Dirichlet series
| L(s) = 1 | + 2·3-s + 9·5-s + 13·9-s − 6·11-s + 2·13-s + 18·15-s + 17-s + 12·23-s + 36·25-s + 24·27-s − 20·29-s + 28·31-s − 12·33-s + 4·39-s − 5·41-s + 2·43-s + 117·45-s − 36·47-s + 35·49-s + 2·51-s − 10·53-s − 54·55-s − 10·59-s + 16·61-s + 18·65-s − 4·67-s + 24·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4.02·5-s + 13/3·9-s − 1.80·11-s + 0.554·13-s + 4.64·15-s + 0.242·17-s + 2.50·23-s + 36/5·25-s + 4.61·27-s − 3.71·29-s + 5.02·31-s − 2.08·33-s + 0.640·39-s − 0.780·41-s + 0.304·43-s + 17.4·45-s − 5.25·47-s + 5·49-s + 0.280·51-s − 1.37·53-s − 7.28·55-s − 1.30·59-s + 2.04·61-s + 2.23·65-s − 0.488·67-s + 2.88·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 5^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 5^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{54} \cdot 5^{18} \cdot 37^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.02161\times 10^{19}\) |
| Root analytic conductor: | \(3.43771\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{54} \cdot 5^{18} \cdot 37^{18} ,\ ( \ : [1/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(376.2544117\) |
| \(L(\frac12)\) | \(\approx\) | \(376.2544117\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{9} \) | |
| 37 | \( 1 + 90 T^{2} - 470 T^{3} + 1449 T^{4} - 41670 T^{5} + 30338 T^{6} - 670437 T^{7} + 8329023 T^{8} + 16966273 T^{9} + 8329023 p T^{10} - 670437 p^{2} T^{11} + 30338 p^{3} T^{12} - 41670 p^{4} T^{13} + 1449 p^{5} T^{14} - 470 p^{6} T^{15} + 90 p^{7} T^{16} + p^{9} T^{18} \) | |
| good | 3 | \( 1 - 2 T - p^{2} T^{2} + 20 T^{3} + 41 T^{4} - 107 T^{5} - 47 p T^{6} + 439 T^{7} + 359 T^{8} - 1483 T^{9} - 338 T^{10} + 4267 T^{11} - 2683 T^{12} - 10169 T^{13} + 20074 T^{14} + 6061 p T^{15} - 87821 T^{16} - 5882 p T^{17} + 98870 p T^{18} - 5882 p^{2} T^{19} - 87821 p^{2} T^{20} + 6061 p^{4} T^{21} + 20074 p^{4} T^{22} - 10169 p^{5} T^{23} - 2683 p^{6} T^{24} + 4267 p^{7} T^{25} - 338 p^{8} T^{26} - 1483 p^{9} T^{27} + 359 p^{10} T^{28} + 439 p^{11} T^{29} - 47 p^{13} T^{30} - 107 p^{13} T^{31} + 41 p^{14} T^{32} + 20 p^{15} T^{33} - p^{18} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) |
| 7 | \( 1 - 5 p T^{2} + 48 T^{3} + 608 T^{4} - 1520 T^{5} - 6161 T^{6} + 23405 T^{7} + 37278 T^{8} - 229405 T^{9} - 120793 T^{10} + 1625161 T^{11} - 25745 T^{12} - 9227019 T^{13} + 3499757 T^{14} + 42393397 T^{15} - 38588364 T^{16} - 104565519 T^{17} + 317054193 T^{18} - 104565519 p T^{19} - 38588364 p^{2} T^{20} + 42393397 p^{3} T^{21} + 3499757 p^{4} T^{22} - 9227019 p^{5} T^{23} - 25745 p^{6} T^{24} + 1625161 p^{7} T^{25} - 120793 p^{8} T^{26} - 229405 p^{9} T^{27} + 37278 p^{10} T^{28} + 23405 p^{11} T^{29} - 6161 p^{12} T^{30} - 1520 p^{13} T^{31} + 608 p^{14} T^{32} + 48 p^{15} T^{33} - 5 p^{17} T^{34} + p^{18} T^{36} \) | |
