Dirichlet series
| L(s) = 1 | − 14·2-s + 105·4-s − 2·5-s + 4·7-s − 560·8-s + 28·10-s − 2·11-s + 4·13-s − 56·14-s + 2.38e3·16-s − 9·17-s + 14·19-s − 210·20-s + 28·22-s − 30·23-s − 24·25-s − 56·26-s + 420·28-s − 6·29-s + 11·31-s − 8.56e3·32-s + 126·34-s − 8·35-s + 13·37-s − 196·38-s + 1.12e3·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 9.89·2-s + 52.5·4-s − 0.894·5-s + 1.51·7-s − 197.·8-s + 8.85·10-s − 0.603·11-s + 1.10·13-s − 14.9·14-s + 595·16-s − 2.18·17-s + 3.21·19-s − 46.9·20-s + 5.96·22-s − 6.25·23-s − 4.79·25-s − 10.9·26-s + 79.3·28-s − 1.11·29-s + 1.97·31-s − 1.51e3·32-s + 21.6·34-s − 1.35·35-s + 2.13·37-s − 31.7·38-s + 177.·40-s + 0.312·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{42} \cdot 149^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{42} \cdot 149^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 3^{42} \cdot 149^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.04173\times 10^{25}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 3^{42} \cdot 149^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2003985784\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2003985784\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{14} \) |
| 3 | \( 1 \) | |
| 149 | \( ( 1 + T )^{14} \) | |
| good | 5 | \( 1 + 2 T + 28 T^{2} + 12 p T^{3} + 87 p T^{4} + 932 T^{5} + 4641 T^{6} + 9874 T^{7} + 7544 p T^{8} + 15896 p T^{9} + 49854 p T^{10} + 522542 T^{11} + 1421451 T^{12} + 2950348 T^{13} + 7372677 T^{14} + 2950348 p T^{15} + 1421451 p^{2} T^{16} + 522542 p^{3} T^{17} + 49854 p^{5} T^{18} + 15896 p^{6} T^{19} + 7544 p^{7} T^{20} + 9874 p^{7} T^{21} + 4641 p^{8} T^{22} + 932 p^{9} T^{23} + 87 p^{11} T^{24} + 12 p^{12} T^{25} + 28 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) |
| 7 | \( 1 - 4 T + 41 T^{2} - 141 T^{3} + 886 T^{4} - 386 p T^{5} + 13194 T^{6} - 36749 T^{7} + 152933 T^{8} - 396432 T^{9} + 30229 p^{2} T^{10} - 74066 p^{2} T^{11} + 12465202 T^{12} - 28914434 T^{13} + 92675285 T^{14} - 28914434 p T^{15} + 12465202 p^{2} T^{16} - 74066 p^{5} T^{17} + 30229 p^{6} T^{18} - 396432 p^{5} T^{19} + 152933 p^{6} T^{20} - 36749 p^{7} T^{21} + 13194 p^{8} T^{22} - 386 p^{10} T^{23} + 886 p^{10} T^{24} - 141 p^{11} T^{25} + 41 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 + 2 T + 61 T^{2} + 136 T^{3} + 2029 T^{4} + 4541 T^{5} + 47979 T^{6} + 9351 p T^{7} + 894464 T^{8} + 1797910 T^{9} + 1260939 p T^{10} + 26073088 T^{11} + 184810378 T^{12} + 326416620 T^{13} + 2161407870 T^{14} + 326416620 p T^{15} + 184810378 p^{2} T^{16} + 26073088 p^{3} T^{17} + 1260939 p^{5} T^{18} + 1797910 p^{5} T^{19} + 894464 p^{6} T^{20} + 9351 p^{8} T^{21} + 47979 p^{8} T^{22} + 4541 p^{9} T^{23} + 2029 p^{10} T^{24} + 136 p^{11} T^{25} + 61 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( 1 - 4 T + 55 T^{2} - 164 T^{3} + 1841 T^{4} - 5834 T^{5} + 51222 T^{6} - 152916 T^{7} + 1105419 T^{8} - 3168570 T^{9} + 20685432 T^{10} - 57342178 T^{11} + 333039468 T^{12} - 858558930 T^{13} + 4596254473 T^{14} - 858558930 p T^{15} + 333039468 p^{2} T^{16} - 57342178 p^{3} T^{17} + 20685432 p^{4} T^{18} - 3168570 p^{5} T^{19} + 1105419 p^{6} T^{20} - 152916 p^{7} T^{21} + 51222 p^{8} T^{22} - 5834 p^{9} T^{23} + 1841 p^{10} T^{24} - 164 p^{11} T^{25} + 55 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 