Dirichlet series
| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 4·7-s − 2·8-s + 8·10-s + 4·11-s + 2·13-s + 8·14-s + 3·16-s − 4·17-s − 4·19-s − 8·20-s − 8·22-s + 8·23-s + 6·25-s − 4·26-s − 8·28-s − 26·29-s − 10·31-s − 2·32-s + 8·34-s + 16·35-s + 22·37-s + 8·38-s + 8·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 1.51·7-s − 0.707·8-s + 2.52·10-s + 1.20·11-s + 0.554·13-s + 2.13·14-s + 3/4·16-s − 0.970·17-s − 0.917·19-s − 1.78·20-s − 1.70·22-s + 1.66·23-s + 6/5·25-s − 0.784·26-s − 1.51·28-s − 4.82·29-s − 1.79·31-s − 0.353·32-s + 1.37·34-s + 2.70·35-s + 3.61·37-s + 1.29·38-s + 1.26·40-s − 0.937·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.54908\times 10^{9}\) |
| Root analytic conductor: | \(1.98811\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4905526167\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4905526167\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) | |
| 11 | \( 1 - 4 T + 5 T^{2} + 30 T^{3} - 20 p T^{4} + 618 T^{5} - 577 T^{6} - 6140 T^{7} + 39185 T^{8} - 6140 p T^{9} - 577 p^{2} T^{10} + 618 p^{3} T^{11} - 20 p^{5} T^{12} + 30 p^{5} T^{13} + 5 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 + p T + p T^{2} + p T^{3} + T^{4} - p T^{5} - 3 T^{6} - p^{3} T^{7} - p^{2} T^{8} + 3 p T^{9} - T^{10} + 5 p^{2} T^{11} + 35 T^{12} - p^{3} T^{13} + 3 T^{14} + 51 p T^{15} + 125 T^{16} + 51 p^{2} T^{17} + 3 p^{2} T^{18} - p^{6} T^{19} + 35 p^{4} T^{20} + 5 p^{7} T^{21} - p^{6} T^{22} + 3 p^{8} T^{23} - p^{10} T^{24} - p^{12} T^{25} - 3 p^{10} T^{26} - p^{12} T^{27} + p^{12} T^{28} + p^{14} T^{29} + p^{15} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 + 4 T - 11 T^{2} - 72 T^{3} - 25 T^{4} + 16 p^{2} T^{5} + 2402 T^{6} - 3242 T^{7} - 29427 T^{8} - 30550 T^{9} + 173300 T^{10} + 643408 T^{11} - 16608 T^{12} - 5179852 T^{13} - 1482911 p T^{14} + 15483442 T^{15} + 103694551 T^{16} + 15483442 p T^{17} - 1482911 p^{3} T^{18} - 5179852 p^{3} T^{19} - 16608 p^{4} T^{20} + 643408 p^{5} T^{21} + 173300 p^{6} T^{22} - 30550 p^{7} T^{23} - 29427 p^{8} T^{24} - 3242 p^{9} T^{25} + 2402 p^{10} T^{26} + 16 p^{13} T^{27} - 25 p^{12} T^{28} - 72 p^{13} T^{29} - 11 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 2 T - 30 T^{2} + 72 T^{3} + 129 T^{4} - 194 T^{5} + 6870 T^{6} - 33528 T^{7} - 98660 T^{8} + 591296 T^{9} - 46700 p T^{10} + 2499634 T^{11} + 21995307 T^{12} - 178104982 T^{13} - 8235904 T^{14} + 107746742 p T^{15} - 3042189333 T^{16} + 107746742 p^{2} T^{17} - 8235904 p^{2} T^{18} - 178104982 p^{3} T^{19} + 21995307 p^{4} T^{20} + 2499634 p^{5} T^{21} - 46700 p^{7} T^{22} + 591296 p^{7} T^{23} - 98660 p^{8} T^{24} - 33528 p^{9} T^{25} + 6870 p^{10} T^{26} - 194 p^{11} T^{27} + 129 p^{12} T^{28} + 72 p^{13} T^{29} - 30 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 4 T - 42 T^{2} - 236 T^{3} + 543 T^{4} + 6716 T^{5} + 4846 T^{6} - 101164 T^{7} - 270622 T^{8} + 567856 T^{9} + 5021204 T^{10} + 14091268 T^{11} - 20383721 T^{12} - 463031744 T^{13} - 1621946578 T^{14} + 4290676392 T^{15} + 45918322451 T^{16} + 4290676392 p T^{17} - 1621946578 p^{2} T^{18} - 463031744 p^{3} T^{19} - 20383721 p^{4} T^{20} + 