Dirichlet series
| L(s) = 1 | + 4·2-s + 3-s + 6·4-s + 5-s + 4·6-s − 12·7-s + 4·8-s + 6·9-s + 4·10-s + 4·11-s + 6·12-s − 12·13-s − 48·14-s + 15-s + 16-s + 6·17-s + 24·18-s − 4·19-s + 6·20-s − 12·21-s + 16·22-s − 26·23-s + 4·24-s + 27·25-s − 48·26-s + 9·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 3·4-s + 0.447·5-s + 1.63·6-s − 4.53·7-s + 1.41·8-s + 2·9-s + 1.26·10-s + 1.20·11-s + 1.73·12-s − 3.32·13-s − 12.8·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 5.65·18-s − 0.917·19-s + 1.34·20-s − 2.61·21-s + 3.41·22-s − 5.42·23-s + 0.816·24-s + 27/5·25-s − 9.41·26-s + 1.73·27-s − 13.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 11^{16} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.37276\times 10^{8}\) |
| Root analytic conductor: | \(1.82695\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 11^{16} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(37.18212763\) |
| \(L(\frac12)\) | \(\approx\) | \(37.18212763\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 11 | \( 1 - 4 T - 4 T^{2} - 58 T^{3} + 337 T^{4} + 16 T^{5} + 2602 T^{6} - 13238 T^{7} + 11091 T^{8} - 13238 p T^{9} + 2602 p^{2} T^{10} + 16 p^{3} T^{11} + 337 p^{4} T^{12} - 58 p^{5} T^{13} - 4 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) | |
| good | 3 | \( 1 - T - 5 T^{2} + 2 T^{3} + 5 p T^{4} + 4 T^{5} - 49 T^{6} - 32 T^{7} + 92 T^{8} + 256 T^{9} - 43 T^{10} - 947 T^{11} - 52 p T^{12} + 2116 T^{13} + 158 p^{2} T^{14} - 3136 T^{15} - 4811 T^{16} - 3136 p T^{17} + 158 p^{4} T^{18} + 2116 p^{3} T^{19} - 52 p^{5} T^{20} - 947 p^{5} T^{21} - 43 p^{6} T^{22} + 256 p^{7} T^{23} + 92 p^{8} T^{24} - 32 p^{9} T^{25} - 49 p^{10} T^{26} + 4 p^{11} T^{27} + 5 p^{13} T^{28} + 2 p^{13} T^{29} - 5 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 - T - 26 T^{2} + 28 T^{3} + 269 T^{4} - 414 T^{5} - 1049 T^{6} + 3982 T^{7} - 4019 T^{8} - 4748 p T^{9} + 13148 p T^{10} + 2633 p^{2} T^{11} - 61627 p T^{12} + 2836 p^{2} T^{13} + 18962 p^{2} T^{14} - 23492 p^{2} T^{15} + 27506 p^{2} T^{16} - 23492 p^{3} T^{17} + 18962 p^{4} T^{18} + 2836 p^{5} T^{19} - 61627 p^{5} T^{20} + 2633 p^{7} T^{21} + 13148 p^{7} T^{22} - 4748 p^{8} T^{23} - 4019 p^{8} T^{24} + 3982 p^{9} T^{25} - 1049 p^{10} T^{26} - 414 p^{11} T^{27} + 269 p^{12} T^{28} + 28 p^{13} T^{29} - 26 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 + 12 T + 46 T^{2} - 32 T^{3} - 829 T^{4} - 2220 T^{5} + 3618 T^{6} + 35706 T^{7} + 8579 p T^{8} - 178034 T^{9} - 967290 T^{10} - 1035068 T^{11} + 4572625 T^{12} + 19111376 T^{13} + 19440602 T^{14} - 72315248 T^{15} - 336635444 T^{16} - 72315248 p T^{17} + 19440602 p^{2} T^{18} + 19111376 p^{3} T^{19} + 4572625 p^{4} T^{20} - 1035068 p^{5} T^{21} - 967290 p^{6} T^{22} - 178034 p^{7} T^{23} + 8579 p^{9} T^{24} + 35706 p^{9} T^{25} + 3618 p^{10} T^{26} - 2220 p^{11} T^{27} - 829 p^{12} T^{28} - 32 p^{13} T^{29} + 46 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 12 T + 68 T^{2} + 202 T^{3} + 94 T^{4} - 1244 T^{5} - 476 T^{6} + 29726 T^{7} + 120487 T^{8} - 106400 T^{9} - 2587772 T^{10} - 9527850 T^{11} - 5891196 T^{12} + 64163586 T^{13} + 112431256 T^{14} - 