9 T + 116 T^{2} + 734 T^{3} + 5894 T^{4} + 33374 T^{5} + 217597 T^{6} + 1158998 T^{7} + 6467809 T^{8} + 31916274 T^{9} + 160080026 T^{10} + 734570171 T^{11} + 3403169256 T^{12} + 14440220552 T^{13} + 62194715466 T^{14} + 14440220552 p T^{15} + 3403169256 p^{2} T^{16} + 734570171 p^{3} T^{17} + 160080026 p^{4} T^{18} + 31916274 p^{5} T^{19} + 6467809 p^{6} T^{20} + 1158998 p^{7} T^{21} + 217597 p^{8} T^{22} + 33374 p^{9} T^{23} + 5894 p^{10} T^{24} + 734 p^{11} T^{25} + 116 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - 14 T + 169 T^{2} - 1403 T^{3} + 11488 T^{4} - 78175 T^{5} + 522320 T^{6} - 3035681 T^{7} + 17497829 T^{8} - 90997285 T^{9} + 473230201 T^{10} - 2251728068 T^{11} + 10805853754 T^{12} - 48136741502 T^{13} + 11453571060 p T^{14} - 48136741502 p T^{15} + 10805853754 p^{2} T^{16} - 2251728068 p^{3} T^{17} + 473230201 p^{4} T^{18} - 90997285 p^{5} T^{19} + 17497829 p^{6} T^{20} - 3035681 p^{7} T^{21} + 522320 p^{8} T^{22} - 78175 p^{9} T^{23} + 11488 p^{10} T^{24} - 1403 p^{11} T^{25} + 169 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 + 30 T + 584 T^{2} + 8287 T^{3} + 96925 T^{4} + 963908 T^{5} + 369441 p T^{6} + 67352281 T^{7} + 489924256 T^{8} + 3292775643 T^{9} + 20674520004 T^{10} + 121569332525 T^{11} + 673577436637 T^{12} + 3516529515759 T^{13} + 17358630554967 T^{14} + 3516529515759 p T^{15} + 673577436637 p^{2} T^{16} + 121569332525 p^{3} T^{17} + 20674520004 p^{4} T^{18} + 3292775643 p^{5} T^{19} + 489924256 p^{6} T^{20} + 67352281 p^{7} T^{21} + 369441 p^{9} T^{22} + 963908 p^{9} T^{23} + 96925 p^{10} T^{24} + 8287 p^{11} T^{25} + 584 p^{12} T^{26} + 30 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 + 6 T + 181 T^{2} + 1020 T^{3} + 14668 T^{4} + 73882 T^{5} + 710119 T^{6} + 2807471 T^{7} + 22480794 T^{8} + 45950648 T^{9} + 431717642 T^{10} - 836466161 T^{11} + 2151226620 T^{12} - 73261063515 T^{13} - 88428928137 T^{14} - 73261063515 p T^{15} + 2151226620 p^{2} T^{16} - 836466161 p^{3} T^{17} + 431717642 p^{4} T^{18} + 45950648 p^{5} T^{19} + 22480794 p^{6} T^{20} + 2807471 p^{7} T^{21} + 710119 p^{8} T^{22} + 73882 p^{9} T^{23} + 14668 p^{10} T^{24} + 1020 p^{11} T^{25} + 181 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 31 | \( 1 - 11 T + 293 T^{2} - 2401 T^{3} + 37974 T^{4} - 249434 T^{5} + 3040390 T^{6} - 16528313 T^{7} + 172961258 T^{8} - 792069810 T^{9} + 7626311273 T^{10} - 30088144642 T^{11} + 280194802802 T^{12} - 994265734303 T^{13} + 9112906214803 T^{14} - 994265734303 p T^{15} + 280194802802 p^{2} T^{16} - 30088144642 p^{3} T^{17} + 7626311273 p^{4} T^{18} - 792069810 p^{5} T^{19} + 172961258 p^{6} T^{20} - 16528313 p^{7} T^{21} + 3040390 p^{8} T^{22} - 249434 p^{9} T^{23} + 37974 p^{10} T^{24} - 2401 p^{11} T^{25} + 293 p^{12} T^{26} - 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 - 13 T + 281 T^{2} - 3111 T^{3} + 40264 T^{4} - 367589 T^{5} + 3638571 T^{6} - 28152253 T^{7} + 231032713 T^{8} - 1552251607 T^{9} + 11156663741 T^{10} - 67286788305 T^{11} + 447921138414 T^{12} - 2557243358182 T^{13} + 16700844235102 T^{14} - 2557243358182 p T^{15} + 447921138414 p^{2} T^{16} - 67286788305 p^{3} T^{17} + 11156663741 p^{4} T^{18} - 1552251607 p^{5} T^{19} + 231032713 p^{6} T^{20} - 28152253 p^{7} T^{21} + 3638571 p^{8} T^{22} - 367589 