14091268 p^{5} T^{21} + 5021204 p^{6} T^{22} + 567856 p^{7} T^{23} - 270622 p^{8} T^{24} - 101164 p^{9} T^{25} + 4846 p^{10} T^{26} + 6716 p^{11} T^{27} + 543 p^{12} T^{28} - 236 p^{13} T^{29} - 42 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 4 T + 6 T^{2} - 20 T^{3} + 53 T^{4} + 3604 T^{5} + 12786 T^{6} + 17092 T^{7} - 4672 p T^{8} + 216684 T^{9} + 4864652 T^{10} + 16930880 T^{11} + 31232351 T^{12} - 74466428 T^{13} + 265439608 T^{14} + 3776635100 T^{15} + 18320619623 T^{16} + 3776635100 p T^{17} + 265439608 p^{2} T^{18} - 74466428 p^{3} T^{19} + 31232351 p^{4} T^{20} + 16930880 p^{5} T^{21} + 4864652 p^{6} T^{22} + 216684 p^{7} T^{23} - 4672 p^{9} T^{24} + 17092 p^{9} T^{25} + 12786 p^{10} T^{26} + 3604 p^{11} T^{27} + 53 p^{12} T^{28} - 20 p^{13} T^{29} + 6 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 4 T + 119 T^{2} - 26 p T^{3} + 7151 T^{4} - 36104 T^{5} + 12528 p T^{6} - 1249998 T^{7} + 8024929 T^{8} - 1249998 p T^{9} + 12528 p^{3} T^{10} - 36104 p^{3} T^{11} + 7151 p^{4} T^{12} - 26 p^{6} T^{13} + 119 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 26 T + 208 T^{2} - 568 T^{3} - 19803 T^{4} - 3166 p T^{5} + 464950 T^{6} + 5643560 T^{7} + 1429936 T^{8} - 195532736 T^{9} - 492099014 T^{10} + 5973472314 T^{11} + 31890905753 T^{12} - 4100159438 p T^{13} - 1245664106312 T^{14} + 974560299946 T^{15} + 36899615810343 T^{16} + 974560299946 p T^{17} - 1245664106312 p^{2} T^{18} - 4100159438 p^{4} T^{19} + 31890905753 p^{4} T^{20} + 5973472314 p^{5} T^{21} - 492099014 p^{6} T^{22} - 195532736 p^{7} T^{23} + 1429936 p^{8} T^{24} + 5643560 p^{9} T^{25} + 464950 p^{10} T^{26} - 3166 p^{12} T^{27} - 19803 p^{12} T^{28} - 568 p^{13} T^{29} + 208 p^{14} T^{30} + 26 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 10 T - 101 T^{2} - 1452 T^{3} + 797 T^{4} + 92772 T^{5} + 16325 p T^{6} - 2219106 T^{7} - 41369673 T^{8} - 98905294 T^{9} + 1418784466 T^{10} + 10876333234 T^{11} - 2768460216 T^{12} - 427660698620 T^{13} - 1935145383850 T^{14} + 6184151376676 T^{15} + 90920519220763 T^{16} + 6184151376676 p T^{17} - 1935145383850 p^{2} T^{18} - 427660698620 p^{3} T^{19} - 2768460216 p^{4} T^{20} + 10876333234 p^{5} T^{21} + 1418784466 p^{6} T^{22} - 98905294 p^{7} T^{23} - 41369673 p^{8} T^{24} - 2219106 p^{9} T^{25} + 16325 p^{11} T^{26} + 92772 p^{11} T^{27} + 797 p^{12} T^{28} - 1452 p^{13} T^{29} - 101 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 22 T + 197 T^{2} - 1016 T^{3} + 7369 T^{4} - 60716 T^{5} + 99609 T^{6} + 49446 p T^{7} - 7003073 T^{8} + 11123524 T^{9} - 621377802 T^{10} + 3822038534 T^{11} + 13051248536 T^{12} - 106886171600 T^{13} - 196157513294 T^{14} - 3497389007868 T^{15} + 58828862455255 T^{16} - 3497389007868 p T^{17} - 196157513294 p^{2} T^{18} - 106886171600 p^{3} T^{19} + 13051248536 p^{4} T^{20} + 3822038534 p^{5} T^{21} - 621377802 p^{6} T^{22} + 11123524 p^{7} T^{23} - 7003073 p^{8} T^{24} + 49446 p^{10} T^{25} + 99609 p^{10} T^{26} - 60716 p^{11} T^{27} + 7369 p^{12} T^{28} - 1016 p^{13} T^{29} + 197 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 6 T - 33 T^{2} - 312 T^{3} - 1631 T^{4} - 