Dirichlet series
| L(s) = 1 | + 4·4-s − 2·5-s + 6·11-s + 2·13-s + 6·16-s − 16·17-s − 8·20-s − 6·23-s − 5·25-s − 4·27-s − 6·29-s − 20·37-s − 44·41-s + 24·44-s + 52·47-s + 27·49-s + 8·52-s − 24·53-s − 12·55-s − 46·59-s + 6·61-s − 4·65-s + 12·67-s − 64·68-s + 6·71-s − 12·80-s − 12·81-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.894·5-s + 1.80·11-s + 0.554·13-s + 3/2·16-s − 3.88·17-s − 1.78·20-s − 1.25·23-s − 25-s − 0.769·27-s − 1.11·29-s − 3.28·37-s − 6.87·41-s + 3.61·44-s + 7.58·47-s + 27/7·49-s + 1.10·52-s − 3.29·53-s − 1.61·55-s − 5.98·59-s + 0.768·61-s − 0.496·65-s + 1.46·67-s − 7.76·68-s + 0.712·71-s − 1.34·80-s − 4/3·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.81772\) |
| Root analytic conductor: | \(1.01884\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2778753714\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2778753714\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 5 | \( ( 1 + T + 4 T^{2} + 19 T^{3} + 18 T^{4} + 19 p T^{5} + 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 2 T - 14 T^{3} + 40 T^{4} - 12 T^{5} - 2206 T^{6} + 8900 T^{7} + 3303 T^{8} + 8900 p T^{9} - 2206 p^{2} T^{10} - 12 p^{3} T^{11} + 40 p^{4} T^{12} - 14 p^{5} T^{13} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + 4 T^{3} + 4 p T^{4} + 2 T^{5} + 8 T^{6} + 28 T^{7} + 26 p T^{8} - 40 T^{9} + 2 p^{2} T^{10} + 14 T^{11} - 1156 T^{12} - 938 T^{13} - 472 T^{14} - 4268 T^{15} - 14165 T^{16} - 4268 p T^{17} - 472 p^{2} T^{18} - 938 p^{3} T^{19} - 1156 p^{4} T^{20} + 14 p^{5} T^{21} + 2 p^{8} T^{22} - 40 p^{7} T^{23} + 26 p^{9} T^{24} + 28 p^{9} T^{25} + 8 p^{10} T^{26} + 2 p^{11} T^{27} + 4 p^{13} T^{28} + 4 p^{13} T^{29} + p^{16} T^{32} \) |
| 7 | \( 1 - 27 T^{2} + 340 T^{4} - 2283 T^{6} + 7069 T^{8} - 2952 T^{10} + 375850 T^{12} - 8501826 T^{14} + 82424344 T^{16} - 8501826 p^{2} T^{18} + 375850 p^{4} T^{20} - 2952 p^{6} T^{22} + 7069 p^{8} T^{24} - 2283 p^{10} T^{26} + 340 p^{12} T^{28} - 27 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 6 T + 9 T^{2} + 54 T^{3} - 126 T^{4} + 192 T^{5} - 171 p T^{6} + 894 p T^{7} + 14585 T^{8} - 130974 T^{9} + 342996 T^{10} - 674106 T^{11} + 4292502 T^{12} - 3170472 T^{13} - 34558152 T^{14} + 137683686 T^{15} - 171563688 T^{16} + 137683686 p T^{17} - 34558152 p^{2} T^{18} - 3170472 p^{3} T^{19} + 4292502 p^{4} T^{20} - 674106 p^{5} T^{21} + 342996 p^{6} T^{22} - 130974 p^{7} T^{23} + 14585 p^{8} T^{24} + 894 p^{10} T^{25} - 171 p^{11} T^{26} + 192 p^{11} T^{27} - 126 p^{12} T^{28} + 54 p^{13} T^{29} + 9 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 16 T + 131 T^{2} + 664 T^{3} + 2621 T^{4} + 12030 T^{5} + 73006 T^{6} + 376698 T^{7} + 1318525 T^{8} + 2893186 T^{9} + 5325729 T^{10} + 14363242 T^{11} - 44621642 T^{12} - 1090579870 T^{13} - 7367918553 T^{14} - 31234892458 T^{15} - 120018415660 T^{16} - 31234892458 p T^{17} - 7367918553 p^{2} T^{18} - 1090579870 p^{3} T^{19} - 44621642 p^{4} T^{20} + 14363242 p^{5} T^{21} + 5325729 p^{6} T^{22} + 2893186 p^{7} T^{23} + 1318525 p^{8} T^{24} + 376698 p^{9} T^{25} + 73006 p^{10} T^{26} + 12030 p^{11} T^{27} + 2621 p^{12} T^{28} + 664 