Dirichlet series
| L(s) = 1 | + 6·3-s + 4·4-s + 2·7-s + 9·9-s + 24·12-s + 23·13-s + 16·16-s + 26·19-s + 12·21-s − 50·25-s + 8·28-s − 13·31-s + 36·36-s + 26·37-s + 138·39-s + 122·43-s + 96·48-s − 45·49-s + 92·52-s + 156·57-s + 47·61-s + 18·63-s + 109·67-s − 924·73-s − 300·75-s + 104·76-s + 252·79-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4-s + 2/7·7-s + 9-s + 2·12-s + 1.76·13-s + 16-s + 1.36·19-s + 4/7·21-s − 2·25-s + 2/7·28-s − 0.419·31-s + 36-s + 0.702·37-s + 3.53·39-s + 2.83·43-s + 2·48-s − 0.918·49-s + 1.76·52-s + 2.73·57-s + 0.770·61-s + 2/7·63-s + 1.62·67-s − 12.6·73-s − 4·75-s + 1.36·76-s + 3.18·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20} \cdot 67^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.89687\times 10^{14}\) |
| Root analytic conductor: | \(2.34026\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} \cdot 67^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(10.05388160\) |
| \(L(\frac12)\) | \(\approx\) | \(10.05388160\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} - p^{5} T^{5} + p^{6} T^{6} - p^{7} T^{7} + p^{8} T^{8} - p^{9} T^{9} + p^{10} T^{10} )^{2} \) |
| 67 | \( 1 - 109 T + 7392 T^{2} - 316427 T^{3} + 1307855 T^{4} + 1277884608 T^{5} - 145160383367 T^{6} + 10086057781691 T^{7} - 447755337269856 T^{8} + 3529018380403405 T^{9} + 1625310705540412416 T^{10} + 3529018380403405 p^{2} T^{11} - 447755337269856 p^{4} T^{12} + 10086057781691 p^{6} T^{13} - 145160383367 p^{8} T^{14} + 1277884608 p^{10} T^{15} + 1307855 p^{12} T^{16} - 316427 p^{14} T^{17} + 7392 p^{16} T^{18} - 109 p^{18} T^{19} + p^{20} T^{20} \) | |
| good | 2 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) |
| 5 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} - p^{5} T^{5} + p^{6} T^{6} - p^{7} T^{7} + p^{8} T^{8} - p^{9} T^{9} + p^{10} T^{10} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} + p^{5} T^{5} + p^{6} T^{6} + p^{7} T^{7} + p^{8} T^{8} + p^{9} T^{9} + p^{10} T^{10} )^{2} \) | |
| 7 | \( ( 1 - 13 T + 120 T^{2} - 923 T^{3} + 6119 T^{4} - 34320 T^{5} + 146329 T^{6} - 220597 T^{7} - 4302360 T^{8} + 66739933 T^{9} - 656803489 T^{10} + 66739933 p^{2} T^{11} - 4302360 p^{4} T^{12} - 220597 p^{6} T^{13} + 146329 p^{8} T^{14} - 34320 p^{10} T^{15} + 6119 p^{12} T^{16} - 923 p^{14} T^{17} + 120 p^{16} T^{18} - 13 p^{18} T^{19} + p^{20} T^{20} )( 1 + 11 T + 72 T^{2} + 253 T^{3} - 745 T^{4} - 20592 T^{5} - 190007 T^{6} - 1081069 T^{7} - 2581416 T^{8} + 24576805 T^{9} + 396834239 T^{10} + 24576805 p^{2} T^{11} - 2581416 p^{4} T^{12} - 1081069 p^{6} T^{13} - 190007 p^{8} T^{14} - 20592 p^{10} T^{15} - 745 p^{12} T^{16} + 253 p^{14} T^{17} + 72 p^{16} T^{18} + 11 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 11 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 13 | \( ( 1 - 22 T + 315 T^{2} - 3212 T^{3} + 17429 T^{4} + 159390 T^{5} - 6452081 T^{6} + 115008872 T^{7} - 1439793495 T^{8} + 12238957522 T^{9} - 25931964829 T^{10} + 12238957522 p^{2} T^{11} - 1439793495 p^{4} T^{12} + 115008872 p^{6} T^{13} - 6452081 p^{8} T^{14} + 159390 p^{10} T^{15} + 17429 p^{12} T^{16} - 3212 p^{14} T^{17} + 315 p^{16} T^{18} - 22 p^{18} T^{19} + p^{20} T^{20} )( 1 - T - 168 T^{2} + 337 T^{3} + 28055 T^{4} - 85008 T^{5} - 4656287 T^{6} + 