Dirichlet series
| L(s) = 1 | + 12·5-s − 8·7-s + 4·11-s − 16·13-s + 2·16-s − 12·17-s − 8·19-s + 32·23-s + 56·25-s − 24·31-s − 96·35-s − 8·37-s − 24·43-s − 24·47-s + 58·49-s + 44·53-s + 48·55-s + 8·59-s + 24·61-s − 192·65-s + 36·67-s − 32·71-s − 40·73-s − 32·77-s + 12·79-s + 24·80-s + 2·81-s + ⋯ |
| L(s) = 1 | + 5.36·5-s − 3.02·7-s + 1.20·11-s − 4.43·13-s + 1/2·16-s − 2.91·17-s − 1.83·19-s + 6.67·23-s + 56/5·25-s − 4.31·31-s − 16.2·35-s − 1.31·37-s − 3.65·43-s − 3.50·47-s + 58/7·49-s + 6.04·53-s + 6.47·55-s + 1.04·59-s + 3.07·61-s − 23.8·65-s + 4.39·67-s − 3.79·71-s − 4.68·73-s − 3.64·77-s + 1.35·79-s + 2.68·80-s + 2/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3907.89\) |
| Root analytic conductor: | \(1.29493\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.507864921\) |
| \(L(\frac12)\) | \(\approx\) | \(4.507864921\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 3 | \( ( 1 - T^{4} + T^{8} )^{2} \) | |
| 5 | \( 1 - 12 T + 88 T^{2} - 472 T^{3} + 2033 T^{4} - 7336 T^{5} + 22744 T^{6} - 61572 T^{7} + 146544 T^{8} - 61572 p T^{9} + 22744 p^{2} T^{10} - 7336 p^{3} T^{11} + 2033 p^{4} T^{12} - 472 p^{5} T^{13} + 88 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 8 T + 6 T^{2} - 120 T^{3} - 275 T^{4} + 160 p T^{5} + 4770 T^{6} - 3216 T^{7} - 40872 T^{8} - 3216 p T^{9} + 4770 p^{2} T^{10} + 160 p^{4} T^{11} - 275 p^{4} T^{12} - 120 p^{5} T^{13} + 6 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 11 | \( 1 - 4 T - 42 T^{2} + 72 T^{3} + 1354 T^{4} - 52 T^{5} - 25972 T^{6} - 39092 T^{7} + 336125 T^{8} + 1014028 T^{9} - 218332 p T^{10} - 15137644 T^{11} - 156262 p T^{12} + 141984000 T^{13} + 309093362 T^{14} - 600162532 T^{15} - 4715588236 T^{16} - 600162532 p T^{17} + 309093362 p^{2} T^{18} + 141984000 p^{3} T^{19} - 156262 p^{5} T^{20} - 15137644 p^{5} T^{21} - 218332 p^{7} T^{22} + 1014028 p^{7} T^{23} + 336125 p^{8} T^{24} - 39092 p^{9} T^{25} - 25972 p^{10} T^{26} - 52 p^{11} T^{27} + 1354 p^{12} T^{28} + 72 p^{13} T^{29} - 42 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) |
| 13 | \( 1 + 16 T + 128 T^{2} + 744 T^{3} + 3314 T^{4} + 10768 T^{5} + 24864 T^{6} + 28864 T^{7} - 4655 T^{8} + 572216 T^{9} + 6801728 T^{10} + 43610328 T^{11} + 198012786 T^{12} + 641713232 T^{13} + 1520861600 T^{14} + 2393411960 T^{15} + 3297590084 T^{16} + 2393411960 p T^{17} + 1520861600 p^{2} T^{18} + 641713232 p^{3} T^{19} + 198012786 p^{4} T^{20} + 43610328 p^{5} T^{21} + 6801728 p^{6} T^{22} + 572216 p^{7} T^{23} - 4655 p^{8} T^{24} + 28864 p^{9} T^{25} + 24864 p^{10} T^{26} + 10768 p^{11} T^{27} + 3314 p^{12} T^{28} + 744 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 12 T + 144 T^{2} + 968 T^{3} + 6368 T^{4} + 26396 T^{5} + 107888 T^{6} + 145332 T^{7} - 233838 T^{8} - 8508352 T^{9} - 40945088 T^{10} - 232534236 T^{11} - 555402720 T^{12} - 1622229852 T^{13} + 5913536576 T^{14} + 33762615104 T^{15} + 261999162867 T^{16} + 33762615104 p T^{17} + 5913536576 p^{2} T^{18} - 1622229852 p^{3} T^{19} - 555402720 p^{4} T^{20} - 232534236 p^{5} T^{21} - 40945088 p^{6} T^{22} - 8508352 p^{7} T^{23} - 233838 p^{8} T^{24} + 