p^{13} T^{29} + 131 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 3 T^{2} - 32 T^{3} - 112 T^{4} + 1278 T^{5} + 167 T^{6} + 19292 T^{7} - 19397 T^{8} - 16266 p T^{9} + 3072688 T^{10} + 308974 T^{11} + 23227688 T^{12} - 100927476 T^{13} - 316827070 T^{14} + 5123849298 T^{15} + 2714752516 T^{16} + 5123849298 p T^{17} - 316827070 p^{2} T^{18} - 100927476 p^{3} T^{19} + 23227688 p^{4} T^{20} + 308974 p^{5} T^{21} + 3072688 p^{6} T^{22} - 16266 p^{8} T^{23} - 19397 p^{8} T^{24} + 19292 p^{9} T^{25} + 167 p^{10} T^{26} + 1278 p^{11} T^{27} - 112 p^{12} T^{28} - 32 p^{13} T^{29} + 3 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 6 T + 18 T^{2} + 216 T^{3} + 1644 T^{4} + 13602 T^{5} + 3276 p T^{6} + 388614 T^{7} + 2675330 T^{8} + 16213800 T^{9} + 90550242 T^{10} + 448011582 T^{11} + 2538033600 T^{12} + 13632286746 T^{13} + 70133541894 T^{14} + 342648141504 T^{15} + 1496778883995 T^{16} + 342648141504 p T^{17} + 70133541894 p^{2} T^{18} + 13632286746 p^{3} T^{19} + 2538033600 p^{4} T^{20} + 448011582 p^{5} T^{21} + 90550242 p^{6} T^{22} + 16213800 p^{7} T^{23} + 2675330 p^{8} T^{24} + 388614 p^{9} T^{25} + 3276 p^{11} T^{26} + 13602 p^{11} T^{27} + 1644 p^{12} T^{28} + 216 p^{13} T^{29} + 18 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 6 T + 173 T^{2} + 966 T^{3} + 15541 T^{4} + 74796 T^{5} + 893326 T^{6} + 3749280 T^{7} + 36955695 T^{8} + 141673374 T^{9} + 1230219831 T^{10} + 4866503730 T^{11} + 37766323214 T^{12} + 5916829434 p T^{13} + 1168540322547 T^{14} + 5807948805702 T^{15} + 35123636326012 T^{16} + 5807948805702 p T^{17} + 1168540322547 p^{2} T^{18} + 5916829434 p^{4} T^{19} + 37766323214 p^{4} T^{20} + 4866503730 p^{5} T^{21} + 1230219831 p^{6} T^{22} + 141673374 p^{7} T^{23} + 36955695 p^{8} T^{24} + 3749280 p^{9} T^{25} + 893326 p^{10} T^{26} + 74796 p^{11} T^{27} + 15541 p^{12} T^{28} + 966 p^{13} T^{29} + 173 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 464 T^{3} - 1816 T^{4} - 12756 T^{5} + 107648 T^{6} - 957044 T^{7} - 1873904 T^{8} + 40462620 T^{9} - 167221880 T^{10} + 578785940 T^{11} + 9885672764 T^{12} - 52047320472 T^{13} + 108429188120 T^{14} + 767715254880 T^{15} - 13955861058266 T^{16} + 767715254880 p T^{17} + 108429188120 p^{2} T^{18} - 52047320472 p^{3} T^{19} + 9885672764 p^{4} T^{20} + 578785940 p^{5} T^{21} - 167221880 p^{6} T^{22} + 40462620 p^{7} T^{23} - 1873904 p^{8} T^{24} - 957044 p^{9} T^{25} + 107648 p^{10} T^{26} - 12756 p^{11} T^{27} - 1816 p^{12} T^{28} + 464 p^{13} T^{29} + p^{16} T^{32} \) | |
| 37 | \( 1 + 20 T + 104 T^{2} - 1180 T^{3} - 19392 T^{4} - 75194 T^{5} + 647344 T^{6} + 9227242 T^{7} + 33299813 T^{8} - 247196384 T^{9} - 3433165666 T^{10} - 11705620612 T^{11} + 80692177824 T^{12} + 1026977965202 T^{13} + 3196337912430 T^{14} - 20647034768838 T^{15} - 248539049903972 T^{16} - 20647034768838 p T^{17} + 3196337912430 p^{2} T^{18} + 1026977965202 p^{3} T^{19} + 80692177824 p^{4} T^{20} - 11705620612 p^{5} T^{21} - 3433165666 p^{6} T^{22} - 247196384 p^{7} T^{23} + 33299813 p^{8} T^{24} + 9227242 p^{9} T^{25} + 647344 p^{10} T^{26} - 75194 p^{11} T^{27} - 19392 p^{12} T^{28} - 1180 