27340 T^{5} - 34422 T^{6} + 1033022 T^{7} + 1954463 T^{8} + 18618904 T^{9} + 325760028 T^{10} - 224751624 T^{11} + 1954094140 T^{12} + 76019270152 T^{13} - 388098926849 T^{14} - 4047081914288 T^{15} - 2823426142793 T^{16} - 4047081914288 p T^{17} - 388098926849 p^{2} T^{18} + 76019270152 p^{3} T^{19} + 1954094140 p^{4} T^{20} - 224751624 p^{5} T^{21} + 325760028 p^{6} T^{22} + 18618904 p^{7} T^{23} + 1954463 p^{8} T^{24} + 1033022 p^{9} T^{25} - 34422 p^{10} T^{26} - 27340 p^{11} T^{27} - 1631 p^{12} T^{28} - 312 p^{13} T^{29} - 33 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 14 T + 4 p T^{2} - 1202 T^{3} + 206 p T^{4} - 59172 T^{5} + 521607 T^{6} - 4081424 T^{7} + 30196845 T^{8} - 4081424 p T^{9} + 521607 p^{2} T^{10} - 59172 p^{3} T^{11} + 206 p^{5} T^{12} - 1202 p^{5} T^{13} + 4 p^{7} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 20 T - 15 T^{2} - 3086 T^{3} - 15411 T^{4} + 204258 T^{5} + 1952786 T^{6} - 6396178 T^{7} - 134733021 T^{8} - 95974552 T^{9} + 6191744836 T^{10} + 25083179148 T^{11} - 153993477580 T^{12} - 1482527057440 T^{13} - 1858610581491 T^{14} + 31954141927224 T^{15} + 292271740933835 T^{16} + 31954141927224 p T^{17} - 1858610581491 p^{2} T^{18} - 1482527057440 p^{3} T^{19} - 153993477580 p^{4} T^{20} + 25083179148 p^{5} T^{21} + 6191744836 p^{6} T^{22} - 95974552 p^{7} T^{23} - 134733021 p^{8} T^{24} - 6396178 p^{9} T^{25} + 1952786 p^{10} T^{26} + 204258 p^{11} T^{27} - 15411 p^{12} T^{28} - 3086 p^{13} T^{29} - 15 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 14 T - 81 T^{2} + 1868 T^{3} - 2978 T^{4} - 37362 T^{5} + 274556 T^{6} - 5645154 T^{7} + 19392662 T^{8} + 303904008 T^{9} - 1710252337 T^{10} + 2978921760 T^{11} - 4694041762 T^{12} - 645953772194 T^{13} + 4627617340722 T^{14} + 15785489640460 T^{15} - 290151692249725 T^{16} + 15785489640460 p T^{17} + 4627617340722 p^{2} T^{18} - 645953772194 p^{3} T^{19} - 4694041762 p^{4} T^{20} + 2978921760 p^{5} T^{21} - 1710252337 p^{6} T^{22} + 303904008 p^{7} T^{23} + 19392662 p^{8} T^{24} - 5645154 p^{9} T^{25} + 274556 p^{10} T^{26} - 37362 p^{11} T^{27} - 2978 p^{12} T^{28} + 1868 p^{13} T^{29} - 81 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 16 T + 47 T^{2} - 64 T^{3} + 7877 T^{4} + 72976 T^{5} - 25379 T^{6} + 28428 T^{7} + 32478681 T^{8} + 153612480 T^{9} - 232138564 T^{10} + 10600211772 T^{11} + 158190932608 T^{12} + 356463904786 T^{13} - 497196898382 T^{14} + 40262560604270 T^{15} + 544821369740383 T^{16} + 40262560604270 p T^{17} - 497196898382 p^{2} T^{18} + 356463904786 p^{3} T^{19} + 158190932608 p^{4} T^{20} + 10600211772 p^{5} T^{21} - 232138564 p^{6} T^{22} + 153612480 p^{7} T^{23} + 32478681 p^{8} T^{24} + 28428 p^{9} T^{25} - 25379 p^{10} T^{26} + 72976 p^{11} T^{27} + 7877 p^{12} T^{28} - 64 p^{13} T^{29} + 47 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 38 T + 629 T^{2} + 6550 T^{3} + 59910 T^{4} + 611100 T^{5} + 6363273 T^{6} + 61633604 T^{7} + 577790970 T^{8} + 5350372458 T^{9} + 47767475840 T^{10} + 404819084138 T^{11} + 3297940508920 T^{12} + 27119510381122 T^{13} + 228654972008018 T^{14} + 1895515913825050 T^{15} + 