145332 p^{9} T^{25} + 107888 p^{10} T^{26} + 26396 p^{11} T^{27} + 6368 p^{12} T^{28} + 968 p^{13} T^{29} + 144 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 8 T - 22 T^{2} + 144 T^{3} + 3371 T^{4} - 4000 T^{5} - 7722 T^{6} + 652040 T^{7} - 370487 T^{8} - 2996624 T^{9} + 89400308 T^{10} - 59920656 T^{11} - 371517522 T^{12} + 10373644960 T^{13} - 8723924296 T^{14} - 39539357168 T^{15} + 977996314130 T^{16} - 39539357168 p T^{17} - 8723924296 p^{2} T^{18} + 10373644960 p^{3} T^{19} - 371517522 p^{4} T^{20} - 59920656 p^{5} T^{21} + 89400308 p^{6} T^{22} - 2996624 p^{7} T^{23} - 370487 p^{8} T^{24} + 652040 p^{9} T^{25} - 7722 p^{10} T^{26} - 4000 p^{11} T^{27} + 3371 p^{12} T^{28} + 144 p^{13} T^{29} - 22 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 32 T + 536 T^{2} - 6152 T^{3} + 54743 T^{4} - 405160 T^{5} + 2601288 T^{6} - 14731984 T^{7} + 73350129 T^{8} - 313225808 T^{9} + 1063218752 T^{10} - 84467920 p T^{11} - 9374200334 T^{12} + 140082082224 T^{13} - 1076056625056 T^{14} + 6480812802528 T^{15} - 33308810016754 T^{16} + 6480812802528 p T^{17} - 1076056625056 p^{2} T^{18} + 140082082224 p^{3} T^{19} - 9374200334 p^{4} T^{20} - 84467920 p^{6} T^{21} + 1063218752 p^{6} T^{22} - 313225808 p^{7} T^{23} + 73350129 p^{8} T^{24} - 14731984 p^{9} T^{25} + 2601288 p^{10} T^{26} - 405160 p^{11} T^{27} + 54743 p^{12} T^{28} - 6152 p^{13} T^{29} + 536 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 13351487849248 p^{2} T^{18} + 370628380738 p^{4} T^{20} - 8752284864 p^{6} T^{22} + 171374865 p^{8} T^{24} - 2677416 p^{10} T^{26} + 31418 p^{12} T^{28} - 248 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 7075762017840 p T^{17} + 1349761682520 p^{2} T^{18} + 259935770400 p^{3} T^{19} + 48339918958 p^{4} T^{20} + 8526426768 p^{5} T^{21} + 1474786764 p^{6} T^{22} + 250780944 p^{7} T^{23} + 43727785 p^{8} T^{24} + 7589688 p^{9} T^{25} + 1273586 p^{10} T^{26} + 199680 p^{11} T^{27} + 27899 p^{12} T^{28} + 3408 p^{13} T^{29} + 334 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 8 T - 184 T^{2} - 1704 T^{3} + 16375 T^{4} + 186288 T^{5} - 883880 T^{6} - 13446752 T^{7} + 28690305 T^{8} + 694378184 T^{9} - 355698912 T^{10} - 26449712536 T^{11} - 12887758 p^{2} T^{12} + 717223895656 T^{13} + 1384593306720 T^{14} - 9602300166392 T^{15} - 59171912839138 T^{16} - 9602300166392 p T^{17} + 1384593306720 p^{2} T^{18} + 717223895656 p^{3} T^{19} - 12887758 p^{6} T^{20} - 26449712536 p^{5} T^{21} - 355698912 p^{6} T^{22} + 694378184 p^{7} T^{23} + 28690305 p^{8} T^{24} - 13446752 p^{9} T^{25} - 883880 p^{10} T^{26} + 186288 p^{11} T^{27} + 16375 p^{12} T^{28} - 1704 p^{13} T^{29} - 184 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 172 T^{2} + 13974 T^{4} - 19304 p T^{6} + 43993793 T^{8} - 63241672 p T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 5711936189732 p^{2} T^{18} + 134337801798 p^{4} T^{20} - 63241672 p^{7} T^{22} + 43993793 p^{8} T^{24} - 19304 p^{11} T^{26} + 13974 p^{12} T^{28} - 172 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 + 24 T + 288 T^{2} + 2456 T^{3} + 18144 T^{4} + 133752 