p^{13} T^{29} + 104 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 44 T + 719 T^{2} + 3084 T^{3} - 61949 T^{4} - 966820 T^{5} - 2954968 T^{6} + 49408728 T^{7} + 537517475 T^{8} + 769798232 T^{9} - 20824887873 T^{10} - 146149086496 T^{11} - 1881378 p T^{12} + 4623617160484 T^{13} + 20402976447851 T^{14} - 47752975679704 T^{15} - 862601243892536 T^{16} - 47752975679704 p T^{17} + 20402976447851 p^{2} T^{18} + 4623617160484 p^{3} T^{19} - 1881378 p^{5} T^{20} - 146149086496 p^{5} T^{21} - 20824887873 p^{6} T^{22} + 769798232 p^{7} T^{23} + 537517475 p^{8} T^{24} + 49408728 p^{9} T^{25} - 2954968 p^{10} T^{26} - 966820 p^{11} T^{27} - 61949 p^{12} T^{28} + 3084 p^{13} T^{29} + 719 p^{14} T^{30} + 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 132 T^{2} - 696 T^{3} + 6652 T^{4} - 112080 T^{5} + 353616 T^{6} - 7378416 T^{7} + 39752602 T^{8} - 288340560 T^{9} + 3322017252 T^{10} - 11962899288 T^{11} + 169457775568 T^{12} - 840625735752 T^{13} + 5743391981172 T^{14} - 55351270278336 T^{15} + 192018494244931 T^{16} - 55351270278336 p T^{17} + 5743391981172 p^{2} T^{18} - 840625735752 p^{3} T^{19} + 169457775568 p^{4} T^{20} - 11962899288 p^{5} T^{21} + 3322017252 p^{6} T^{22} - 288340560 p^{7} T^{23} + 39752602 p^{8} T^{24} - 7378416 p^{9} T^{25} + 353616 p^{10} T^{26} - 112080 p^{11} T^{27} + 6652 p^{12} T^{28} - 696 p^{13} T^{29} + 132 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 26 T + 543 T^{2} - 170 p T^{3} + 100989 T^{4} - 1065000 T^{5} + 9931030 T^{6} - 80995036 T^{7} + 590415066 T^{8} - 80995036 p T^{9} + 9931030 p^{2} T^{10} - 1065000 p^{3} T^{11} + 100989 p^{4} T^{12} - 170 p^{6} T^{13} + 543 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 24 T + 288 T^{2} + 2160 T^{3} + 10596 T^{4} + 47580 T^{5} + 423072 T^{6} + 3923940 T^{7} + 20716670 T^{8} + 23880828 T^{9} - 268493400 T^{10} - 857615460 T^{11} - 10850803956 T^{12} - 333283834956 T^{13} - 3811014479016 T^{14} - 24616158134484 T^{15} - 146461023029253 T^{16} - 24616158134484 p T^{17} - 3811014479016 p^{2} T^{18} - 333283834956 p^{3} T^{19} - 10850803956 p^{4} T^{20} - 857615460 p^{5} T^{21} - 268493400 p^{6} T^{22} + 23880828 p^{7} T^{23} + 20716670 p^{8} T^{24} + 3923940 p^{9} T^{25} + 423072 p^{10} T^{26} + 47580 p^{11} T^{27} + 10596 p^{12} T^{28} + 2160 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 46 T + 986 T^{2} + 12952 T^{3} + 122768 T^{4} + 1065798 T^{5} + 10469620 T^{6} + 102242826 T^{7} + 811281850 T^{8} + 5382246268 T^{9} + 40177378590 T^{10} + 371425234282 T^{11} + 3114261615520 T^{12} + 21434234861390 T^{13} + 156495376933662 T^{14} + 1437908050730036 T^{15} + 12420334440038843 T^{16} + 1437908050730036 p T^{17} + 156495376933662 p^{2} T^{18} + 21434234861390 p^{3} T^{19} + 3114261615520 p^{4} T^{20} + 371425234282 p^{5} T^{21} + 40177378590 p^{6} T^{22} + 5382246268 p^{7} T^{23} + 811281850 p^{8} T^{24} + 102242826 p^{9} T^{25} + 10469620 p^{10} T^{26} + 1065798 p^{11} T^{27} + 122768 p^{12} T^{28} + 12952 p^{13} T^{29} + 986 p^{14} T^{30} + 46 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 6 T - 323 T^{2} + 3106 T^{3} + 54921 T^{4} - 