Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 8·4-s − 5·5-s + 4·6-s + 4·7-s − 17·8-s + 7·9-s + 10·10-s − 16·12-s − 13·13-s − 8·14-s + 10·15-s + 39·16-s − 10·17-s − 14·18-s + 6·19-s − 40·20-s − 8·21-s + 32·23-s + 34·24-s + 11·25-s + 26·26-s − 16·27-s + 32·28-s + 12·29-s − 20·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4·4-s − 2.23·5-s + 1.63·6-s + 1.51·7-s − 6.01·8-s + 7/3·9-s + 3.16·10-s − 4.61·12-s − 3.60·13-s − 2.13·14-s + 2.58·15-s + 39/4·16-s − 2.42·17-s − 3.29·18-s + 1.37·19-s − 8.94·20-s − 1.74·21-s + 6.67·23-s + 6.94·24-s + 11/5·25-s + 5.09·26-s − 3.07·27-s + 6.04·28-s + 2.22·29-s − 3.65·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{16} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.91676\times 10^{13}\) |
| Root analytic conductor: | \(2.60064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{16} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.285036219\) |
| \(L(\frac12)\) | \(\approx\) | \(1.285036219\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p T - p^{2} T^{2} - 7 T^{3} + 13 T^{4} + 9 p T^{5} - 25 T^{6} - 39 T^{7} + 13 p^{2} T^{8} + 109 T^{9} - 115 T^{10} - 99 p T^{11} + 303 T^{12} + 287 T^{13} - 209 p T^{14} - 37 p^{2} T^{15} + 625 T^{16} - 37 p^{3} T^{17} - 209 p^{3} T^{18} + 287 p^{3} T^{19} + 303 p^{4} T^{20} - 99 p^{6} T^{21} - 115 p^{6} T^{22} + 109 p^{7} T^{23} + 13 p^{10} T^{24} - 39 p^{9} T^{25} - 25 p^{10} T^{26} + 9 p^{12} T^{27} + 13 p^{12} T^{28} - 7 p^{13} T^{29} - p^{16} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + 2 T - p T^{2} - 4 T^{3} + 2 T^{4} - 2 p T^{5} - 8 T^{6} - 86 T^{7} - 38 p T^{8} + 76 p T^{9} + 367 T^{10} + 2 p^{2} T^{11} - 37 p^{2} T^{12} + 1016 T^{13} + 2720 T^{14} - 2368 T^{15} - 10592 T^{16} - 2368 p T^{17} + 2720 p^{2} T^{18} + 1016 p^{3} T^{19} - 37 p^{6} T^{20} + 2 p^{7} T^{21} + 367 p^{6} T^{22} + 76 p^{8} T^{23} - 38 p^{9} T^{24} - 86 p^{9} T^{25} - 8 p^{10} T^{26} - 2 p^{12} T^{27} + 2 p^{12} T^{28} - 4 p^{13} T^{29} - p^{15} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 + p T + 14 T^{2} + 44 T^{3} + 109 T^{4} + 292 T^{5} + 994 T^{6} + 2416 T^{7} + 5986 T^{8} + 16176 T^{9} + 35286 T^{10} + 91101 T^{11} + 234456 T^{12} + 492396 T^{13} + 1194866 T^{14} + 2681212 T^{15} + 5341741 T^{16} + 2681212 p T^{17} + 1194866 p^{2} T^{18} + 492396 p^{3} T^{19} + 234456 p^{4} T^{20} + 91101 p^{5} T^{21} + 35286 p^{6} T^{22} + 16176 p^{7} T^{23} + 5986 p^{8} T^{24} + 2416 p^{9} T^{25} + 994 p^{10} T^{26} + 292 p^{11} T^{27} + 109 p^{12} T^{28} + 44 p^{13} T^{29} + 14 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + p T + 72 T^{2} + 244 T^{3} + 737 T^{4} + 2006 T^{5} + 3492 T^{6} + 7936 T^{7} + 3122 p T^{8} + 125222 T^{9} + 55652 T^{10} - 2894553 T^{11} - 1765676 p T^{12} - 102379926 T^{13} - 28728950 p T^{14} - 1282762402 T^{15} - 4399561547 T^{16} - 1282762402 p T^{17} - 28728950 p^{3} T^{18} - 102379926 p^{3} T^{19} - 1765676 p^{5} T^{20} - 2894553 p^{5} T^{21} + 55652 p^{6} T^{22} + 125222 p^{7} T^{23} + 3122 p^{9} T^{24} + 7936 p^{9} T^{25} + 3492 p^{10} T^{26} + 2006 p^{11} T^{27} + 737 p^{12} T^{28} + 244 p^{13} T^{29} + 72 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 