| 11 | \( ( 1 + 3 T + 50 T^{2} + 14 p T^{3} + 1371 T^{4} + 4021 T^{5} + 25395 T^{6} + 70881 T^{7} + 32353 p T^{8} + 901238 T^{9} + 32353 p^{2} T^{10} + 70881 p^{2} T^{11} + 25395 p^{3} T^{12} + 4021 p^{4} T^{13} + 1371 p^{5} T^{14} + 14 p^{7} T^{15} + 50 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 13 | \( 1 - 2 T - 50 T^{2} + 72 T^{3} + 92 p T^{4} - 1331 T^{5} - 17898 T^{6} + 27402 T^{7} + 180026 T^{8} - 743818 T^{9} - 1081217 T^{10} + 15937112 T^{11} - 6182534 T^{12} - 216328491 T^{13} + 414371655 T^{14} + 1781090242 T^{15} - 9650415458 T^{16} - 7027576178 T^{17} + 149345346422 T^{18} - 7027576178 p T^{19} - 9650415458 p^{2} T^{20} + 1781090242 p^{3} T^{21} + 414371655 p^{4} T^{22} - 216328491 p^{5} T^{23} - 6182534 p^{6} T^{24} + 15937112 p^{7} T^{25} - 1081217 p^{8} T^{26} - 743818 p^{9} T^{27} + 180026 p^{10} T^{28} + 27402 p^{11} T^{29} - 17898 p^{12} T^{30} - 1331 p^{13} T^{31} + 92 p^{15} T^{32} + 72 p^{15} T^{33} - 50 p^{16} T^{34} - 2 p^{17} T^{35} + p^{18} T^{36} \) | |
| 17 | \( 1 - T - 65 T^{2} - 80 T^{3} + 2379 T^{4} + 7859 T^{5} - 49818 T^{6} - 333260 T^{7} + 410050 T^{8} + 8801745 T^{9} + 11775716 T^{10} - 9538465 p T^{11} - 542673370 T^{12} + 2084752108 T^{13} + 12344416012 T^{14} - 18032905149 T^{15} - 212004283325 T^{16} + 81069298567 T^{17} + 3477069960385 T^{18} + 81069298567 p T^{19} - 212004283325 p^{2} T^{20} - 18032905149 p^{3} T^{21} + 12344416012 p^{4} T^{22} + 2084752108 p^{5} T^{23} - 542673370 p^{6} T^{24} - 9538465 p^{8} T^{25} + 11775716 p^{8} T^{26} + 8801745 p^{9} T^{27} + 410050 p^{10} T^{28} - 333260 p^{11} T^{29} - 49818 p^{12} T^{30} + 7859 p^{13} T^{31} + 2379 p^{14} T^{32} - 80 p^{15} T^{33} - 65 p^{16} T^{34} - p^{17} T^{35} + p^{18} T^{36} \) | |
| 19 | \( 1 - 47 T^{2} - 106 T^{3} + 731 T^{4} + 4719 T^{5} + 585 p T^{6} - 60474 T^{7} - 586680 T^{8} - 691896 T^{9} + 5655254 T^{10} + 21900804 T^{11} + 99756850 T^{12} + 144863362 T^{13} - 1842125940 T^{14} - 12438823186 T^{15} - 34882167969 T^{16} + 136677531777 T^{17} + 1408040423457 T^{18} + 136677531777 p T^{19} - 34882167969 p^{2} T^{20} - 12438823186 p^{3} T^{21} - 1842125940 p^{4} T^{22} + 144863362 p^{5} T^{23} + 99756850 p^{6} T^{24} + 21900804 p^{7} T^{25} + 5655254 p^{8} T^{26} - 691896 p^{9} T^{27} - 586680 p^{10} T^{28} - 60474 p^{11} T^{29} + 585 p^{13} T^{30} + 4719 p^{13} T^{31} + 731 p^{14} T^{32} - 106 p^{15} T^{33} - 47 p^{16} T^{34} + p^{18} T^{36} \) | |
