Dirichlet series
| L(s) = 1 | − 12·5-s − 4·7-s − 4·11-s + 16·13-s + 2·16-s − 12·17-s + 8·19-s + 40·23-s + 80·25-s − 24·31-s + 48·35-s − 8·37-s − 24·43-s − 18·49-s + 28·53-s + 48·55-s + 8·59-s + 24·61-s − 192·65-s − 84·67-s + 32·71-s + 16·73-s + 16·77-s − 12·79-s − 24·80-s − 16·83-s + 144·85-s + ⋯ |
| L(s) = 1 | − 5.36·5-s − 1.51·7-s − 1.20·11-s + 4.43·13-s + 1/2·16-s − 2.91·17-s + 1.83·19-s + 8.34·23-s + 16·25-s − 4.31·31-s + 8.11·35-s − 1.31·37-s − 3.65·43-s − 2.57·49-s + 3.84·53-s + 6.47·55-s + 1.04·59-s + 3.07·61-s − 23.8·65-s − 10.2·67-s + 3.79·71-s + 1.87·73-s + 1.82·77-s − 1.35·79-s − 2.68·80-s − 1.75·83-s + 15.6·85-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.68221\times 10^{11}\) |
| Root analytic conductor: | \(2.24289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.717674486\) |
| \(L(\frac12)\) | \(\approx\) | \(1.717674486\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( 1 + 12 T + 64 T^{2} + 8 p^{2} T^{3} + 398 T^{4} + 76 p T^{5} - 992 T^{6} - 6636 T^{7} - 19341 T^{8} - 6636 p T^{9} - 992 p^{2} T^{10} + 76 p^{4} T^{11} + 398 p^{4} T^{12} + 8 p^{7} T^{13} + 64 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 4 T + 34 T^{2} + 80 T^{3} + 445 T^{4} + 704 T^{5} + 3926 T^{6} + 5044 T^{7} + 29688 T^{8} + 5044 p T^{9} + 3926 p^{2} T^{10} + 704 p^{3} T^{11} + 445 p^{4} T^{12} + 80 p^{5} T^{13} + 34 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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 | \( 1 + 12 T - 760 T^{3} - 4000 T^{4} + 9644 T^{5} + 164912 T^{6} + 417780 T^{7} - 2033646 T^{8} - 15406336 T^{9} - 21694928 T^{10} + 141136884 T^{11} + 768607008 T^{12} + 1405601268 T^{13} - 1667112976 T^{14} - 24205669888 T^{15} - 123458993805 T^{16} - 24205669888 p T^{17} - 1667112976 p^{2} T^{18} + 1405601268 p^{3} T^{19} + 768607008 p^{4} T^{20} + 141136884 p^{5} T^{21} - 21694928 p^{6} T^{22} - 15406336 p^{7} T^{23} - 2033646 p^{8} T^{24} + 417780 p^{9} T^{25} + 164912 p^{10} T^{26} + 9644 p^{11} T^{27} - 4000 p^{12} T^{28} - 760 p^{13} T^{29} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 - 40 T + 776 T^{2} - 9784 T^{3} + 91799 T^{4} - 705368 T^{5} + 4787160 T^{6} - 29946728 T^{7} + 172895985 T^{8} - 902021104 T^{9} + 4178360768 T^{10} - 17091640336 T^{11} + 61289409778 T^{12} - 7918826496 p T^{13} + 342869112800 T^{14} + 372347426016 T^{15} - 5577275049586 T^{16} + 372347426016 p T^{17} + 342869112800 p^{2} T^{18} - 7918826496 p^{4} T^{19} + 61289409778 p^{4} T^{20} - 17091640336 p^{5} T^{21} + 4178360768 p^{6} T^{22} - 902021104 p^{7} T^{23} + 172895985 p^{8} T^{24} - 29946728 p^{9} T^{25} + 4787160 p^{10} T^{26} - 705368 p^{11} T^{27} + 91799 p^{12} T^{28} - 9784 p^{13} T^{29} + 776 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 + 8 T + 248 T^{2} + 1752 T^{3} + 30199 T^{4} + 206976 T^{5} + 2540200 T^{6} + 17745904 T^{7} + 169033089 T^{8} + 1209867128 T^{9} + 9485737632 T^{10} + 68571970520 T^{11} + 465855584354 T^{12} + 3322795052152 T^{13} + 20345020109280 T^{14} + 139940970795928 T^{15} + 794780046531614 T^{16} + 139940970795928 p T^{17} + 20345020109280 p^{2} T^{18} + 3322795052152 p^{3} T^{19} + 465855584354 p^{4} T^{20} + 68571970520 p^{5} T^{21} + 9485737632 p^{6} T^{22} + 1209867128 p^{7} T^{23} + 169033089 p^{8} T^{24} + 17745904 p^{9} T^{25} + 2540200 p^{10} T^{26} + 206976 p^{11} T^{27} + 30199 p^{12} T^{28} + 1752 p^{13} T^{29} + 248 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 + 72 T^{2} - 616 T^{3} + 3239 T^{4} - 50872 T^{5} + 298520 T^{6} - 2523744 T^{7} + 15099153 T^{8} - 113209408 T^{9} + 814540096 T^{10} - 3210267840 T^{11} + 24971489874 T^{12} - 159883078560 T^{13} + 379015320032 T^{14} - 3041257505584 T^{15} + 39450787699278 T^{16} - 3041257505584 p T^{17} + 379015320032 p^{2} T^{18} - 159883078560 p^{3} T^{19} + 24971489874 p^{4} T^{20} - 3210267840 p^{5} T^{21} + 814540096 p^{6} T^{22} - 113209408 p^{7} T^{23} + 15099153 p^{8} T^{24} - 2523744 p^{9} T^{25} + 298520 p^{10} T^{26} - 50872 p^{11} T^{27} + 3239 p^{12} T^{28} - 616 p^{13} T^{29} + 72 