} + 13576225585107 T^{16} - 953132684712 p T^{17} - 194373541052 p^{2} T^{18} + 11264846372 p^{3} T^{19} + 3387912176 p^{4} T^{20} + 582115324 p^{5} T^{21} - 100178956 p^{6} T^{22} - 15498616 p^{7} T^{23} + 3201786 p^{8} T^{24} - 192012 p^{9} T^{25} - 992 p^{11} T^{26} + 14676 p^{11} T^{27} - 244 p^{12} T^{28} + 128 p^{13} T^{29} - 28 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 32 T + 542 T^{2} + 6640 T^{3} + 70013 T^{4} + 673936 T^{5} + 5819058 T^{6} + 44704640 T^{7} + 315760584 T^{8} + 44704640 p T^{9} + 5819058 p^{2} T^{10} + 673936 p^{3} T^{11} + 70013 p^{4} T^{12} + 6640 p^{5} T^{13} + 542 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 32 T + 512 T^{2} - 6104 T^{3} + 65908 T^{4} - 651672 T^{5} + 5738016 T^{6} - 45790792 T^{7} + 336961338 T^{8} - 2268876552 T^{9} + 13743878240 T^{10} - 69976369712 T^{11} + 230071189040 T^{12} + 377350627856 T^{13} - 15095347636576 T^{14} + 177561823926392 T^{15} - 1497496155946237 T^{16} + 177561823926392 p T^{17} - 15095347636576 p^{2} T^{18} + 377350627856 p^{3} T^{19} + 230071189040 p^{4} T^{20} - 69976369712 p^{5} T^{21} + 13743878240 p^{6} T^{22} - 2268876552 p^{7} T^{23} + 336961338 p^{8} T^{24} - 45790792 p^{9} T^{25} + 5738016 p^{10} T^{26} - 651672 p^{11} T^{27} + 65908 p^{12} T^{28} - 6104 p^{13} T^{29} + 512 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 4 T + 8 T^{2} + 512 T^{3} - 112 T^{4} - 24372 T^{5} + 34480 T^{6} - 1927732 T^{7} - 32675214 T^{8} - 83882232 T^{9} - 179485864 T^{10} - 7658987164 T^{11} + 15211787616 T^{12} + 522611172172 T^{13} + 86458103544 T^{14} + 22589414609432 T^{15} + 463290423709299 T^{16} + 22589414609432 p T^{17} + 86458103544 p^{2} T^{18} + 522611172172 p^{3} T^{19} + 15211787616 p^{4} T^{20} - 7658987164 p^{5} T^{21} - 179485864 p^{6} T^{22} - 83882232 p^{7} T^{23} - 32675214 p^{8} T^{24} - 1927732 p^{9} T^{25} + 34480 p^{10} T^{26} - 24372 p^{11} T^{27} - 112 p^{12} T^{28} + 512 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 4 T - 246 T^{2} + 1728 T^{3} + 29047 T^{4} - 312296 T^{5} - 2013298 T^{6} + 35897948 T^{7} + 54702357 T^{8} - 3101398808 T^{9} + 7100399196 T^{10} + 206250119576 T^{11} - 1301680214486 T^{12} - 9720109911144 T^{13} + 124251220620856 T^{14} + 219823404322984 T^{15} - 8625873473692886 T^{16} + 219823404322984 p T^{17} + 124251220620856 p^{2} T^{18} - 9720109911144 p^{3} T^{19} - 1301680214486 p^{4} T^{20} + 206250119576 p^{5} T^{21} + 7100399196 p^{6} T^{22} - 3101398808 p^{7} T^{23} + 54702357 p^{8} T^{24} + 35897948 p^{9} T^{25} - 2013298 p^{10} T^{26} - 312296 p^{11} T^{27} + 29047 p^{12} T^{28} + 1728 p^{13} T^{29} - 246 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 36 T + 812 T^{2} + 13680 T^{3} + 187508 T^{4} + 2283012 T^{5} + 25146824 T^{6} + 261014556 T^{7} + 2568969658 T^{8} + 24123735816 T^{9} + 219449135204 T^{10} + 1912641036636 T^{11} + 16267761269584 T^{12} + 134315259973188 T^{13} + 1093075200611988 T^{14} + 8906161165884504 T^{15} + 72308197379104083 T^{16} + 8906161165884504 p T^{17} + 1093075200611988 p^{2} T^{18} + 134315259973188 p^{3} T^{19} + 16267761269584 p^{4} T^{20} + 1912641036636 p^{5} T^{21} + 219449135204 p^{6} T^{22} + 24123735816 p^{7} T^{23} + 2568969658 p^{8} T^{24} + 261014556 p^{9} T^{25} + 25146824 p^{10} T^{26} + 2283012 p^{11} T^{27} + 187508 p^{12} T^{28} + 13680 p^{13} T^{29} + 812 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 36 T + 612 T^{2} + 6480 T^{3} + 57884 T^{4} + 686556 T^{5} + 9493344 T^{6} + 107972748 T^{7} + 996670522 T^{8} + 8955733032 T^{9} + 88672509684 T^{10} + 883651458708 T^{11} + 8136897483440 T^{12} + 71191387753020 T^{13} + 621645519669732 T^{14} + 76178447260200 p T^{15} + 649729440379061 p T^{16} + 76178447260200 p^{2} T^{17} + 621645519669732 p^{2} T^{18} + 71191387753020 p^{3} T^{19} + 8136897483440 p^{4} T^{20} + 883651458708 p^{5} T^{21} + 88672509684 p^{6} T^{22} + 8955733032 p^{7} T^{23} + 996670522 p^{8} T^{24} + 107972748 p^{9} T^{25} + 9493344 p^{10} T^{26} + 686556 p^{11} T^{27} + 57884 p^{12} T^{28} + 6480 p^{13} T^{29} + 612 p^{14} T^{30} + 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 804 T^{2} + 298534 T^{4} - 67348872 T^{6} + 10219900017 T^{8} - 1094107932888 T^{10} + 