19022639 T^{7} + 767889864 T^{8} - 3982715855 T^{9} - 125790671161 T^{10} - 3982715855 p^{2} T^{11} + 767889864 p^{4} T^{12} + 19022639 p^{6} T^{13} - 4656287 p^{8} T^{14} - 85008 p^{10} T^{15} + 28055 p^{12} T^{16} + 337 p^{14} T^{17} - 168 p^{16} T^{18} - p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 17 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 19 | \( ( 1 - 37 T + 1008 T^{2} - 23939 T^{3} + 521855 T^{4} - 10666656 T^{5} + 206276617 T^{6} - 3781572013 T^{7} + 65452305744 T^{8} - 1056587815835 T^{9} + 15465466812311 T^{10} - 1056587815835 p^{2} T^{11} + 65452305744 p^{4} T^{12} - 3781572013 p^{6} T^{13} + 206276617 p^{8} T^{14} - 10666656 p^{10} T^{15} + 521855 p^{12} T^{16} - 23939 p^{14} T^{17} + 1008 p^{16} T^{18} - 37 p^{18} T^{19} + p^{20} T^{20} )( 1 + 11 T - 240 T^{2} - 6611 T^{3} + 13919 T^{4} + 2539680 T^{5} + 22911721 T^{6} - 664795549 T^{7} - 15583882320 T^{8} + 68568487669 T^{9} + 6380034881879 T^{10} + 68568487669 p^{2} T^{11} - 15583882320 p^{4} T^{12} - 664795549 p^{6} T^{13} + 22911721 p^{8} T^{14} + 2539680 p^{10} T^{15} + 13919 p^{12} T^{16} - 6611 p^{14} T^{17} - 240 p^{16} T^{18} + 11 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 23 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 29 | \( ( 1 - p T + p^{2} T^{2} )^{10}( 1 + p T + p^{2} T^{2} )^{10} \) | |
| 31 | \( ( 1 - 46 T + 1155 T^{2} - 8924 T^{3} - 699451 T^{4} + 40750710 T^{5} - 1202360249 T^{6} + 16147139144 T^{7} + 412699798665 T^{8} - 34501591455974 T^{9} + 1190468700457739 T^{10} - 34501591455974 p^{2} T^{11} + 412699798665 p^{4} T^{12} + 16147139144 p^{6} T^{13} - 1202360249 p^{8} T^{14} + 40750710 p^{10} T^{15} - 699451 p^{12} T^{16} - 8924 p^{14} T^{17} + 1155 p^{16} T^{18} - 46 p^{18} T^{19} + p^{20} T^{20} )( 1 + 59 T + 2520 T^{2} + 91981 T^{3} + 3005159 T^{4} + 88910640 T^{5} + 2357769961 T^{6} + 53665302659 T^{7} + 900435924360 T^{8} + 1553363681941 T^{9} - 773670466075441 T^{10} + 1553363681941 p^{2} T^{11} + 900435924360 p^{4} T^{12} + 53665302659 p^{6} T^{13} + 2357769961 p^{8} T^{14} + 88910640 p^{10} T^{15} + 3005159 p^{12} T^{16} + 91981 p^{14} T^{17} + 2520 p^{16} T^{18} + 59 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 37 | \( ( 1 - 73 T + 3960 T^{2} - 189143 T^{3} + 8386199 T^{4} - 353255760 T^{5} + 14306964049 T^{6} - 560801240137 T^{7} + 21352256746920 T^{8} - 790977844777607 T^{9} + 28510143182231831 T^{10} - 790977844777607 p^{2} T^{11} + 21352256746920 p^{4} T^{12} - 560801240137 p^{6} T^{13} + 14306964049 p^{8} T^{14} - 353255760 p^{10} T^{15} + 8386199 p^{12} T^{16} - 189143 p^{14} T^{17} + 3960 p^{16} T^{18} - 73 p^{18} T^{19} + p^{20} T^{20} )( 1 + 47 T + 840 T^{2} - 24863 T^{3} - 2318521 T^{4} - 74933040 T^{5} - 347797631 T^{6} + 86236843103 T^{7} + 4529266582680 T^{8} + 94817291177953 T^{9} - 1744153266325129 T^{10} + 94817291177953 p^{2} T^{11} + 4529266582680 p^{4} T^{12} + 86236843103 p^{6} T^{13} - 347797631 p^{8} T^{14} - 74933040 p^{10} T^{15} - 2318521 p^{12} T^{16} - 24863 p^{14} T^{17} + 840 p^{16} T^{18} + 47 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 41 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 43 | \( ( 1 - 61 T + 1872 T^{2} - 1403 T^{3} - 3375745 T^{4} + 208514592 T^{5} - 6477637607 T^{6} + 9592413419 T^{7} + 11392014716784 T^{8} - 712649270135555 T^{9} + 22407770266935239 T^{10} - 712649270135555 p^{2} T^{11} + 11392014716784 p^{4} T^{12} + 9592413419 p^{6} T^{13} - 6477637607 p^{8} T^{14} + 208514592 p^{10} T^{15} - 3375745 p^{12} T^{16} - 1403 p^{14} T^{17} + 1872 p^{16} T^{18} - 61 p^{18} T^{19} + p^{20} T^{20} )^{2} \) | |