15095733069399999 T^{16} + 1895515913825050 p T^{17} + 228654972008018 p^{2} T^{18} + 27119510381122 p^{3} T^{19} + 3297940508920 p^{4} T^{20} + 404819084138 p^{5} T^{21} + 47767475840 p^{6} T^{22} + 5350372458 p^{7} T^{23} + 577790970 p^{8} T^{24} + 61633604 p^{9} T^{25} + 6363273 p^{10} T^{26} + 611100 p^{11} T^{27} + 59910 p^{12} T^{28} + 6550 p^{13} T^{29} + 629 p^{14} T^{30} + 38 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 10 T + 363 T^{2} - 3452 T^{3} + 65848 T^{4} - 571942 T^{5} + 7668881 T^{6} - 57892138 T^{7} + 613900435 T^{8} - 57892138 p T^{9} + 7668881 p^{2} T^{10} - 571942 p^{3} T^{11} + 65848 p^{4} T^{12} - 3452 p^{5} T^{13} + 363 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 54 T + 1157 T^{2} + 11932 T^{3} + 61679 T^{4} + 421860 T^{5} + 6917853 T^{6} + 58689338 T^{7} + 250322433 T^{8} + 2080917136 T^{9} + 17296659008 T^{10} + 45843186634 T^{11} + 1293150719710 T^{12} + 24026071058708 T^{13} + 164209894901826 T^{14} + 880553120615108 T^{15} + 7466806408550847 T^{16} + 880553120615108 p T^{17} + 164209894901826 p^{2} T^{18} + 24026071058708 p^{3} T^{19} + 1293150719710 p^{4} T^{20} + 45843186634 p^{5} T^{21} + 17296659008 p^{6} T^{22} + 2080917136 p^{7} T^{23} + 250322433 p^{8} T^{24} + 58689338 p^{9} T^{25} + 6917853 p^{10} T^{26} + 421860 p^{11} T^{27} + 61679 p^{12} T^{28} + 11932 p^{13} T^{29} + 1157 p^{14} T^{30} + 54 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 2 T - 245 T^{2} + 652 T^{3} + 29494 T^{4} - 169814 T^{5} - 2045545 T^{6} + 31748412 T^{7} + 59982430 T^{8} - 3825016474 T^{9} + 6756484540 T^{10} + 308420786874 T^{11} - 1689838925268 T^{12} - 16318498821682 T^{13} + 220955728853566 T^{14} + 414652181174906 T^{15} - 19372282507696953 T^{16} + 414652181174906 p T^{17} + 220955728853566 p^{2} T^{18} - 16318498821682 p^{3} T^{19} - 1689838925268 p^{4} T^{20} + 308420786874 p^{5} T^{21} + 6756484540 p^{6} T^{22} - 3825016474 p^{7} T^{23} + 59982430 p^{8} T^{24} + 31748412 p^{9} T^{25} - 2045545 p^{10} T^{26} - 169814 p^{11} T^{27} + 29494 p^{12} T^{28} + 652 p^{13} T^{29} - 245 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 12 T - 14 T^{2} - 8 p T^{3} + 11815 T^{4} + 150580 T^{5} - 574028 T^{6} - 9778672 T^{7} + 112174351 T^{8} + 1108825864 T^{9} - 8016595792 T^{10} - 46100265652 T^{11} + 1109037136181 T^{12} + 4031116524896 T^{13} - 86215149870550 T^{14} - 73129089032892 T^{15} + 7542253760285352 T^{16} - 73129089032892 p T^{17} - 86215149870550 p^{2} T^{18} + 4031116524896 p^{3} T^{19} + 1109037136181 p^{4} T^{20} - 46100265652 p^{5} T^{21} - 8016595792 p^{6} T^{22} + 1108825864 p^{7} T^{23} + 112174351 p^{8} T^{24} - 9778672 p^{9} T^{25} - 574028 p^{10} T^{26} + 150580 p^{11} T^{27} + 11815 p^{12} T^{28} - 8 p^{14} T^{29} - 14 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 28 T + 279 T^{2} + 1464 T^{3} + 5015 T^{4} + 75612 T^{5} + 2958495 T^{6} + 43065708 T^{7} + 392902009 T^{8} + 3215612882 T^{9} + 25051007664 T^{10} + 274056979712 T^{11} + 3009168574798 T^{12} + 27606439012398 T^{13} + 279838222164326 T^{14} + 2842217228426526 T^{15} + 26585845636144695 T^{16} + 2842217228426526 p