T^{5} + 1000544 T^{6} + 6828856 T^{7} + 39963644 T^{8} + 227672888 T^{9} + 1517218592 T^{10} + 11117129912 T^{11} + 81274307360 T^{12} + 589957036568 T^{13} + 4310182316384 T^{14} + 31623166948824 T^{15} + 218467749390534 T^{16} + 31623166948824 p T^{17} + 4310182316384 p^{2} T^{18} + 589957036568 p^{3} T^{19} + 81274307360 p^{4} T^{20} + 11117129912 p^{5} T^{21} + 1517218592 p^{6} T^{22} + 227672888 p^{7} T^{23} + 39963644 p^{8} T^{24} + 6828856 p^{9} T^{25} + 1000544 p^{10} T^{26} + 133752 p^{11} T^{27} + 18144 p^{12} T^{28} + 2456 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 24 T + 216 T^{2} - 40 T^{3} - 18457 T^{4} - 150040 T^{5} + 44552 T^{6} + 9386184 T^{7} + 1315263 p T^{8} - 96146848 T^{9} - 3977203520 T^{10} - 21778214496 T^{11} + 50868959058 T^{12} + 1343760123600 T^{13} + 6047140835168 T^{14} - 23428375060624 T^{15} - 394415861681970 T^{16} - 23428375060624 p T^{17} + 6047140835168 p^{2} T^{18} + 1343760123600 p^{3} T^{19} + 50868959058 p^{4} T^{20} - 21778214496 p^{5} T^{21} - 3977203520 p^{6} T^{22} - 96146848 p^{7} T^{23} + 1315263 p^{9} T^{24} + 9386184 p^{9} T^{25} + 44552 p^{10} T^{26} - 150040 p^{11} T^{27} - 18457 p^{12} T^{28} - 40 p^{13} T^{29} + 216 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 44 T + 1016 T^{2} - 15712 T^{3} + 178130 T^{4} - 1540980 T^{5} + 10390336 T^{6} - 58117132 T^{7} + 357423825 T^{8} - 3614994372 T^{9} + 45716027504 T^{10} - 505985826004 T^{11} + 4503741642258 T^{12} - 32039845594664 T^{13} + 184020453561240 T^{14} - 916952178658276 T^{15} + 5330817007952772 T^{16} - 916952178658276 p T^{17} + 184020453561240 p^{2} T^{18} - 32039845594664 p^{3} T^{19} + 4503741642258 p^{4} T^{20} - 505985826004 p^{5} T^{21} + 45716027504 p^{6} T^{22} - 3614994372 p^{7} T^{23} + 357423825 p^{8} T^{24} - 58117132 p^{9} T^{25} + 10390336 p^{10} T^{26} - 1540980 p^{11} T^{27} + 178130 p^{12} T^{28} - 15712 p^{13} T^{29} + 1016 p^{14} T^{30} - 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 8 T - 4 p T^{2} + 1632 T^{3} + 30563 T^{4} - 163648 T^{5} - 2682716 T^{6} + 9836488 T^{7} + 178469613 T^{8} - 332320192 T^{9} - 10946443072 T^{10} + 3817987168 T^{11} + 756189977838 T^{12} + 175777223888 T^{13} - 56160273686696 T^{14} - 6297587928448 T^{15} + 3675723794639094 T^{16} - 6297587928448 p T^{17} - 56160273686696 p^{2} T^{18} + 175777223888 p^{3} T^{19} + 756189977838 p^{4} T^{20} + 3817987168 p^{5} T^{21} - 10946443072 p^{6} T^{22} - 332320192 p^{7} T^{23} + 178469613 p^{8} T^{24} + 9836488 p^{9} T^{25} - 2682716 p^{10} T^{26} - 163648 p^{11} T^{27} + 30563 p^{12} T^{28} + 1632 p^{13} T^{29} - 4 p^{15} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 1465410452960880 p T^{17} - 160222534806948 p^{2} T^{18} + 15272951044440 p^{3} T^{19} - 1139116635712 p^{4} T^{20} + 36652670904 p^{5} T^{21} + 8493304812 p^{6} T^{22} - 2460066384 p^{7} T^{23} + 432435906 p^{8} T^{24} - 61686168 p^{9} T^{25} + 7576872 p^{10} T^{26} - 829512 p^{11} T^{27} + 79480 p^{12} T^{28} - 6624 p^{13} T^{29} + 468 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 36 T + 960 T^{2} - 19808 T^{3} + 351456 T^{4} - 