| 47 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 53 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} - p^{5} T^{5} + p^{6} T^{6} - p^{7} T^{7} + p^{8} T^{8} - p^{9} T^{9} + p^{10} T^{10} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} + p^{5} T^{5} + p^{6} T^{6} + p^{7} T^{7} + p^{8} T^{8} + p^{9} T^{9} + p^{10} T^{10} )^{2} \) | |
| 59 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} - p^{5} T^{5} + p^{6} T^{6} - p^{7} T^{7} + p^{8} T^{8} - p^{9} T^{9} + p^{10} T^{10} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} + p^{5} T^{5} + p^{6} T^{6} + p^{7} T^{7} + p^{8} T^{8} + p^{9} T^{9} + p^{10} T^{10} )^{2} \) | |
| 61 | \( ( 1 - 121 T + 10920 T^{2} - 871079 T^{3} + 64767239 T^{4} - 4595550960 T^{5} + 315062769841 T^{6} - 21022550028601 T^{7} + 1371379986882360 T^{8} - 87712069756341239 T^{9} + 5510255509328028359 T^{10} - 87712069756341239 p^{2} T^{11} + 1371379986882360 p^{4} T^{12} - 21022550028601 p^{6} T^{13} + 315062769841 p^{8} T^{14} - 4595550960 p^{10} T^{15} + 64767239 p^{12} T^{16} - 871079 p^{14} T^{17} + 10920 p^{16} T^{18} - 121 p^{18} T^{19} + p^{20} T^{20} )( 1 + 74 T + 1755 T^{2} - 145484 T^{3} - 17296171 T^{4} - 738570690 T^{5} + 9704821231 T^{6} + 3466378308584 T^{7} + 220400355034665 T^{8} + 3411232586324146 T^{9} - 567678509696001661 T^{10} + 3411232586324146 p^{2} T^{11} + 220400355034665 p^{4} T^{12} + 3466378308584 p^{6} T^{13} + 9704821231 p^{8} T^{14} - 738570690 p^{10} T^{15} - 17296171 p^{12} T^{16} - 145484 p^{14} T^{17} + 1755 p^{16} T^{18} + 74 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 71 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 73 | \( ( 1 + 97 T + p^{2} T^{2} )^{10}( 1 - 46 T - 3213 T^{2} + 392932 T^{3} - 952795 T^{4} - 2050106058 T^{5} + 99382323223 T^{6} + 6353428314824 T^{7} - 821866102937271 T^{8} + 3948421245417370 T^{9} + 4198097085263518139 T^{10} + 3948421245417370 p^{2} T^{11} - 821866102937271 p^{4} T^{12} + 6353428314824 p^{6} T^{13} + 99382323223 p^{8} T^{14} - 2050106058 p^{10} T^{15} - 952795 p^{12} T^{16} + 392932 p^{14} T^{17} - 3213 p^{16} T^{18} - 46 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 79 | \( ( 1 - 11 T + p^{2} T^{2} )^{10}( 1 - 142 T + 13923 T^{2} - 1090844 T^{3} + 68006405 T^{4} - 2848952106 T^{5} - 19876774553 T^{6} + 20602812080072 T^{7} - 2801548365384951 T^{8} + 269237717692933690 T^{9} - 20747292564029104789 T^{10} + 269237717692933690 p^{2} T^{11} - 2801548365384951 p^{4} T^{12} + 20602812080072 p^{6} T^{13} - 19876774553 p^{8} T^{14} - 2848952106 p^{10} T^{15} + 68006405 p^{12} T^{16} - 1090844 p^{14} T^{17} + 13923 p^{16} T^{18} - 142 p^{18} T^{19} + p^{20} T^{20} ) \) | |
| 83 | \( ( 1 - p T + p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} - p^{7} T^{7} + p^{9} T^{9} - p^{10} T^{10} + p^{11} T^{11} - p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} + p^{17} T^{17} - p^{19} T^{19} + p^{20} T^{20} )( 1 + p T - p^{3} T^{3} - p^{4} T^{4} + p^{6} T^{6} + p^{7} T^{7} - p^{9} T^{9} - p^{10} T^{10} - p^{11} T^{11} + p^{13} T^{13} + p^{14} T^{14} - p^{16} T^{16} - p^{17} T^{17} + p^{19} T^{19} + p^{20} T^{20} ) \) | |