T^{17} + 279838222164326 p^{2} T^{18} + 27606439012398 p^{3} T^{19} + 3009168574798 p^{4} T^{20} + 274056979712 p^{5} T^{21} + 25051007664 p^{6} T^{22} + 3215612882 p^{7} T^{23} + 392902009 p^{8} T^{24} + 43065708 p^{9} T^{25} + 2958495 p^{10} T^{26} + 75612 p^{11} T^{27} + 5015 p^{12} T^{28} + 1464 p^{13} T^{29} + 279 p^{14} T^{30} + 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 38 T + 1150 T^{2} - 22786 T^{3} + 390897 T^{4} - 5296342 T^{5} + 65582590 T^{6} - 692743768 T^{7} + 6971358239 T^{8} - 692743768 p T^{9} + 65582590 p^{2} T^{10} - 5296342 p^{3} T^{11} + 390897 p^{4} T^{12} - 22786 p^{5} T^{13} + 1150 p^{6} T^{14} - 38 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 18 T + 15 T^{2} - 1118 T^{3} - 3756 T^{4} - 286848 T^{5} - 46542 p T^{6} + 18425414 T^{7} + 504297382 T^{8} + 307514118 T^{9} + 19682670087 T^{10} + 449194274196 T^{11} - 4421738359066 T^{12} - 66509821177502 T^{13} + 3302827121668 p T^{14} + 2588736546159896 T^{15} - 31633422570533229 T^{16} + 2588736546159896 p T^{17} + 3302827121668 p^{3} T^{18} - 66509821177502 p^{3} T^{19} - 4421738359066 p^{4} T^{20} + 449194274196 p^{5} T^{21} + 19682670087 p^{6} T^{22} + 307514118 p^{7} T^{23} + 504297382 p^{8} T^{24} + 18425414 p^{9} T^{25} - 46542 p^{11} T^{26} - 286848 p^{11} T^{27} - 3756 p^{12} T^{28} - 1118 p^{13} T^{29} + 15 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.07602813881436782163193591716, −3.06514970826954468580048257686, −2.97829641114062456153844185632, −2.85535301101758411832376947373, −2.53874793215669118979363997236, −2.47184100453563590030032535264, −2.45464331959825082327598058744, −2.36515753299758807474312513104, −2.26797967690494632961977346026, −2.14419102909652436867729686334, −2.04013256358837471058544562983, −1.93439612979499160636113882985, −1.85926568597048715360017170908, −1.73692072742484326813826045931, −1.71147405060159010456523058955, −1.64441345291449648318294270770, −1.29311068719447377141473918223, −1.23864878560500929907227351682, −1.13862353739453269143888065578, −1.11817659032922006235046654564, −0.941204061834723070306934833303, −0.56318639587929078878575942604, −0.39256072666732757350151535913, −0.38799912125712207092650640108, −0.20481243555999796316986383133, 0.20481243555999796316986383133, 0.38799912125712207092650640108, 0.39256072666732757350151535913, 0.56318639587929078878575942604, 0.941204061834723070306934833303, 1.11817659032922006235046654564, 1.13862353739453269143888065578, 1.23864878560500929907227351682, 1.29311068719447377141473918223, 1.64441345291449648318294270770, 1.71147405060159010456523058955, 1.73692072742484326813826045931, 1.85926568597048715360017170908, 1.93439612979499160636113882985, 2.04013256358837471058544562983, 2.14419102909652436867729686334, 2.26797967690494632961977346026, 2.36515753299758807474312513104, 2.45464331959825082327598058744, 2.47184100453563590030032535264, 2.53874793215669118979363997236, 2.85535301101758411832376947373, 2.97829641114062456153844185632, 3.06514970826954468580048257686, 3.07602813881436782163193591716
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.