10 T - 37 T^{2} - 633 T^{3} - 168 T^{4} + 13271 T^{5} + 17626 T^{6} + 16005 T^{7} + 351109 T^{8} - 5653873 T^{9} - 25003323 T^{10} + 95686688 T^{11} + 315645412 T^{12} - 305068606 T^{13} + 5822669428 T^{14} - 3664060760 T^{15} - 218663265475 T^{16} - 3664060760 p T^{17} + 5822669428 p^{2} T^{18} - 305068606 p^{3} T^{19} + 315645412 p^{4} T^{20} + 95686688 p^{5} T^{21} - 25003323 p^{6} T^{22} - 5653873 p^{7} T^{23} + 351109 p^{8} T^{24} + 16005 p^{9} T^{25} + 17626 p^{10} T^{26} + 13271 p^{11} T^{27} - 168 p^{12} T^{28} - 633 p^{13} T^{29} - 37 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 6 T + 3 T^{2} - 85 T^{3} + 1184 T^{4} + 2399 T^{5} - 34320 T^{6} - 47453 T^{7} + 19149 T^{8} + 5712061 T^{9} - 3409115 T^{10} - 100225110 T^{11} - 227341598 T^{12} + 1375692546 T^{13} + 11005678798 T^{14} - 19570163952 T^{15} - 191338098209 T^{16} - 19570163952 p T^{17} + 11005678798 p^{2} T^{18} + 1375692546 p^{3} T^{19} - 227341598 p^{4} T^{20} - 100225110 p^{5} T^{21} - 3409115 p^{6} T^{22} + 5712061 p^{7} T^{23} + 19149 p^{8} T^{24} - 47453 p^{9} T^{25} - 34320 p^{10} T^{26} + 2399 p^{11} T^{27} + 1184 p^{12} T^{28} - 85 p^{13} T^{29} + 3 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 40290 p T^{6} - 5230592 T^{7} + 26816927 T^{8} - 5230592 p T^{9} + 40290 p^{3} T^{10} - 141712 p^{3} T^{11} + 19567 p^{4} T^{12} - 2234 p^{5} T^{13} + 224 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 12 T - 70 T^{2} + 1390 T^{3} - 774 T^{4} - 64772 T^{5} + 304755 T^{6} + 772470 T^{7} - 15507999 T^{8} + 61115248 T^{9} + 297254044 T^{10} - 3599964238 T^{11} + 3985868241 T^{12} + 93662943228 T^{13} - 421310359901 T^{14} - 1028919750178 T^{15} + 15133672321246 T^{16} - 1028919750178 p T^{17} - 421310359901 p^{2} T^{18} + 93662943228 p^{3} T^{19} + 3985868241 p^{4} T^{20} - 3599964238 p^{5} T^{21} + 297254044 p^{6} T^{22} + 61115248 p^{7} T^{23} - 15507999 p^{8} T^{24} + 772470 p^{9} T^{25} + 304755 p^{10} T^{26} - 64772 p^{11} T^{27} - 774 p^{12} T^{28} + 1390 p^{13} T^{29} - 70 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 2 T - 75 T^{2} - 352 T^{3} + 2587 T^{4} + 15784 T^{5} - 75864 T^{6} - 435376 T^{7} + 3782312 T^{8} + 14561418 T^{9} - 150890523 T^{10} - 457844234 T^{11} + 4912277406 T^{12} + 5133296146 T^{13} - 182512228746 T^{14} + 17734057088 T^{15} + 6474843737997 T^{16} + 17734057088 p T^{17} - 182512228746 p^{2} T^{18} + 5133296146 p^{3} T^{19} + 4912277406 p^{4} T^{20} - 457844234 p^{5} T^{21} - 150890523 p^{6} T^{22} + 14561418 p^{7} T^{23} + 3782312 p^{8} T^{24} - 435376 p^{9} T^{25} - 75864 p^{10} T^{26} + 15784 p^{11} T^{27} + 2587 p^{12} T^{28} - 352 p^{13} T^{29} - 75 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 11 T - 2 p T^{2} - 1323 T^{3} + 1923 T^{4} + 97285 T^{5} + 90914 T^{6} - 5713113 T^{7} - 17349512 T^{8} + 265295915 T^{9} + 1411267662 T^{10} - 9905347845 T^{11} - 84217332664 T^{12} + 276882360296 T^{13} + 4072917985018 T^{14} - 3748231783796 T^{15} - 163433617792703 T^{16} - 3748231783796 p T^{17} + 4072917985018 p^{2} T^{18} + 276882360296 p^{3} T^{19} - 84217332664 p^{4} T^{20} - 9905347845 p^{5} T^{21} + 1411267662 p^{6} T^{22} + 265295915 