1077842606 T^{15} - 6853277211 T^{16} - 1077842606 p T^{17} + 112431256 p^{2} T^{18} + 64163586 p^{3} T^{19} - 5891196 p^{4} T^{20} - 9527850 p^{5} T^{21} - 2587772 p^{6} T^{22} - 106400 p^{7} T^{23} + 120487 p^{8} T^{24} + 29726 p^{9} T^{25} - 476 p^{10} T^{26} - 1244 p^{11} T^{27} + 94 p^{12} T^{28} + 202 p^{13} T^{29} + 68 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 6 T + 3 T^{2} - 121 T^{3} + 662 T^{4} - 1837 T^{5} + 26179 T^{6} - 56686 T^{7} + 132912 T^{8} - 2578311 T^{9} + 3943080 T^{10} - 26103102 T^{11} + 218083744 T^{12} - 32152744 T^{13} + 2348968386 T^{14} - 14609580633 T^{15} + 7008088210 T^{16} - 14609580633 p T^{17} + 2348968386 p^{2} T^{18} - 32152744 p^{3} T^{19} + 218083744 p^{4} T^{20} - 26103102 p^{5} T^{21} + 3943080 p^{6} T^{22} - 2578311 p^{7} T^{23} + 132912 p^{8} T^{24} - 56686 p^{9} T^{25} + 26179 p^{10} T^{26} - 1837 p^{11} T^{27} + 662 p^{12} T^{28} - 121 p^{13} T^{29} + 3 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 13 T + 137 T^{2} + 947 T^{3} + 6622 T^{4} + 38197 T^{5} + 234334 T^{6} + 52826 p T^{7} + 6404443 T^{8} + 52826 p^{2} T^{9} + 234334 p^{2} T^{10} + 38197 p^{3} T^{11} + 6622 p^{4} T^{12} + 947 p^{5} T^{13} + 137 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 42 T^{2} - 176 T^{3} + 2006 T^{4} - 5100 T^{5} - 43362 T^{6} + 138670 T^{7} + 3367263 T^{8} - 11111500 T^{9} - 37988878 T^{10} + 9893286 T^{11} + 1441529760 T^{12} - 11240167356 T^{13} + 6341917884 T^{14} - 3484089358 T^{15} + 1713908933797 T^{16} - 3484089358 p T^{17} + 6341917884 p^{2} T^{18} - 11240167356 p^{3} T^{19} + 1441529760 p^{4} T^{20} + 9893286 p^{5} T^{21} - 37988878 p^{6} T^{22} - 11111500 p^{7} T^{23} + 3367263 p^{8} T^{24} + 138670 p^{9} T^{25} - 43362 p^{10} T^{26} - 5100 p^{11} T^{27} + 2006 p^{12} T^{28} - 176 p^{13} T^{29} - 42 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 11 T - 37 T^{2} + 772 T^{3} + 419 T^{4} - 32668 T^{5} + 40371 T^{6} + 1096668 T^{7} - 4927174 T^{8} - 18760654 T^{9} + 184476589 T^{10} + 318565201 T^{11} - 5342767992 T^{12} - 4534140536 T^{13} + 120907103572 T^{14} - 66631402812 T^{15} - 1836907228027 T^{16} - 66631402812 p T^{17} + 120907103572 p^{2} T^{18} - 4534140536 p^{3} T^{19} - 5342767992 p^{4} T^{20} + 318565201 p^{5} T^{21} + 184476589 p^{6} T^{22} - 18760654 p^{7} T^{23} - 4927174 p^{8} T^{24} + 1096668 p^{9} T^{25} + 40371 p^{10} T^{26} - 32668 p^{11} T^{27} + 419 p^{12} T^{28} + 772 p^{13} T^{29} - 37 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 48 T^{2} - 188 T^{3} + 4632 T^{4} + 5932 T^{5} - 229328 T^{6} - 483478 T^{7} + 13069897 T^{8} + 20880328 T^{9} - 16050348 p T^{10} - 600364202 T^{11} + 717261096 p T^{12} + 25343587240 T^{13} - 1083064815692 T^{14} - 112347274666 T^{15} + 40178342879549 T^{16} - 112347274666 p T^{17} - 1083064815692 p^{2} T^{18} + 25343587240 p^{3} T^{19} + 717261096 p^{5} T^{20} - 600364202 p^{5} T^{21} - 16050348 p^{7} T^{22} + 20880328 p^{7} T^{23} + 13069897 p^{8} T^{24} - 483478 p^{9} T^{25} - 229328 p^{10} T^{26} + 5932 p^{11} T^{27} + 4632 p^{12} T^{28} - 188 p^{13} T^{29} - 48 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 + 7 T - 58 T^{2} - 14 p T^{3} + 3643 T^{4} + 62286 T^{5} - 96537 T^{6} - 3844860 T^{7} - 3184611 T^{8} + 197908882 T^{9} + 740381832 T^{10} - 8654446355 T^{11} - 52349132257 T^{12} + 273510670218 T^{13} + 3129227787882 T^{14} - 2933385562624 T^{15} - 140040425705494 T^{16} - 2933385562624 p T^{17} + 3129227787882 p^{2} T^{18} + 273510670218 p^{3} T^{19} - 52349132257 p^{4} T^{20} - 8654446355 p^{5} T^{21} + 740381832 p^{6} T^{22} + 197908882 p^{7} T^{23} - 3184611 p^{8} T^{24} - 3844860 p^{9} T^{25} - 96537 p^{10} T^{26} + 62286 p^{11} T^{27} + 3643 p^{12} T^{28} - 14 p^{14} T^{29} - 58 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 33 T + 680 T^{2} - 9588 T^{3} + 105494 T^{4} - 927683 T^{5} + 7000175 T^{6} - 47258994 T^{7} + 311555680 T^{8} - 47258994 p T^{9} + 7000175 p^{2} T^{10} - 927683 p^{3} T^{11} + 105494 p^{4} T^{12} - 9588 p^{5} T^{13} + 680 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - p T + 1135 T^{2} - 18484 T^{3} + 225279 T^{4} - 2149828 T^{5} + 16123005 T^{6} - 89652998 T^{7} + 271392932 T^{8} + 1083707442 T^{9} - 23240597565 T^{10} + 185635205355 T^{11} - 747694104594 T^{12} - 2338099668010 T^{13} + 76404630750274 T^{14} - 853969382577434 T^{15} + 6716389487023675 T^{16} - 853969382577434 p T^{17} + 76404630750274 p^{2} T^{18} - 2338099668010 p^{3} T^{19} - 747694104594 p^{4} T^{20} + 185635205355 p^{5} T^{21} - 23240597565 p^{6} T^{22} + 1083707442 p^{7} T^{23} + 271392932 p^{8} T^{24} - 89652998 p^{9} T^{25} + 16123005 p^{10} T^{26} - 2149828 p^{11} T^{27} + 225279 p^{12} T^{28} - 18484 p^{13} T^{29} + 1135 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 15 T + 99 T^{2} - 2 T^{3} - 1771 T^{4} - 60622 T^{5} + 850283 T^{6} - 4546726 T^{7} - 3904842 T^{8} + 46549128 T^{9} + 2699180241 T^{10} - 33072883425 T^{11} + 172114872162 T^{12} + 108861456380 T^{13} - 1596381326192 T^{14} - 66989421518744 T^{15} + 780553759965631 T^{16} - 66989421518744 p T^{17} - 1596381326192 p^{2} T^{18} + 108861456380 p^{3} T^{19} + 172114872162 p^{4} T^{20} - 33072883425 p^{5} T^{21} + 2699180241 p^{6} T^{22} + 46549128 p^{7} T^{23} - 3904842 p^{8} T^{24} - 4546726 p^{9} T^{25} + 850283 p^{10} T^{26} - 60622 p^{11} T^{27} - 1771 p^{12} T^{28} - 2 p^{13} T^{29} + 99 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 18 T + 102 T^{2} - 314 T^{3} - 2422 T^{4} - 652 T^{5} - 421044 T^{6} - 3978890 T^{7} + 32562089 T^{8} + 424829532 T^{9} + 268821002 T^{10} - 814258236 T^{11} + 170803394130 T^{12} + 412659411006 T^{13} - 9767101147792 T^{14} - 19739620190238 T^{15} + 513472595553643 T^{16} - 19739620190238 p T^{17} - 9767101147792 p^{2} T^{18} + 412659411006 p^{3} T^{19} + 170803394130 p^{4} T^{20} - 814258236 p^{5} T^{21} + 268821002 p^{6} T^{22} + 424829532 p^{7} T^{23} + 32562089 p^{8} T^{24} - 3978890 p^{9} T^{25} - 421044 p^{10} T^{26} - 652 p^{11} T^{27} - 2422 p^{12} T^{28} - 314 p^{13} T^{29} + 102 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 43 T + 631 T^{2} + 1043 T^{3} - 76541 T^{4} - 870354 T^{5} - 321552 T^{6} + 77676218 T^{7} + 697367805 T^{8} - 951348209 T^{9} - 65744596119 T^{10} - 434026521239 T^{11} + 1566918001511 