| 23 | \( ( 1 - 6 T + 118 T^{2} - 684 T^{3} + 313 p T^{4} - 38569 T^{5} + 294416 T^{6} - 1426631 T^{7} + 8839900 T^{8} - 38098816 T^{9} + 8839900 p T^{10} - 1426631 p^{2} T^{11} + 294416 p^{3} T^{12} - 38569 p^{4} T^{13} + 313 p^{6} T^{14} - 684 p^{6} T^{15} + 118 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 29 | \( ( 1 + 10 T + 253 T^{2} + 1970 T^{3} + 27954 T^{4} + 177424 T^{5} + 1814421 T^{6} + 9566525 T^{7} + 77113975 T^{8} + 338381870 T^{9} + 77113975 p T^{10} + 9566525 p^{2} T^{11} + 1814421 p^{3} T^{12} + 177424 p^{4} T^{13} + 27954 p^{5} T^{14} + 1970 p^{6} T^{15} + 253 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 31 | \( ( 1 - 14 T + 249 T^{2} - 2654 T^{3} + 28422 T^{4} - 238216 T^{5} + 1936060 T^{6} - 13242755 T^{7} + 87110253 T^{8} - 15983091 p T^{9} + 87110253 p T^{10} - 13242755 p^{2} T^{11} + 1936060 p^{3} T^{12} - 238216 p^{4} T^{13} + 28422 p^{5} T^{14} - 2654 p^{6} T^{15} + 249 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 41 | \( 1 + 5 T - 177 T^{2} - 1076 T^{3} + 15357 T^{4} + 111624 T^{5} - 842921 T^{6} - 7332184 T^{7} + 30909844 T^{8} + 350066544 T^{9} - 546934545 T^{10} - 13187252700 T^{11} - 27051623874 T^{12} + 411551131892 T^{13} + 3571725275350 T^{14} - 10321859441086 T^{15} - 232382500168859 T^{16} + 142209043314961 T^{17} + 10936030316896632 T^{18} + 142209043314961 p T^{19} - 232382500168859 p^{2} T^{20} - 10321859441086 p^{3} T^{21} + 3571725275350 p^{4} T^{22} + 411551131892 p^{5} T^{23} - 27051623874 p^{6} T^{24} - 13187252700 p^{7} T^{25} - 546934545 p^{8} T^{26} + 350066544 p^{9} T^{27} + 30909844 p^{10} T^{28} - 7332184 p^{11} T^{29} - 842921 p^{12} T^{30} + 111624 p^{13} T^{31} + 15357 p^{14} T^{32} - 1076 p^{15} T^{33} - 177 p^{16} T^{34} + 5 p^{17} T^{35} + p^{18} T^{36} \) | |
| 43 | \( ( 1 - T + 173 T^{2} - 152 T^{3} + 14847 T^{4} - 11701 T^{5} + 870983 T^{6} - 610669 T^{7} + 944540 p T^{8} - 26579737 T^{9} + 944540 p^{2} T^{10} - 610669 p^{2} T^{11} + 870983 p^{3} T^{12} - 11701 p^{4} T^{13} + 14847 p^{5} T^{14} - 152 p^{6} T^{15} + 173 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 47 | \( ( 1 + 18 T + 200 T^{2} + 1199 T^{3} + 6440 T^{4} + 33039 T^{5} + 290461 T^{6} + 745794 T^{7} - 5828318 T^{8} - 109707200 T^{9} - 5828318 p T^{10} + 745794 p^{2} T^{11} + 290461 p^{3} T^{12} + 33039 p^{4} T^{13} + 6440 p^{5} T^{14} + 1199 p^{6} T^{15} + 200 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 53 | \( 1 + 10 T - 108 T^{2} - 1782 T^{3} + 3826 T^{4} + 146436 T^{5} + 239855 T^{6} - 6000011 T^{7} - 33696080 T^{8} - 22521280 T^{9} + 1833421805 T^{10} + 22957576915 T^{11} - 36154871785 T^{12} - 1801501128477 T^{13} - 2198329948664 T^{14} + 81143305633838 T^{15} + 254167240745489 T^{16} - 1624968883008449 T^{17} - 15458862198323262 T^{18} - 1624968883008449 p T^{19} + 254167240745489 p^{2} T^{20} + 81143305633838 p^{3} T^{21} - 2198329948664 p^{4} T^{22} - 1801501128477 p^{5} T^{23} - 36154871785 p^{6} T^{24} + 22957576915 p^{7} T^{25} + 1833421805 p^{8} T^{26} - 22521280 p^{9} T^{27} - 33696080 p^{10} T^{28} - 6000011 p^{11} T^{29} + 239855 p^{12} T^{30} + 146436 p^{13} T^{31} + 3826 p^{14} T^{32} - 1782 p^{15} T^{33} - 108 p^{16} T^{34} + 10 p^{17} T^{35} + p^{18} T^{36} \) | |