p^{9} T^{23} + 40264 p^{10} T^{24} - 3111 p^{11} T^{25} + 281 p^{12} T^{26} - 13 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 - 2 T + 310 T^{2} - 1067 T^{3} + 48772 T^{4} - 213511 T^{5} + 5293181 T^{6} - 25200647 T^{7} + 441132414 T^{8} - 2099126214 T^{9} + 29405656103 T^{10} - 134309132098 T^{11} + 1599620490390 T^{12} - 6841742496901 T^{13} + 71944474959847 T^{14} - 6841742496901 p T^{15} + 1599620490390 p^{2} T^{16} - 134309132098 p^{3} T^{17} + 29405656103 p^{4} T^{18} - 2099126214 p^{5} T^{19} + 441132414 p^{6} T^{20} - 25200647 p^{7} T^{21} + 5293181 p^{8} T^{22} - 213511 p^{9} T^{23} + 48772 p^{10} T^{24} - 1067 p^{11} T^{25} + 310 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 - 12 T + 313 T^{2} - 2941 T^{3} + 1078 p T^{4} - 364249 T^{5} + 4500456 T^{6} - 30371096 T^{7} + 323732139 T^{8} - 1907284425 T^{9} + 18605092085 T^{10} - 97945613665 T^{11} + 915170309538 T^{12} - 4461374153680 T^{13} + 40823650298033 T^{14} - 4461374153680 p T^{15} + 915170309538 p^{2} T^{16} - 97945613665 p^{3} T^{17} + 18605092085 p^{4} T^{18} - 1907284425 p^{5} T^{19} + 323732139 p^{6} T^{20} - 30371096 p^{7} T^{21} + 4500456 p^{8} T^{22} - 364249 p^{9} T^{23} + 1078 p^{11} T^{24} - 2941 p^{11} T^{25} + 313 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 + 21 T + 642 T^{2} + 9579 T^{3} + 173282 T^{4} + 2058501 T^{5} + 28081277 T^{6} + 281260080 T^{7} + 3164831869 T^{8} + 27593886580 T^{9} + 267463992132 T^{10} + 2066070951610 T^{11} + 17645268146416 T^{12} + 121770185396311 T^{13} + 926763873038362 T^{14} + 121770185396311 p T^{15} + 17645268146416 p^{2} T^{16} + 2066070951610 p^{3} T^{17} + 267463992132 p^{4} T^{18} + 27593886580 p^{5} T^{19} + 3164831869 p^{6} T^{20} + 281260080 p^{7} T^{21} + 28081277 p^{8} T^{22} + 2058501 p^{9} T^{23} + 173282 p^{10} T^{24} + 9579 p^{11} T^{25} + 642 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 + 22 T + 600 T^{2} + 8891 T^{3} + 149025 T^{4} + 1767428 T^{5} + 23162405 T^{6} + 236054523 T^{7} + 2622232050 T^{8} + 23676784217 T^{9} + 231170051836 T^{10} + 1878471829095 T^{11} + 16435898992365 T^{12} + 121114546059821 T^{13} + 959063276750599 T^{14} + 121114546059821 p T^{15} + 16435898992365 p^{2} T^{16} + 1878471829095 p^{3} T^{17} + 231170051836 p^{4} T^{18} + 23676784217 p^{5} T^{19} + 2622232050 p^{6} T^{20} + 236054523 p^{7} T^{21} + 23162405 p^{8} T^{22} + 1767428 p^{9} T^{23} + 149025 p^{10} T^{24} + 8891 p^{11} T^{25} + 600 p^{12} T^{26} + 22 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 14 T + 581 T^{2} + 7377 T^{3} + 164702 T^{4} + 1883247 T^{5} + 30220988 T^{6} + 310284265 T^{7} + 4012968527 T^{8} + 37055439705 T^{9} + 408802361169 T^{10} + 3406965871622 T^{11} + 33055223215074 T^{12} + 249122296374202 T^{13} + 2161859424644700 T^{14} + 249122296374202 p T^{15} + 33055223215074 p^{2} T^{16} + 3406965871622 p^{3} T^{17} + 408802361169 p^{4} T^{18} + 37055439705 p^{5} T^{19} + 4012968527 p^{6} T^{20} + 310284265 p^{7} T^{21} + 30220988 p^{8} T^{22} + 1883247 p^{9} T^{23} + 164702 p^{10} T^{24} + 7377 p^{11} T^{25} + 581 p^{12} T^{26} + 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 31 T + 799 T^{2} - 14275 T^{3} + 232164 T^{4} - 3132074 T^{5} + 39576049 T^{6} - 440162687 T^{7} + 4672193963 T^{8} - 738173930 p T^{9} + 421062079995 T^{10} - 