p^{7} T^{23} - 17349512 p^{8} T^{24} - 5713113 p^{9} T^{25} + 90914 p^{10} T^{26} + 97285 p^{11} T^{27} + 1923 p^{12} T^{28} - 1323 p^{13} T^{29} - 2 p^{15} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 20 T + 7 T^{2} - 2422 T^{3} - 13592 T^{4} + 89190 T^{5} + 972960 T^{6} + 955362 T^{7} - 17703016 T^{8} - 127968546 T^{9} - 596716619 T^{10} + 1461391100 T^{11} + 26017347843 T^{12} + 43282378760 T^{13} - 65439843920 T^{14} - 655546700040 T^{15} - 12524609697280 T^{16} - 655546700040 p T^{17} - 65439843920 p^{2} T^{18} + 43282378760 p^{3} T^{19} + 26017347843 p^{4} T^{20} + 1461391100 p^{5} T^{21} - 596716619 p^{6} T^{22} - 127968546 p^{7} T^{23} - 17703016 p^{8} T^{24} + 955362 p^{9} T^{25} + 972960 p^{10} T^{26} + 89190 p^{11} T^{27} - 13592 p^{12} T^{28} - 2422 p^{13} T^{29} + 7 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 4 T + 239 T^{2} - 936 T^{3} + 27462 T^{4} - 98508 T^{5} + 2007760 T^{6} - 6271668 T^{7} + 102278657 T^{8} - 6271668 p T^{9} + 2007760 p^{2} T^{10} - 98508 p^{3} T^{11} + 27462 p^{4} T^{12} - 936 p^{5} T^{13} + 239 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 7 T - 89 T^{2} + 202 T^{3} + 4968 T^{4} + 21827 T^{5} + 11495 T^{6} - 2393311 T^{7} - 19649733 T^{8} + 35705166 T^{9} + 1391558095 T^{10} + 7499381673 T^{11} - 30821621912 T^{12} - 590826226892 T^{13} - 1975676885408 T^{14} + 14803300228298 T^{15} + 164951174929905 T^{16} + 14803300228298 p T^{17} - 1975676885408 p^{2} T^{18} - 590826226892 p^{3} T^{19} - 30821621912 p^{4} T^{20} + 7499381673 p^{5} T^{21} + 1391558095 p^{6} T^{22} + 35705166 p^{7} T^{23} - 19649733 p^{8} T^{24} - 2393311 p^{9} T^{25} + 11495 p^{10} T^{26} + 21827 p^{11} T^{27} + 4968 p^{12} T^{28} + 202 p^{13} T^{29} - 89 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 41 T + 698 T^{2} + 5865 T^{3} + 18415 T^{4} - 17787 T^{5} + 1184038 T^{6} + 25764329 T^{7} + 182959840 T^{8} + 399954485 T^{9} - 220252678 T^{10} + 21803507967 T^{11} + 208503614596 T^{12} - 375622782850 T^{13} - 10836301461310 T^{14} - 2007212797106 T^{15} + 469856653928949 T^{16} - 2007212797106 p T^{17} - 10836301461310 p^{2} T^{18} - 375622782850 p^{3} T^{19} + 208503614596 p^{4} T^{20} + 21803507967 p^{5} T^{21} - 220252678 p^{6} T^{22} + 399954485 p^{7} T^{23} + 182959840 p^{8} T^{24} + 25764329 p^{9} T^{25} + 1184038 p^{10} T^{26} - 17787 p^{11} T^{27} + 18415 p^{12} T^{28} + 5865 p^{13} T^{29} + 698 p^{14} T^{30} + 41 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 18 T + 25 T^{2} - 860 T^{3} + 7311 T^{4} + 177528 T^{5} + 4750 p T^{6} - 157320 p T^{7} - 6546264 T^{8} + 769859398 T^{9} + 2676500529 T^{10} - 31642972128 T^{11} - 147191476614 T^{12} + 1958975281938 T^{13} + 18358900572134 T^{14} + 6395455699032 T^{15} - 550800860184419 T^{16} + 6395455699032 p T^{17} + 18358900572134 p^{2} T^{18} + 1958975281938 p^{3} T^{19} - 147191476614 p^{4} T^{20} - 31642972128 p^{5} T^{21} + 2676500529 p^{6} T^{22} + 769859398 p^{7} T^{23} - 6546264 p^{8} T^{24} - 157320 p^{10} T^{25} + 4750 p^{11} T^{26} + 177528 p^{11} T^{27} + 7311 p^{12} T^{28} - 860 p^{13} T^{29} + 25 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 12 T - 195 T^{2} + 3592 T^{3} + 