| 59 | \( 1 + 10 T - 227 T^{2} - 2252 T^{3} + 27307 T^{4} + 240019 T^{5} - 2409971 T^{6} - 15544266 T^{7} + 176472220 T^{8} + 582659682 T^{9} - 10780675458 T^{10} + 4744842652 T^{11} + 564606740524 T^{12} - 2543230225028 T^{13} - 27784553984366 T^{14} + 182306041719034 T^{15} + 1414084600794929 T^{16} - 4744440684547935 T^{17} - 78456812762598383 T^{18} - 4744440684547935 p T^{19} + 1414084600794929 p^{2} T^{20} + 182306041719034 p^{3} T^{21} - 27784553984366 p^{4} T^{22} - 2543230225028 p^{5} T^{23} + 564606740524 p^{6} T^{24} + 4744842652 p^{7} T^{25} - 10780675458 p^{8} T^{26} + 582659682 p^{9} T^{27} + 176472220 p^{10} T^{28} - 15544266 p^{11} T^{29} - 2409971 p^{12} T^{30} + 240019 p^{13} T^{31} + 27307 p^{14} T^{32} - 2252 p^{15} T^{33} - 227 p^{16} T^{34} + 10 p^{17} T^{35} + p^{18} T^{36} \) | |
| 61 | \( 1 - 16 T - 135 T^{2} + 3654 T^{3} - 1217 T^{4} - 311737 T^{5} + 728495 T^{6} + 11108092 T^{7} + 61915834 T^{8} - 668993098 T^{9} - 12551944974 T^{10} + 104528692176 T^{11} + 448506722220 T^{12} - 5600026123352 T^{13} + 4781645551194 T^{14} - 97720205272902 T^{15} + 1605080023074733 T^{16} + 10409840156565273 T^{17} - 228115272580359973 T^{18} + 10409840156565273 p T^{19} + 1605080023074733 p^{2} T^{20} - 97720205272902 p^{3} T^{21} + 4781645551194 p^{4} T^{22} - 5600026123352 p^{5} T^{23} + 448506722220 p^{6} T^{24} + 104528692176 p^{7} T^{25} - 12551944974 p^{8} T^{26} - 668993098 p^{9} T^{27} + 61915834 p^{10} T^{28} + 11108092 p^{11} T^{29} + 728495 p^{12} T^{30} - 311737 p^{13} T^{31} - 1217 p^{14} T^{32} + 3654 p^{15} T^{33} - 135 p^{16} T^{34} - 16 p^{17} T^{35} + p^{18} T^{36} \) | |
| 67 | \( 1 + 4 T - 212 T^{2} - 2764 T^{3} + 14134 T^{4} + 462258 T^{5} + 2204602 T^{6} - 33230947 T^{7} - 473404888 T^{8} - 574450312 T^{9} + 37293410268 T^{10} + 300452040119 T^{11} - 845286010235 T^{12} - 26127908638343 T^{13} - 95648208847815 T^{14} + 1182493304699202 T^{15} + 11503623385155704 T^{16} - 22934012630039767 T^{17} - 843922695765809705 T^{18} - 22934012630039767 p T^{19} + 11503623385155704 p^{2} T^{20} + 1182493304699202 p^{3} T^{21} - 95648208847815 p^{4} T^{22} - 26127908638343 p^{5} T^{23} - 845286010235 p^{6} T^{24} + 300452040119 p^{7} T^{25} + 37293410268 p^{8} T^{26} - 574450312 p^{9} T^{27} - 473404888 p^{10} T^{28} - 33230947 p^{11} T^{29} + 2204602 p^{12} T^{30} + 462258 p^{13} T^{31} + 14134 p^{14} T^{32} - 2764 p^{15} T^{33} - 212 p^{16} T^{34} + 4 p^{17} T^{35} + p^{18} T^{36} \) | |