5525148 T^{5} + 79071728 T^{6} - 1045655452 T^{7} + 12919151378 T^{8} - 150131218232 T^{9} + 24647355920 p T^{10} - 17248158507668 T^{11} + 171756254361824 T^{12} - 1633413949285436 T^{13} + 14868335084466800 T^{14} - 129646495999196328 T^{15} + 1083758100261010803 T^{16} - 129646495999196328 p T^{17} + 14868335084466800 p^{2} T^{18} - 1633413949285436 p^{3} T^{19} + 171756254361824 p^{4} T^{20} - 17248158507668 p^{5} T^{21} + 24647355920 p^{7} T^{22} - 150131218232 p^{7} T^{23} + 12919151378 p^{8} T^{24} - 1045655452 p^{9} T^{25} + 79071728 p^{10} T^{26} - 5525148 p^{11} T^{27} + 351456 p^{12} T^{28} - 19808 p^{13} T^{29} + 960 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 146278 p^{2} T^{8} + 71051216 p T^{9} + 8835896 p^{2} T^{10} + 697872 p^{3} T^{11} + 72412 p^{4} T^{12} + 4432 p^{5} T^{13} + 392 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 40 T + 512 T^{2} - 1704 T^{3} - 1580 p T^{4} - 995264 T^{5} + 4224032 T^{6} + 134702512 T^{7} + 706330218 T^{8} - 4851409768 T^{9} - 89219066272 T^{10} - 434676875784 T^{11} + 2356262737872 T^{12} + 56320191577336 T^{13} + 325034384414112 T^{14} - 1853571681237960 T^{15} - 38795165571506349 T^{16} - 1853571681237960 p T^{17} + 325034384414112 p^{2} T^{18} + 56320191577336 p^{3} T^{19} + 2356262737872 p^{4} T^{20} - 434676875784 p^{5} T^{21} - 89219066272 p^{6} T^{22} - 4851409768 p^{7} T^{23} + 706330218 p^{8} T^{24} + 134702512 p^{9} T^{25} + 4224032 p^{10} T^{26} - 995264 p^{11} T^{27} - 1580 p^{13} T^{28} - 1704 p^{13} T^{29} + 512 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 12 T + 518 T^{2} - 5640 T^{3} + 137639 T^{4} - 1484376 T^{5} + 26147402 T^{6} - 279843540 T^{7} + 3950719429 T^{8} - 41115674928 T^{9} + 499791767516 T^{10} - 4982987471808 T^{11} + 54520483633690 T^{12} - 515559074594184 T^{13} + 5198103978036912 T^{14} - 46390749479828928 T^{15} + 436708099187124858 T^{16} - 46390749479828928 p T^{17} + 5198103978036912 p^{2} T^{18} - 515559074594184 p^{3} T^{19} + 54520483633690 p^{4} T^{20} - 4982987471808 p^{5} T^{21} + 499791767516 p^{6} T^{22} - 41115674928 p^{7} T^{23} + 3950719429 p^{8} T^{24} - 279843540 p^{9} T^{25} + 26147402 p^{10} T^{26} - 1484376 p^{11} T^{27} + 137639 p^{12} T^{28} - 5640 p^{13} T^{29} + 518 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 16 T + 128 T^{2} + 48 T^{3} - 16022 T^{4} - 55056 T^{5} + 1171072 T^{6} + 29664624 T^{7} + 191234977 T^{8} - 1519529248 T^{9} - 27088145920 T^{10} - 210706506976 T^{11} + 1209312464722 T^{12} + 24316714882016 T^{13} + 89859490185984 T^{14} - 1324590918805248 T^{15} - 28693493560058396 T^{16} - 1324590918805248 p T^{17} + 89859490185984 p^{2} T^{18} + 24316714882016 p^{3} T^{19} + 1209312464722 p^{4} T^{20} - 210706506976 p^{5} T^{21} - 27088145920 p^{6} T^{22} - 1519529248 p^{7} T^{23} + 191234977 p^{8} T^{24} + 29664624 p^{9} T^{25} + 1171072 p^{10} T^{26} - 55056 p^{11} T^{27} - 16022 p^{12} T^{28} + 48 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 16 T - 440 T^{2} - 6624 T^{3} + 131636 T^{4} + 1629968 T^{5} - 29449232 T^{6} - 280328720 T^{7} + 5329291050 T^{8} + 