9457 T^{4} - 507244 T^{5} + 1595716 T^{6} + 40490936 T^{7} - 318233418 T^{8} - 1680265148 T^{9} + 27894939097 T^{10} - 2641977826 T^{11} - 1448376512614 T^{12} + 4191432114084 T^{13} + 46058803271574 T^{14} - 147290895640268 T^{15} - 1443941235745343 T^{16} - 147290895640268 p T^{17} + 46058803271574 p^{2} T^{18} + 4191432114084 p^{3} T^{19} - 1448376512614 p^{4} T^{20} - 2641977826 p^{5} T^{21} + 27894939097 p^{6} T^{22} - 1680265148 p^{7} T^{23} - 318233418 p^{8} T^{24} + 40490936 p^{9} T^{25} + 1595716 p^{10} T^{26} - 507244 p^{11} T^{27} + 9457 p^{12} T^{28} + 3592 p^{13} T^{29} - 195 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 150444400 p T^{9} + 15702408 p^{2} T^{10} + 1415171 p^{3} T^{11} + 119005 p^{4} T^{12} + 7751 p^{5} T^{13} + 520 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - T - 5 p T^{2} - 347 T^{3} + 62162 T^{4} + 107731 T^{5} - 7363645 T^{6} - 8830711 T^{7} + 730398853 T^{8} + 111771627 T^{9} - 68716020299 T^{10} + 17483084955 T^{11} + 5903862179484 T^{12} - 219949697282 T^{13} - 448386307599448 T^{14} - 29021605851186 T^{15} + 32028099319201737 T^{16} - 29021605851186 p T^{17} - 448386307599448 p^{2} T^{18} - 219949697282 p^{3} T^{19} + 5903862179484 p^{4} T^{20} + 17483084955 p^{5} T^{21} - 68716020299 p^{6} T^{22} + 111771627 p^{7} T^{23} + 730398853 p^{8} T^{24} - 8830711 p^{9} T^{25} - 7363645 p^{10} T^{26} + 107731 p^{11} T^{27} + 62162 p^{12} T^{28} - 347 p^{13} T^{29} - 5 p^{15} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 60 T + 1487 T^{2} + 18394 T^{3} + 89657 T^{4} - 411322 T^{5} - 5441626 T^{6} + 37899490 T^{7} + 846703154 T^{8} + 1296063306 T^{9} - 89886244257 T^{10} - 1038415881434 T^{11} - 2204058435198 T^{12} + 53506579646092 T^{13} + 424723915258082 T^{14} - 2140532344138680 T^{15} - 48779036759889075 T^{16} - 2140532344138680 p T^{17} + 424723915258082 p^{2} T^{18} + 53506579646092 p^{3} T^{19} - 2204058435198 p^{4} T^{20} - 1038415881434 p^{5} T^{21} - 89886244257 p^{6} T^{22} + 1296063306 p^{7} T^{23} + 846703154 p^{8} T^{24} + 37899490 p^{9} T^{25} - 5441626 p^{10} T^{26} - 411322 p^{11} T^{27} + 89657 p^{12} T^{28} + 18394 p^{13} T^{29} + 1487 p^{14} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 15 T - 206 T^{2} + 3010 T^{3} + 29105 T^{4} - 124250 T^{5} - 5253080 T^{6} - 18691300 T^{7} + 581808430 T^{8} + 4760179440 T^{9} - 24079531838 T^{10} - 673703703905 T^{11} - 1814209779892 T^{12} + 52895538994030 T^{13} + 475566405446690 T^{14} - 1654552484719050 T^{15} - 51014747327008815 T^{16} - 1654552484719050 p T^{17} + 475566405446690 p^{2} T^{18} + 52895538994030 p^{3} T^{19} - 1814209779892 p^{4} T^{20} - 673703703905 p^{5} T^{21} - 24079531838 p^{6} T^{22} + 4760179440 p^{7} T^{23} + 581808430 p^{8} T^{24} - 18691300 p^{9} T^{25} - 5253080 p^{10} T^{26} - 124250 p^{11} T^{27} + 29105 p^{12} T^{28} + 3010 p^{13} T^{29} - 206 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 20 T + 57 T^{2} + 124 T^{3} + 26767 T^{4} + 258208 T^{5} + 403724 T^{6} + 19732220 T^{7} + 380894284 T^{8} + 2094980066 T^{9} + 16014449293 T^{10} + 312895475826 T^{11} + 2857383169742 T^{12} + 20223947588302 T^{13} + 234825461586902 