| 71 | \( 1 - 17 T - 300 T^{2} + 4107 T^{3} + 86868 T^{4} - 679540 T^{5} - 16243911 T^{6} + 55505375 T^{7} + 2345569400 T^{8} - 1007378546 T^{9} - 248995137015 T^{10} - 489844924333 T^{11} + 20698661341805 T^{12} + 71845723388448 T^{13} - 1313669709111476 T^{14} - 5578634072673011 T^{15} + 72300958451106108 T^{16} + 163980535332853327 T^{17} - 4241912047551272861 T^{18} + 163980535332853327 p T^{19} + 72300958451106108 p^{2} T^{20} - 5578634072673011 p^{3} T^{21} - 1313669709111476 p^{4} T^{22} + 71845723388448 p^{5} T^{23} + 20698661341805 p^{6} T^{24} - 489844924333 p^{7} T^{25} - 248995137015 p^{8} T^{26} - 1007378546 p^{9} T^{27} + 2345569400 p^{10} T^{28} + 55505375 p^{11} T^{29} - 16243911 p^{12} T^{30} - 679540 p^{13} T^{31} + 86868 p^{14} T^{32} + 4107 p^{15} T^{33} - 300 p^{16} T^{34} - 17 p^{17} T^{35} + p^{18} T^{36} \) | |
| 73 | \( ( 1 - 28 T + 809 T^{2} - 14666 T^{3} + 254777 T^{4} - 3461886 T^{5} + 44615441 T^{6} - 481572855 T^{7} + 4927649454 T^{8} - 43235190634 T^{9} + 4927649454 p T^{10} - 481572855 p^{2} T^{11} + 44615441 p^{3} T^{12} - 3461886 p^{4} T^{13} + 254777 p^{5} T^{14} - 14666 p^{6} T^{15} + 809 p^{7} T^{16} - 28 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
| 79 | \( 1 + 23 T - 41 T^{2} - 2536 T^{3} + 29012 T^{4} + 355802 T^{5} - 3419797 T^{6} - 11240734 T^{7} + 340997378 T^{8} - 97825383 T^{9} - 1685475826 T^{10} + 58526477843 T^{11} - 1530915834722 T^{12} + 12806228395192 T^{13} + 171533135372322 T^{14} - 1666211971056667 T^{15} - 1089646115916926 T^{16} + 81794692536649012 T^{17} - 336255795547181461 T^{18} + 81794692536649012 p T^{19} - 1089646115916926 p^{2} T^{20} - 1666211971056667 p^{3} T^{21} + 171533135372322 p^{4} T^{22} + 12806228395192 p^{5} T^{23} - 1530915834722 p^{6} T^{24} + 58526477843 p^{7} T^{25} - 1685475826 p^{8} T^{26} - 97825383 p^{9} T^{27} + 340997378 p^{10} T^{28} - 11240734 p^{11} T^{29} - 3419797 p^{12} T^{30} + 355802 p^{13} T^{31} + 29012 p^{14} T^{32} - 2536 p^{15} T^{33} - 41 p^{16} T^{34} + 23 p^{17} T^{35} + p^{18} T^{36} \) | |
| 83 | \( 1 - 29 T - 183 T^{2} + 9936 T^{3} + 70741 T^{4} - 2752687 T^{5} - 18244663 T^{6} + 518133132 T^{7} + 4510968742 T^{8} - 79750591432 T^{9} - 844004923227 T^{10} + 9384960493308 T^{11} + 132560349724982 T^{12} - 881517234474019 T^{13} - 16871401115513427 T^{14} + 57897148097844156 T^{15} + 1820273835204960649 T^{16} - 1913487122275787921 T^{17} - \)\(16\!\cdots\!65\)\( T^{18} - 1913487122275787921 p T^{19} + 1820273835204960649 p^{2} T^{20} + 57897148097844156 p^{3} T^{21} - 16871401115513427 p^{4} T^{22} - 881517234474019 p^{5} T^{23} + 132560349724982 p^{6} T^{24} + 9384960493308 p^{7} T^{25} - 844004923227 p^{8} T^{26} - 79750591432 p^{9} T^{27} + 4510968742 p^{10} T^{28} + 518133132 p^{11} T^{29} - 18244663 p^{12} T^{30} - 2752687 p^{13} T^{31} + 70741 p^{14} T^{32} + 9936 p^{15} T^{33} - 183 p^{16} T^{34} - 29 p^{17} T^{35} + p^{18} T^{36} \) | |