36595065152 T^{9} - 800017753480 T^{10} - 3635275214288 T^{11} + 102057968083920 T^{12} + 258547916350832 T^{13} - 11209896420041672 T^{14} - 8808353629704256 T^{15} + 1069443829226298003 T^{16} - 8808353629704256 p T^{17} - 11209896420041672 p^{2} T^{18} + 258547916350832 p^{3} T^{19} + 102057968083920 p^{4} T^{20} - 3635275214288 p^{5} T^{21} - 800017753480 p^{6} T^{22} + 36595065152 p^{7} T^{23} + 5329291050 p^{8} T^{24} - 280328720 p^{9} T^{25} - 29449232 p^{10} T^{26} + 1629968 p^{11} T^{27} + 131636 p^{12} T^{28} - 6624 p^{13} T^{29} - 440 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 44 T + 968 T^{2} - 15000 T^{3} + 231654 T^{4} - 3860388 T^{5} + 58116000 T^{6} - 743419588 T^{7} + 9111376513 T^{8} - 116138978204 T^{9} + 1430090287664 T^{10} - 15964265880436 T^{11} + 171574947065342 T^{12} - 1888109943974960 T^{13} + 20331243756190952 T^{14} - 203613182755137748 T^{15} + 1985416913924716484 T^{16} - 203613182755137748 p T^{17} + 20331243756190952 p^{2} T^{18} - 1888109943974960 p^{3} T^{19} + 171574947065342 p^{4} T^{20} - 15964265880436 p^{5} T^{21} + 1430090287664 p^{6} T^{22} - 116138978204 p^{7} T^{23} + 9111376513 p^{8} T^{24} - 743419588 p^{9} T^{25} + 58116000 p^{10} T^{26} - 3860388 p^{11} T^{27} + 231654 p^{12} T^{28} - 15000 p^{13} T^{29} + 968 p^{14} T^{30} - 44 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.58629041212724973734336450085, −3.44427963987781838015509396645, −3.34987986563469409282303515094, −3.25054712896079769835694908080, −3.16561949050084236796912819948, −3.13962501745289712786095761050, −2.97526938828148939816596502655, −2.95674623297793804247756243829, −2.69422637750338143520831217646, −2.59055967443050078886782986182, −2.58128882892844727584666324334, −2.46662526222410467302949865496, −2.20397868591573535635903639929, −2.19954396857977002512350157580, −2.18939267973319333472040331577, −2.04979528530124723330652741844, −1.98316127493371393164939762444, −1.95645094532689080246629341073, −1.79485023929811537208191051065, −1.73337544208187731407185567743, −1.42156759447870718443304892647, −1.39044946075330768687028721148, −0.70447370768372466001486010477, −0.58981192959008811344652750719, −0.56939563453193576153919577584, 0.56939563453193576153919577584, 0.58981192959008811344652750719, 0.70447370768372466001486010477, 1.39044946075330768687028721148, 1.42156759447870718443304892647, 1.73337544208187731407185567743, 1.79485023929811537208191051065, 1.95645094532689080246629341073, 1.98316127493371393164939762444, 2.04979528530124723330652741844, 2.18939267973319333472040331577, 2.19954396857977002512350157580, 2.20397868591573535635903639929, 2.46662526222410467302949865496, 2.58128882892844727584666324334, 2.59055967443050078886782986182, 2.69422637750338143520831217646, 2.95674623297793804247756243829, 2.97526938828148939816596502655, 3.13962501745289712786095761050, 3.16561949050084236796912819948, 3.25054712896079769835694908080, 3.34987986563469409282303515094, 3.44427963987781838015509396645, 3.58629041212724973734336450085
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.