85638259868918 T^{12} - 5353390750475388 T^{14} + 344693827013400740 T^{16} - 5353390750475388 p^{2} T^{18} + 85638259868918 p^{4} T^{20} - 1094107932888 p^{6} T^{22} + 10219900017 p^{8} T^{24} - 67348872 p^{10} T^{26} + 298534 p^{12} T^{28} - 804 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 432 T^{2} + 105880 T^{4} - 18703760 T^{6} + 2622463324 T^{8} - 307817687856 T^{10} + 31449220334120 T^{12} - 2866887332112784 T^{14} + 237091343146988742 T^{16} - 2866887332112784 p^{2} T^{18} + 31449220334120 p^{4} T^{20} - 307817687856 p^{6} T^{22} + 2622463324 p^{8} T^{24} - 18703760 p^{10} T^{26} + 105880 p^{12} T^{28} - 432 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 - 12 T + 620 T^{2} - 6468 T^{3} + 171660 T^{4} - 1531812 T^{5} + 27698788 T^{6} - 206648460 T^{7} + 2846958518 T^{8} - 206648460 p T^{9} + 27698788 p^{2} T^{10} - 1531812 p^{3} T^{11} + 171660 p^{4} T^{12} - 6468 p^{5} T^{13} + 620 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 24 T + 498 T^{2} - 9472 T^{3} + 144723 T^{4} - 2145840 T^{5} + 29326238 T^{6} - 374649272 T^{7} + 4649475449 T^{8} - 54684169104 T^{9} + 621851871908 T^{10} - 6877907046832 T^{11} + 73432168171230 T^{12} - 763838013100784 T^{13} + 7725931053309368 T^{14} - 75902182001059216 T^{15} + 726511602032636722 T^{16} - 75902182001059216 p T^{17} + 7725931053309368 p^{2} T^{18} - 763838013100784 p^{3} T^{19} + 73432168171230 p^{4} T^{20} - 6877907046832 p^{5} T^{21} + 621851871908 p^{6} T^{22} - 54684169104 p^{7} T^{23} + 4649475449 p^{8} T^{24} - 374649272 p^{9} T^{25} + 29326238 p^{10} T^{26} - 2145840 p^{11} T^{27} + 144723 p^{12} T^{28} - 9472 p^{13} T^{29} + 498 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 60 T + 2056 T^{2} - 51360 T^{3} + 1035728 T^{4} - 17741772 T^{5} + 265962416 T^{6} - 3563385660 T^{7} + 43331225746 T^{8} - 484246905624 T^{9} + 5021626851240 T^{10} - 48740206158828 T^{11} + 446850528035296 T^{12} - 3923010962196492 T^{13} + 33801454739108280 T^{14} - 297445495958970648 T^{15} + 2800959394800939507 T^{16} - 297445495958970648 p T^{17} + 33801454739108280 p^{2} T^{18} - 3923010962196492 p^{3} T^{19} + 446850528035296 p^{4} T^{20} - 48740206158828 p^{5} T^{21} + 5021626851240 p^{6} T^{22} - 484246905624 p^{7} T^{23} + 43331225746 p^{8} T^{24} - 3563385660 p^{9} T^{25} + 265962416 p^{10} T^{26} - 17741772 p^{11} T^{27} + 1035728 p^{12} T^{28} - 51360 p^{13} T^{29} + 2056 p^{14} T^{30} - 60 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.02205165488721833084930689013, −2.98959653680451799933911188613, −2.80522247996991309674877062007, −2.80262817169839905041252031724, −2.76714302399659932802651562808, −2.68330593609035425353406323378, −2.62106265471268997003509163513, −2.62001858020081417702737803680, −2.50184291955474764851707256140, −2.49610627206295452676774690115, −2.11159802674323062282264667645, −2.07385929703355323217538245240, −2.03719159209019725168882966550, −1.85925344111451321956572188594, −1.73908083344788817600136477820, −1.58718228081539179189964308758, −1.54242107241138421945686553428, −1.50005697788257949315816997977, −1.28310884254163375514355069167, −1.24583809940017975522128328297, −1.03746128871270151968926580658, −0.999225838513310593171628839774, −0.969366924413690423518343123371, −0.68547539238969497380705407674, −0.66251158619757081263477840341, 0.66251158619757081263477840341, 0.68547539238969497380705407674, 0.969366924413690423518343123371, 0.999225838513310593171628839774, 1.03746128871270151968926580658, 1.24583809940017975522128328297, 1.28310884254163375514355069167, 1.50005697788257949315816997977, 1.54242107241138421945686553428, 1.58718228081539179189964308758, 1.73908083344788817600136477820, 1.85925344111451321956572188594, 2.03719159209019725168882966550, 2.07385929703355323217538245240, 2.11159802674323062282264667645, 2.49610627206295452676774690115, 2.50184291955474764851707256140, 2.62001858020081417702737803680, 2.62106265471268997003509163513, 2.68330593609035425353406323378, 2.76714302399659932802651562808, 2.80262817169839905041252031724, 2.80522247996991309674877062007, 2.98959653680451799933911188613, 3.02205165488721833084930689013
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.