| 89 | \( 1 + 16 T - 215 T^{2} - 4840 T^{3} + 16293 T^{4} + 7362 p T^{5} + 318398 T^{6} - 36831999 T^{7} - 58303818 T^{8} - 1314667921 T^{9} - 12053465421 T^{10} + 397084981645 T^{11} + 2082992818196 T^{12} - 31462359035499 T^{13} - 9862118806157 T^{14} + 1245367318227483 T^{15} - 28877011354146467 T^{16} - 20602930462417985 T^{17} + 3908954816484087009 T^{18} - 20602930462417985 p T^{19} - 28877011354146467 p^{2} T^{20} + 1245367318227483 p^{3} T^{21} - 9862118806157 p^{4} T^{22} - 31462359035499 p^{5} T^{23} + 2082992818196 p^{6} T^{24} + 397084981645 p^{7} T^{25} - 12053465421 p^{8} T^{26} - 1314667921 p^{9} T^{27} - 58303818 p^{10} T^{28} - 36831999 p^{11} T^{29} + 318398 p^{12} T^{30} + 7362 p^{14} T^{31} + 16293 p^{14} T^{32} - 4840 p^{15} T^{33} - 215 p^{16} T^{34} + 16 p^{17} T^{35} + p^{18} T^{36} \) | |
| 97 | \( ( 1 - 13 T + 414 T^{2} - 5594 T^{3} + 98011 T^{4} - 1201603 T^{5} + 16342757 T^{6} - 173865667 T^{7} + 2061598845 T^{8} - 19031700510 T^{9} + 2061598845 p T^{10} - 173865667 p^{2} T^{11} + 16342757 p^{3} T^{12} - 1201603 p^{4} T^{13} + 98011 p^{5} T^{14} - 5594 p^{6} T^{15} + 414 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.15522695309587337208953690514, −2.11039219095477816182582137324, −2.06786864797071694399303083942, −1.98516195648648984325466260692, −1.89558204608881343987975557047, −1.86748786959321520967714270909, −1.77220822353153436754111965428, −1.74284346260144483397794229663, −1.72678575870165073329390876275, −1.70855671521002817659218394030, −1.59252154339476762935768633739, −1.36233354030578175992672463975, −1.33064636982643103617012424877, −1.30692099418708005772916846648, −1.21345221731811278094164755885, −1.15093557666826625161074539743, −1.02686704344867952778713783208, −0.907324485823840728748147422928, −0.896921057655454555881048249678, −0.77686280416473840589648719261, −0.75675679731280491955378916169, −0.65225777104771375241436044166, −0.41829561459248018934007675577, −0.28696785814632800201047099435, −0.21410792411958247523669407182, 0.21410792411958247523669407182, 0.28696785814632800201047099435, 0.41829561459248018934007675577, 0.65225777104771375241436044166, 0.75675679731280491955378916169, 0.77686280416473840589648719261, 0.896921057655454555881048249678, 0.907324485823840728748147422928, 1.02686704344867952778713783208, 1.15093557666826625161074539743, 1.21345221731811278094164755885, 1.30692099418708005772916846648, 1.33064636982643103617012424877, 1.36233354030578175992672463975, 1.59252154339476762935768633739, 1.70855671521002817659218394030, 1.72678575870165073329390876275, 1.74284346260144483397794229663, 1.77220822353153436754111965428, 1.86748786959321520967714270909, 1.89558204608881343987975557047, 1.98516195648648984325466260692, 2.06786864797071694399303083942, 2.11039219095477816182582137324, 2.15522695309587337208953690514
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.