59847784184 p T^{11} + 31205726643312 T^{12} - 251150250443459 T^{13} + 2015313166590698 T^{14} - 251150250443459 p T^{15} + 31205726643312 p^{2} T^{16} - 59847784184 p^{4} T^{17} + 421062079995 p^{4} T^{18} - 738173930 p^{6} T^{19} + 4672193963 p^{6} T^{20} - 440162687 p^{7} T^{21} + 39576049 p^{8} T^{22} - 3132074 p^{9} T^{23} + 232164 p^{10} T^{24} - 14275 p^{11} T^{25} + 799 p^{12} T^{26} - 31 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 24 T + 645 T^{2} - 10579 T^{3} + 173398 T^{4} - 2229369 T^{5} + 27927266 T^{6} - 300633461 T^{7} + 3134279069 T^{8} - 29546445593 T^{9} + 270345477209 T^{10} - 2329683918478 T^{11} + 19689782826956 T^{12} - 162614886738420 T^{13} + 1334836555870912 T^{14} - 162614886738420 p T^{15} + 19689782826956 p^{2} T^{16} - 2329683918478 p^{3} T^{17} + 270345477209 p^{4} T^{18} - 29546445593 p^{5} T^{19} + 3134279069 p^{6} T^{20} - 300633461 p^{7} T^{21} + 27927266 p^{8} T^{22} - 2229369 p^{9} T^{23} + 173398 p^{10} T^{24} - 10579 p^{11} T^{25} + 645 p^{12} T^{26} - 24 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 28 T + 793 T^{2} + 15570 T^{3} + 289339 T^{4} + 4536466 T^{5} + 66735100 T^{6} + 884178035 T^{7} + 11020890785 T^{8} + 127178786894 T^{9} + 1385246841317 T^{10} + 14162754693270 T^{11} + 137026427434000 T^{12} + 1252415837062688 T^{13} + 10846731641608785 T^{14} + 1252415837062688 p T^{15} + 137026427434000 p^{2} T^{16} + 14162754693270 p^{3} T^{17} + 1385246841317 p^{4} T^{18} + 127178786894 p^{5} T^{19} + 11020890785 p^{6} T^{20} + 884178035 p^{7} T^{21} + 66735100 p^{8} T^{22} + 4536466 p^{9} T^{23} + 289339 p^{10} T^{24} + 15570 p^{11} T^{25} + 793 p^{12} T^{26} + 28 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( 1 - 24 T + 559 T^{2} - 10259 T^{3} + 166212 T^{4} - 2371579 T^{5} + 426638 p T^{6} - 379413212 T^{7} + 4292139247 T^{8} - 45848681295 T^{9} + 466387972197 T^{10} - 4520665874695 T^{11} + 42080295557530 T^{12} - 5174864995182 p T^{13} + 3289457070452829 T^{14} - 5174864995182 p^{2} T^{15} + 42080295557530 p^{2} T^{16} - 4520665874695 p^{3} T^{17} + 466387972197 p^{4} T^{18} - 45848681295 p^{5} T^{19} + 4292139247 p^{6} T^{20} - 379413212 p^{7} T^{21} + 426638 p^{9} T^{22} - 2371579 p^{9} T^{23} + 166212 p^{10} T^{24} - 10259 p^{11} T^{25} + 559 p^{12} T^{26} - 24 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 - 65 T + 2695 T^{2} - 81273 T^{3} + 1994435 T^{4} - 41257851 T^{5} + 745630895 T^{6} - 11965350812 T^{7} + 173460480090 T^{8} - 2292541231594 T^{9} + 353190319937 p T^{10} - 314417424571610 T^{11} + 3300424334374354 T^{12} - 32356362354243161 T^{13} + 297156786296087190 T^{14} - 32356362354243161 p T^{15} + 3300424334374354 p^{2} T^{16} - 314417424571610 p^{3} T^{17} + 353190319937 p^{5} T^{18} - 2292541231594 p^{5} T^{19} + 173460480090 p^{6} T^{20} - 11965350812 p^{7} T^{21} + 745630895 p^{8} T^{22} - 41257851 p^{9} T^{23} + 1994435 p^{10} T^{24} - 81273 p^{11} T^{25} + 2695 p^{12} T^{26} - 65 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 15 T + 316 T^{2} - 3384 T^{3} + 55867 T^{4} - 540953 T^{5} + 8059014 T^{6} - 79440974 T^{7} + 1044683618 T^{8} - 9433281291 T^{9} + 111583188986 T^{10} - 972398968994 T^{11} + 10784971542554 T^{12} - 91456205304469 T^{13} + 961321029078984 T^{14} - 91456205304469 p T^{15} + 10784971542554 