| 89 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} - p^{5} T^{5} + p^{6} T^{6} - p^{7} T^{7} + p^{8} T^{8} - p^{9} T^{9} + p^{10} T^{10} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} + p^{5} T^{5} + p^{6} T^{6} + p^{7} T^{7} + p^{8} T^{8} + p^{9} T^{9} + p^{10} T^{10} )^{2} \) | |
| 97 | \( ( 1 - 169 T + 19152 T^{2} - 1646567 T^{3} + 98068655 T^{4} - 1081053792 T^{5} - 740029884047 T^{6} + 135236685532871 T^{7} - 15892058676056976 T^{8} + 1413315942074845705 T^{9} - 89322014127628836961 T^{10} + 1413315942074845705 p^{2} T^{11} - 15892058676056976 p^{4} T^{12} + 135236685532871 p^{6} T^{13} - 740029884047 p^{8} T^{14} - 1081053792 p^{10} T^{15} + 98068655 p^{12} T^{16} - 1646567 p^{14} T^{17} + 19152 p^{16} T^{18} - 169 p^{18} T^{19} + p^{20} T^{20} )( 1 + 2 T - 9405 T^{2} - 37628 T^{3} + 88416389 T^{4} + 530874630 T^{5} - 830848054841 T^{6} - 6656695503352 T^{7} + 7804135956992265 T^{8} + 78241119905023498 T^{9} - 73272632979530174389 T^{10} + 78241119905023498 p^{2} T^{11} + 7804135956992265 p^{4} T^{12} - 6656695503352 p^{6} T^{13} - 830848054841 p^{8} T^{14} + 530874630 p^{10} T^{15} + 88416389 p^{12} T^{16} - 37628 p^{14} T^{17} - 9405 p^{16} T^{18} + 2 p^{18} T^{19} + p^{20} T^{20} ) \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.91805814152481694691029701837, −2.81036084033474906088025623834, −2.78898676173494927531726530102, −2.52195223964227485483762431286, −2.50363385487842813794118537177, −2.45735362913757941066270957210, −2.38006959552702229807840765279, −2.34436278224257291676683677293, −2.19785984475897998393334362312, −1.93723293411883902598718991359, −1.85651616025767959437945609766, −1.84230372802242121773320224425, −1.82212683216454244125707447464, −1.63141445960952292160365040494, −1.59342509524605268294879027383, −1.58007225538203431605312158414, −1.38262694491995588504529259084, −1.16213626982020024999060404743, −1.13981233585495968221058436785, −1.02602832072111213080566071243, −0.75970266555753019492444962997, −0.72866326343227662894284078974, −0.47030766639054917410723971006, −0.41977627276994826727249569870, −0.07851029381438647948902911721, 0.07851029381438647948902911721, 0.41977627276994826727249569870, 0.47030766639054917410723971006, 0.72866326343227662894284078974, 0.75970266555753019492444962997, 1.02602832072111213080566071243, 1.13981233585495968221058436785, 1.16213626982020024999060404743, 1.38262694491995588504529259084, 1.58007225538203431605312158414, 1.59342509524605268294879027383, 1.63141445960952292160365040494, 1.82212683216454244125707447464, 1.84230372802242121773320224425, 1.85651616025767959437945609766, 1.93723293411883902598718991359, 2.19785984475897998393334362312, 2.34436278224257291676683677293, 2.38006959552702229807840765279, 2.45735362913757941066270957210, 2.50363385487842813794118537177, 2.52195223964227485483762431286, 2.78898676173494927531726530102, 2.81036084033474906088025623834, 2.91805814152481694691029701837
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.