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 28 T + 344 T^{2} - 2912 T^{3} + 25298 T^{4} - 232644 T^{5} + 1917568 T^{6} - 14158268 T^{7} + 104375121 T^{8} - 770687892 T^{9} + 5270701040 T^{10} - 39047642852 T^{11} + 318443399442 T^{12} - 2209044273928 T^{13} + 14320519756920 T^{14} - 111315852463604 T^{15} + 873920983850628 T^{16} - 111315852463604 p T^{17} + 14320519756920 p^{2} T^{18} - 2209044273928 p^{3} T^{19} + 318443399442 p^{4} T^{20} - 39047642852 p^{5} T^{21} + 5270701040 p^{6} T^{22} - 770687892 p^{7} T^{23} + 104375121 p^{8} T^{24} - 14158268 p^{9} T^{25} + 1917568 p^{10} T^{26} - 232644 p^{11} T^{27} + 25298 p^{12} T^{28} - 2912 p^{13} T^{29} + 344 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 + 84 T + 48 p T^{2} + 71776 T^{3} + 957024 T^{4} + 5741436 T^{5} - 40997200 T^{6} - 1158667636 T^{7} - 7828301806 T^{8} + 48210145144 T^{9} + 1314996417536 T^{10} + 8000602781236 T^{11} - 45952509042976 T^{12} - 1085658642853028 T^{13} - 5561139754106368 T^{14} + 34677137718483048 T^{15} + 641528682100762995 T^{16} + 34677137718483048 p T^{17} - 5561139754106368 p^{2} T^{18} - 1085658642853028 p^{3} T^{19} - 45952509042976 p^{4} T^{20} + 8000602781236 p^{5} T^{21} + 1314996417536 p^{6} T^{22} + 48210145144 p^{7} T^{23} - 7828301806 p^{8} T^{24} - 1158667636 p^{9} T^{25} - 40997200 p^{10} T^{26} + 5741436 p^{11} T^{27} + 957024 p^{12} T^{28} + 71776 p^{13} T^{29} + 48 p^{15} T^{30} + 84 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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 | \( 1 - 16 T + 416 T^{2} - 4440 T^{3} + 68404 T^{4} - 584824 T^{5} + 7229216 T^{6} - 53644504 T^{7} + 544685226 T^{8} - 2736110216 T^{9} + 18336160640 T^{10} + 72833715600 T^{11} - 1347659375280 T^{12} + 28574375923760 T^{13} - 284065502536896 T^{14} + 3320804755103832 T^{15} - 26931854637922413 T^{16} + 3320804755103832 p T^{17} - 284065502536896 p^{2} T^{18} + 28574375923760 p^{3} T^{19} - 1347659375280 p^{4} T^{20} + 72833715600 p^{5} T^{21} + 18336160640 p^{6} T^{22} - 2736110216 p^{7} T^{23} + 544685226 p^{8} T^{24} - 53644504 p^{9} T^{25} + 7229216 p^{10} T^{26} - 584824 p^{11} T^{27} + 68404 p^{12} T^{28} - 4440 p^{13} T^{29} + 416 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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
−2.92844936300377012633872368028, −2.79832267159018081331564504691, −2.72641537402117716086094679708, −2.69239156330797954175180233430, −2.68369255687875053486652542001, −2.58884328532527084576493220065, −2.52934796745084247755533399766, −2.35986019349699273703394084447, −2.18903494544880141119399066060, −2.14877337106632243211823355106, −1.73031266323121501952858810195, −1.69995628542305293297711413188, −1.63985649397590908793699511360, −1.57253289669924943304305307576, −1.44607612716723944195448213569, −1.43140931567786697822843815818, −1.34619231184155392973808562389, −1.29866664101411580752051111640, −1.17166899316633644989433414908, −0.78107962031033490045464571649, −0.63771996416968004182443505505, −0.55491287478501010187644513345, −0.49958174351089310508805749214, −0.46322656048925196301774111854, −0.20364312598947976676245266538, 0.20364312598947976676245266538, 0.46322656048925196301774111854, 0.49958174351089310508805749214, 0.55491287478501010187644513345, 0.63771996416968004182443505505, 0.78107962031033490045464571649, 1.17166899316633644989433414908, 1.29866664101411580752051111640, 1.34619231184155392973808562389, 1.43140931567786697822843815818, 1.44607612716723944195448213569, 1.57253289669924943304305307576, 1.63985649397590908793699511360, 1.69995628542305293297711413188, 1.73031266323121501952858810195, 2.14877337106632243211823355106, 2.18903494544880141119399066060, 2.35986019349699273703394084447, 2.52934796745084247755533399766, 2.58884328532527084576493220065, 2.68369255687875053486652542001, 2.69239156330797954175180233430, 2.72641537402117716086094679708, 2.79832267159018081331564504691, 2.92844936300377012633872368028
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.