735530 T^{5} - 5301922 T^{6} + 113463090 T^{7} + 206395019 T^{8} - 12314213104 T^{9} + 24628393761 T^{10} + 969158900984 T^{11} - 5468049848500 T^{12} - 51745797618564 T^{13} + 578771777592989 T^{14} + 1272301612503704 T^{15} - 41587454469955608 T^{16} + 1272301612503704 p T^{17} + 578771777592989 p^{2} T^{18} - 51745797618564 p^{3} T^{19} - 5468049848500 p^{4} T^{20} + 969158900984 p^{5} T^{21} + 24628393761 p^{6} T^{22} - 12314213104 p^{7} T^{23} + 206395019 p^{8} T^{24} + 113463090 p^{9} T^{25} - 5301922 p^{10} T^{26} - 735530 p^{11} T^{27} + 54921 p^{12} T^{28} + 3106 p^{13} T^{29} - 323 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 12 T + 356 T^{2} - 3696 T^{3} + 62748 T^{4} - 563100 T^{5} + 6969720 T^{6} - 52177956 T^{7} + 499001658 T^{8} - 2732130456 T^{9} + 17152922124 T^{10} + 13864914300 T^{11} - 954058645456 T^{12} + 18908535707412 T^{13} - 204244997008004 T^{14} + 2138046932472408 T^{15} - 17645045189998509 T^{16} + 2138046932472408 p T^{17} - 204244997008004 p^{2} T^{18} + 18908535707412 p^{3} T^{19} - 954058645456 p^{4} T^{20} + 13864914300 p^{5} T^{21} + 17152922124 p^{6} T^{22} - 2732130456 p^{7} T^{23} + 499001658 p^{8} T^{24} - 52177956 p^{9} T^{25} + 6969720 p^{10} T^{26} - 563100 p^{11} T^{27} + 62748 p^{12} T^{28} - 3696 p^{13} T^{29} + 356 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 6 T + 150 T^{2} - 192 T^{3} + 4128 T^{4} + 58698 T^{5} - 263916 T^{6} - 694818 T^{7} + 18406010 T^{8} - 756986772 T^{9} + 1957602954 T^{10} - 26634243258 T^{11} - 160523824176 T^{12} + 595653176586 T^{13} + 6736960532106 T^{14} - 99027857209260 T^{15} + 2379255597138411 T^{16} - 99027857209260 p T^{17} + 6736960532106 p^{2} T^{18} + 595653176586 p^{3} T^{19} - 160523824176 p^{4} T^{20} - 26634243258 p^{5} T^{21} + 1957602954 p^{6} T^{22} - 756986772 p^{7} T^{23} + 18406010 p^{8} T^{24} - 694818 p^{9} T^{25} - 263916 p^{10} T^{26} + 58698 p^{11} T^{27} + 4128 p^{12} T^{28} - 192 p^{13} T^{29} + 150 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 562 T^{2} + 152817 T^{4} - 26428854 T^{6} + 3209782926 T^{8} - 285703397178 T^{10} + 19139519352215 T^{12} - 1045006353253598 T^{14} + 62694696227021526 T^{16} - 1045006353253598 p^{2} T^{18} + 19139519352215 p^{4} T^{20} - 285703397178 p^{6} T^{22} + 3209782926 p^{8} T^{24} - 26428854 p^{10} T^{26} + 152817 p^{12} T^{28} - 562 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 996 T^{2} + 476872 T^{4} - 1849860 p T^{6} + 32163924568 T^{8} - 5401010486124 T^{10} + 716709759649948 T^{12} - 76727387928377316 T^{14} + 6699931044322533286 T^{16} - 76727387928377316 p^{2} T^{18} + 716709759649948 p^{4} T^{20} - 5401010486124 p^{6} T^{22} + 32163924568 p^{8} T^{24} - 1849860 p^{11} T^{26} + 476872 p^{12} T^{28} - 996 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 32 T + 784 T^{2} + 13814 T^{3} + 215308 T^{4} + 2807118 T^{5} + 33363590 T^{6} + 348386832 T^{7} + 3367522442 T^{8} + 348386832 p T^{9} + 33363590 p^{2} T^{10} + 2807118 p^{3} T^{11} + 215308 p^{4} T^{12} + 13814 p^{5} T^{13} + 784 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 24 T + 813 T^{2} + 14940 T^{3} + 296112 