T^{12} + 42356882685354 T^{13} + 208895293525664 T^{14} - 1278905976187216 T^{15} - 22481819741660704 T^{16} - 1278905976187216 p T^{17} + 208895293525664 p^{2} T^{18} + 42356882685354 p^{3} T^{19} + 1566918001511 p^{4} T^{20} - 434026521239 p^{5} T^{21} - 65744596119 p^{6} T^{22} - 951348209 p^{7} T^{23} + 697367805 p^{8} T^{24} + 77676218 p^{9} T^{25} - 321552 p^{10} T^{26} - 870354 p^{11} T^{27} - 76541 p^{12} T^{28} + 1043 p^{13} T^{29} + 631 p^{14} T^{30} + 43 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 25 T + 371 T^{2} + 3915 T^{3} + 40282 T^{4} + 413775 T^{5} + 4185913 T^{6} + 36826625 T^{7} + 309794950 T^{8} + 36826625 p T^{9} + 4185913 p^{2} T^{10} + 413775 p^{3} T^{11} + 40282 p^{4} T^{12} + 3915 p^{5} T^{13} + 371 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 5 T - 225 T^{2} - 326 T^{3} + 26883 T^{4} - 19676 T^{5} - 2059199 T^{6} + 2587514 T^{7} + 119962570 T^{8} - 178890472 T^{9} - 7135594085 T^{10} + 4487702027 T^{11} + 487953375170 T^{12} + 862571662220 T^{13} - 45552701007916 T^{14} - 657261530042 p T^{15} + 56685458959163 p T^{16} - 657261530042 p^{2} T^{17} - 45552701007916 p^{2} T^{18} + 862571662220 p^{3} T^{19} + 487953375170 p^{4} T^{20} + 4487702027 p^{5} T^{21} - 7135594085 p^{6} T^{22} - 178890472 p^{7} T^{23} + 119962570 p^{8} T^{24} + 2587514 p^{9} T^{25} - 2059199 p^{10} T^{26} - 19676 p^{11} T^{27} + 26883 p^{12} T^{28} - 326 p^{13} T^{29} - 225 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 13 T - 152 T^{2} - 2786 T^{3} + 5793 T^{4} + 320710 T^{5} + 1152337 T^{6} - 22109754 T^{7} - 224089803 T^{8} + 469226422 T^{9} + 19281800562 T^{10} + 75971609147 T^{11} - 764460397847 T^{12} - 9223830078456 T^{13} - 20175242241986 T^{14} + 328057369452680 T^{15} + 4051800332157358 T^{16} + 328057369452680 p T^{17} - 20175242241986 p^{2} T^{18} - 9223830078456 p^{3} T^{19} - 764460397847 p^{4} T^{20} + 75971609147 p^{5} T^{21} + 19281800562 p^{6} T^{22} + 469226422 p^{7} T^{23} - 224089803 p^{8} T^{24} - 22109754 p^{9} T^{25} + 1152337 p^{10} T^{26} + 320710 p^{11} T^{27} + 5793 p^{12} T^{28} - 2786 p^{13} T^{29} - 152 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 5 T - 199 T^{2} + 1410 T^{3} + 19931 T^{4} - 238730 T^{5} - 1366183 T^{6} + 34141930 T^{7} + 6002252 T^{8} - 3596834300 T^{9} + 11474805951 T^{10} + 281255263425 T^{11} - 1973419692228 T^{12} - 15996971644060 T^{13} + 235387305996362 T^{14} + 442739884387360 T^{15} - 20963546269508819 T^{16} + 442739884387360 p T^{17} + 235387305996362 p^{2} T^{18} - 15996971644060 p^{3} T^{19} - 1973419692228 p^{4} T^{20} + 281255263425 p^{5} T^{21} + 11474805951 p^{6} T^{22} - 3596834300 p^{7} T^{23} + 6002252 p^{8} T^{24} + 34141930 p^{9} T^{25} - 1366183 p^{10} T^{26} - 238730 p^{11} T^{27} + 19931 p^{12} T^{28} + 1410 p^{13} T^{29} - 199 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 7 T - 132 T^{2} - 158 T^{3} + 23355 T^{4} - 78112 T^{5} - 977327 T^{6} + 4150312 T^{7} + 132328847 T^{8} - 1481074294 T^{9} - 220822666 T^{10} + 83266041981 T^{11} - 21034222535 T^{12} - 10536666632260 T^{13} + 41800162777586 T^{14} + 240524340808204 T^{15} - 