T^{14} + 2222853955193260 T^{15} + 17391213981381345 T^{16} + 2222853955193260 p T^{17} + 234825461586902 p^{2} T^{18} + 20223947588302 p^{3} T^{19} + 2857383169742 p^{4} T^{20} + 312895475826 p^{5} T^{21} + 16014449293 p^{6} T^{22} + 2094980066 p^{7} T^{23} + 380894284 p^{8} T^{24} + 19732220 p^{9} T^{25} + 403724 p^{10} T^{26} + 258208 p^{11} T^{27} + 26767 p^{12} T^{28} + 124 p^{13} T^{29} + 57 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 737152 p T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 731972999 p T^{9} + 737152 p^{3} T^{10} - 5152639 p^{3} T^{11} + 357823 p^{4} T^{12} - 20901 p^{5} T^{13} + 1032 p^{6} T^{14} - 37 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 35 T + 248 T^{2} - 5577 T^{3} - 95113 T^{4} + 5579 T^{5} + 8611936 T^{6} + 29293955 T^{7} - 337756866 T^{8} + 1241258413 T^{9} + 46602892672 T^{10} - 183890451593 T^{11} - 7991146758268 T^{12} - 52842171229544 T^{13} + 128069756523188 T^{14} + 4904533562796750 T^{15} + 55268578137339225 T^{16} + 4904533562796750 p T^{17} + 128069756523188 p^{2} T^{18} - 52842171229544 p^{3} T^{19} - 7991146758268 p^{4} T^{20} - 183890451593 p^{5} T^{21} + 46602892672 p^{6} T^{22} + 1241258413 p^{7} T^{23} - 337756866 p^{8} T^{24} + 29293955 p^{9} T^{25} + 8611936 p^{10} T^{26} + 5579 p^{11} T^{27} - 95113 p^{12} T^{28} - 5577 p^{13} T^{29} + 248 p^{14} T^{30} + 35 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.73783737340360321626328338345, −2.73717573270884576272189257844, −2.67009731442544118320455676745, −2.56209231382159549426057675291, −2.26829200597479779112420687600, −2.18052717455140290389212423143, −1.93442962777893463532678717528, −1.87351207836493914157406071735, −1.85743811863806319970783251504, −1.84313639829240507514860273639, −1.77930878213350429628622189889, −1.76348613541605463820961999353, −1.75670325271314239649156372400, −1.67703257947014548991161304814, −1.66210071837211361695198009305, −1.34224924207645234692628939676, −1.34060317763591232341394295975, −1.12865955488084428567882182149, −0.818860908107798424483135732163, −0.76935163373594220574888829748, −0.71485306694356436922930779841, −0.55267068040867357058844035499, −0.51172428521840287852491741829, −0.49217316858746146802944468552, −0.092819770644049395554481536969, 0.092819770644049395554481536969, 0.49217316858746146802944468552, 0.51172428521840287852491741829, 0.55267068040867357058844035499, 0.71485306694356436922930779841, 0.76935163373594220574888829748, 0.818860908107798424483135732163, 1.12865955488084428567882182149, 1.34060317763591232341394295975, 1.34224924207645234692628939676, 1.66210071837211361695198009305, 1.67703257947014548991161304814, 1.75670325271314239649156372400, 1.76348613541605463820961999353, 1.77930878213350429628622189889, 1.84313639829240507514860273639, 1.85743811863806319970783251504, 1.87351207836493914157406071735, 1.93442962777893463532678717528, 2.18052717455140290389212423143, 2.26829200597479779112420687600, 2.56209231382159549426057675291, 2.67009731442544118320455676745, 2.73717573270884576272189257844, 2.73783737340360321626328338345
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.