T^{4} + 4311852 T^{5} + 62848017 T^{6} + 732957672 T^{7} + 8247059417 T^{8} + 73335211608 T^{9} + 582700972344 T^{10} + 2490529813824 T^{11} - 7767047557170 T^{12} - 475690885054032 T^{13} - 7304128030918710 T^{14} - 93400357006546320 T^{15} - 918788119713763392 T^{16} - 93400357006546320 p T^{17} - 7304128030918710 p^{2} T^{18} - 475690885054032 p^{3} T^{19} - 7767047557170 p^{4} T^{20} + 2490529813824 p^{5} T^{21} + 582700972344 p^{6} T^{22} + 73335211608 p^{7} T^{23} + 8247059417 p^{8} T^{24} + 732957672 p^{9} T^{25} + 62848017 p^{10} T^{26} + 4311852 p^{11} T^{27} + 296112 p^{12} T^{28} + 14940 p^{13} T^{29} + 813 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 6 T + 368 T^{2} + 2136 T^{3} + 61758 T^{4} + 334956 T^{5} + 7178820 T^{6} + 23929842 T^{7} + 640898262 T^{8} - 997108116 T^{9} + 29865006630 T^{10} - 465405988452 T^{11} - 1337596835932 T^{12} - 76344553093008 T^{13} - 384963509135300 T^{14} - 10012663858840956 T^{15} - 44271572930188893 T^{16} - 10012663858840956 p T^{17} - 384963509135300 p^{2} T^{18} - 76344553093008 p^{3} T^{19} - 1337596835932 p^{4} T^{20} - 465405988452 p^{5} T^{21} + 29865006630 p^{6} T^{22} - 997108116 p^{7} T^{23} + 640898262 p^{8} T^{24} + 23929842 p^{9} T^{25} + 7178820 p^{10} T^{26} + 334956 p^{11} T^{27} + 61758 p^{12} T^{28} + 2136 p^{13} T^{29} + 368 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−4.04647818674869499568921085872, −4.01646173483925478288921569161, −3.97513260465008365214853955811, −3.97124641265687415784435722979, −3.81518423070482049123399368079, −3.34411594510023973552728246087, −3.30317011769006619153082888243, −3.23840333296428993476056814819, −3.19673721399594053573022496804, −3.18935265302090773214112512821, −3.01410138091956350058261150373, −3.00133869840295706742766351989, −2.93137204397670265916986011849, −2.59148102713572964137404197343, −2.32724899051925819122846576163, −2.22146072120894385260342893036, −2.21355713128005205388728381919, −2.17176010838433026691600888337, −1.85399672613982573458467893935, −1.82595513268569819686332505645, −1.77578400439426050168752820119, −1.53074757636986622497649330778, −1.40398146015036678220760443720, −1.31904551889943623662204075573, −0.20223573074322255795982390145, 0.20223573074322255795982390145, 1.31904551889943623662204075573, 1.40398146015036678220760443720, 1.53074757636986622497649330778, 1.77578400439426050168752820119, 1.82595513268569819686332505645, 1.85399672613982573458467893935, 2.17176010838433026691600888337, 2.21355713128005205388728381919, 2.22146072120894385260342893036, 2.32724899051925819122846576163, 2.59148102713572964137404197343, 2.93137204397670265916986011849, 3.00133869840295706742766351989, 3.01410138091956350058261150373, 3.18935265302090773214112512821, 3.19673721399594053573022496804, 3.23840333296428993476056814819, 3.30317011769006619153082888243, 3.34411594510023973552728246087, 3.81518423070482049123399368079, 3.97124641265687415784435722979, 3.97513260465008365214853955811, 4.01646173483925478288921569161, 4.04647818674869499568921085872
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.