837905537584858 T^{16} + 240524340808204 p T^{17} + 41800162777586 p^{2} T^{18} - 10536666632260 p^{3} T^{19} - 21034222535 p^{4} T^{20} + 83266041981 p^{5} T^{21} - 220822666 p^{6} T^{22} - 1481074294 p^{7} T^{23} + 132328847 p^{8} T^{24} + 4150312 p^{9} T^{25} - 977327 p^{10} T^{26} - 78112 p^{11} T^{27} + 23355 p^{12} T^{28} - 158 p^{13} T^{29} - 132 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 14 T + 479 T^{2} + 6016 T^{3} + 111662 T^{4} + 1285044 T^{5} + 16706177 T^{6} + 171410446 T^{7} + 1754144066 T^{8} + 171410446 p T^{9} + 16706177 p^{2} T^{10} + 1285044 p^{3} T^{11} + 111662 p^{4} T^{12} + 6016 p^{5} T^{13} + 479 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 11 T - 51 T^{2} - 1602 T^{3} + 11311 T^{4} + 115470 T^{5} - 2068055 T^{6} - 19333306 T^{7} + 386122270 T^{8} + 2445863766 T^{9} - 26810985491 T^{10} - 144708305433 T^{11} + 4762678267332 T^{12} + 7983982139830 T^{13} - 400944477086378 T^{14} - 659877747523196 T^{15} + 47740066364638435 T^{16} - 659877747523196 p T^{17} - 400944477086378 p^{2} T^{18} + 7983982139830 p^{3} T^{19} + 4762678267332 p^{4} T^{20} - 144708305433 p^{5} T^{21} - 26810985491 p^{6} T^{22} + 2445863766 p^{7} T^{23} + 386122270 p^{8} T^{24} - 19333306 p^{9} T^{25} - 2068055 p^{10} T^{26} + 115470 p^{11} T^{27} + 11311 p^{12} T^{28} - 1602 p^{13} T^{29} - 51 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.10257744264926081730775369335, −3.09715625750707535685968477340, −2.97942796740581051516023386285, −2.92089932889007177937849030607, −2.76688186896465726775250943602, −2.71530751464004419306391298248, −2.66810702829439705351196737921, −2.60677995529098379544279184253, −2.50615823412926413257155394293, −2.38587261832254410321862113557, −2.18957557511263114733952925374, −2.16743719111378744912606280978, −2.15176540744749943920949493746, −2.13281082225622641374778857359, −2.12527627548866563137543671478, −1.56704064725194415016527443158, −1.50555652346388448444383172295, −1.34581752672268589147311146576, −1.31637325951306153208624672463, −1.21212218661377515893434319282, −1.08662420325137212404678302115, −0.66788481606334140123592530855, −0.58084908189244947081046703754, −0.53518069617203066337701031291, −0.44604735969682308488296490944, 0.44604735969682308488296490944, 0.53518069617203066337701031291, 0.58084908189244947081046703754, 0.66788481606334140123592530855, 1.08662420325137212404678302115, 1.21212218661377515893434319282, 1.31637325951306153208624672463, 1.34581752672268589147311146576, 1.50555652346388448444383172295, 1.56704064725194415016527443158, 2.12527627548866563137543671478, 2.13281082225622641374778857359, 2.15176540744749943920949493746, 2.16743719111378744912606280978, 2.18957557511263114733952925374, 2.38587261832254410321862113557, 2.50615823412926413257155394293, 2.60677995529098379544279184253, 2.66810702829439705351196737921, 2.71530751464004419306391298248, 2.76688186896465726775250943602, 2.92089932889007177937849030607, 2.97942796740581051516023386285, 3.09715625750707535685968477340, 3.10257744264926081730775369335
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.