p^{2} T^{16} - 972398968994 p^{3} T^{17} + 111583188986 p^{4} T^{18} - 9433281291 p^{5} T^{19} + 1044683618 p^{6} T^{20} - 79440974 p^{7} T^{21} + 8059014 p^{8} T^{22} - 540953 p^{9} T^{23} + 55867 p^{10} T^{24} - 3384 p^{11} T^{25} + 316 p^{12} T^{26} - 15 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 - 11 T + 637 T^{2} - 4666 T^{3} + 194340 T^{4} - 988871 T^{5} + 40302057 T^{6} - 140952095 T^{7} + 6481748951 T^{8} - 15564145255 T^{9} + 861119774688 T^{10} - 16647969839 p T^{11} + 96912973093927 T^{12} - 132503710750456 T^{13} + 9321434523205205 T^{14} - 132503710750456 p T^{15} + 96912973093927 p^{2} T^{16} - 16647969839 p^{4} T^{17} + 861119774688 p^{4} T^{18} - 15564145255 p^{5} T^{19} + 6481748951 p^{6} T^{20} - 140952095 p^{7} T^{21} + 40302057 p^{8} T^{22} - 988871 p^{9} T^{23} + 194340 p^{10} T^{24} - 4666 p^{11} T^{25} + 637 p^{12} T^{26} - 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 - 23 T + 725 T^{2} - 11408 T^{3} + 224098 T^{4} - 2793230 T^{5} + 41901368 T^{6} - 415274078 T^{7} + 5095591937 T^{8} - 38237772108 T^{9} + 402889459893 T^{10} - 1768490206825 T^{11} + 18928181782076 T^{12} + 18318177648168 T^{13} + 831994608674812 T^{14} + 18318177648168 p T^{15} + 18928181782076 p^{2} T^{16} - 1768490206825 p^{3} T^{17} + 402889459893 p^{4} T^{18} - 38237772108 p^{5} T^{19} + 5095591937 p^{6} T^{20} - 415274078 p^{7} T^{21} + 41901368 p^{8} T^{22} - 2793230 p^{9} T^{23} + 224098 p^{10} T^{24} - 11408 p^{11} T^{25} + 725 p^{12} T^{26} - 23 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.80800580102500366421074769971, −1.80698255445826888546928782438, −1.77143161716908533344263272542, −1.70532991504670520652445565871, −1.64104143901890803086609887527, −1.58373704487606464610129435670, −1.57510394837898366797643062879, −1.55319751809146881595895460211, −1.54003626322720291778468439485, −1.50474919222535233892914973579, −1.40075286140097712648803990087, −1.04035966786327686969974188151, −0.933543217018592914544018687801, −0.932758106612295087073151752979, −0.835351618716706655164450746929, −0.69515067842685968119980432254, −0.68102466368486236586509160048, −0.64489097421563880820321834632, −0.60854695244027595594082024578, −0.56671807931999729572921487893, −0.42583722552219482862191398947, −0.38042540909278742453760484987, −0.28356569627392562670479817900, −0.24423216496667796374922099052, −0.15479356035186714925286213408, 0.15479356035186714925286213408, 0.24423216496667796374922099052, 0.28356569627392562670479817900, 0.38042540909278742453760484987, 0.42583722552219482862191398947, 0.56671807931999729572921487893, 0.60854695244027595594082024578, 0.64489097421563880820321834632, 0.68102466368486236586509160048, 0.69515067842685968119980432254, 0.835351618716706655164450746929, 0.932758106612295087073151752979, 0.933543217018592914544018687801, 1.04035966786327686969974188151, 1.40075286140097712648803990087, 1.50474919222535233892914973579, 1.54003626322720291778468439485, 1.55319751809146881595895460211, 1.57510394837898366797643062879, 1.58373704487606464610129435670, 1.64104143901890803086609887527, 1.70532991504670520652445565871, 1.77143161716908533344263272542, 1.80698255445826888546928782438, 1.80800580102500366421074769971
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.