Dirichlet series
| L(s) = 1 | + 4·2-s − 4·3-s + 6·4-s − 16·5-s − 16·6-s − 7·7-s + 4·8-s + 6·9-s − 64·10-s + 4·11-s − 24·12-s + 2·13-s − 28·14-s + 64·15-s + 16-s + 5·17-s + 24·18-s − 2·19-s − 96·20-s + 28·21-s + 16·22-s − 4·23-s − 16·24-s + 136·25-s + 8·26-s − 4·27-s − 42·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 3·4-s − 7.15·5-s − 6.53·6-s − 2.64·7-s + 1.41·8-s + 2·9-s − 20.2·10-s + 1.20·11-s − 6.92·12-s + 0.554·13-s − 7.48·14-s + 16.5·15-s + 1/4·16-s + 1.21·17-s + 5.65·18-s − 0.458·19-s − 21.4·20-s + 6.11·21-s + 3.41·22-s − 0.834·23-s − 3.26·24-s + 27.1·25-s + 1.56·26-s − 0.769·27-s − 7.93·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.55383\times 10^{13}\) |
| Root analytic conductor: | \(2.72508\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.001152779368\) |
| \(L(\frac12)\) | \(\approx\) | \(0.001152779368\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 31 | \( 1 + T - 55 T^{2} - 50 T^{3} + 1620 T^{4} + 5123 T^{5} - 49792 T^{6} - 127490 T^{7} + 1742455 T^{8} - 127490 p T^{9} - 49792 p^{2} T^{10} + 5123 p^{3} T^{11} + 1620 p^{4} T^{12} - 50 p^{5} T^{13} - 55 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 + p T + 3 p T^{2} + 36 T^{3} + 18 p T^{4} + 941 T^{5} + 3539 T^{6} + 7549 T^{7} + 2321 p T^{8} + 71356 T^{9} + 294485 T^{10} + 751903 T^{11} + 1675042 T^{12} + 716316 p T^{13} + 2347992 p T^{14} + 6284492 p T^{15} + 106569619 T^{16} + 6284492 p^{2} T^{17} + 2347992 p^{3} T^{18} + 716316 p^{4} T^{19} + 1675042 p^{4} T^{20} + 751903 p^{5} T^{21} + 294485 p^{6} T^{22} + 71356 p^{7} T^{23} + 2321 p^{9} T^{24} + 7549 p^{9} T^{25} + 3539 p^{10} T^{26} + 941 p^{11} T^{27} + 18 p^{13} T^{28} + 36 p^{13} T^{29} + 3 p^{15} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 11 | \( 1 - 4 T - 3 T^{2} - 54 T^{3} + 170 T^{4} + 808 T^{5} + 673 T^{6} - 247 T^{7} - 42975 T^{8} - 63029 T^{9} + 6744 p T^{10} + 61318 p T^{11} + 3846160 T^{12} - 1791863 T^{13} + 5829645 T^{14} - 80465069 T^{15} - 135871254 T^{16} - 80465069 p T^{17} + 5829645 p^{2} T^{18} - 1791863 p^{3} T^{19} + 3846160 p^{4} T^{20} + 61318 p^{6} T^{21} + 6744 p^{7} T^{22} - 63029 p^{7} T^{23} - 42975 p^{8} T^{24} - 247 p^{9} T^{25} + 673 p^{10} T^{26} + 808 p^{11} T^{27} + 170 p^{12} T^{28} - 54 p^{13} T^{29} - 3 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 2 T - 32 T^{2} + 85 T^{3} + 475 T^{4} + 144 T^{5} - 8868 T^{6} - 33073 T^{7} + 221335 T^{8} + 396195 T^{9} - 1942429 T^{10} - 3902862 T^{11} - 11089192 T^{12} + 125612900 T^{13} + 165951870 T^{14} - 1115029719 T^{15} + 619125048 T^{16} - 1115029719 p T^{17} + 165951870 p^{2} T^{18} + 125612900 p^{3} T^{19} - 11089192 p^{4} T^{20} - 3902862 p^{5} T^{21} - 1942429 p^{6} T^{22} + 396195 p^{7} T^{23} + 221335 p^{8} T^{24} - 33073 p^{9} T^{25} - 8868 p^{10} T^{26} + 144 p^{11} T^{27} + 475 p^{12} T^{28} + 85 p^{13} T^{29} - 32 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 5 T - p T^{2} + 117 T^{3} - 78 T^{4} - 4349 T^{5} + 21741 T^{6} + 52165 T^{7} - 401391 T^{8} + 589037 T^{9} + 6760607 T^{10} - 44700127 T^{11} - 43853958 T^{12} + 570705794 T^{13} - 1648903442 T^{14} - 3181003980 T^{15} + 61573206235 T^{16} - 3181003980 p T^{17} - 1648903442 p^{2} T^{18} + 570705794 p^{3} T^{19} - 43853958 p^{4} T^{20} - 44700127 p^{5} T^{21} + 6760607 p^{6} T^{22} + 589037 p^{7} T^{23} - 401391 p^{8} T^{24} + 52165 p^{9} T^{25} + 21741 p^{10} T^{26} - 4349 p^{11} T^{27} - 78 p^{12} T^{28} + 117 p^{13} T^{29} - p^{15} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 2 T - 3 p T^{2} - 104 T^{3} + 1613 T^{4} - 788 T^{5} - 40522 T^{6} + 131820 T^{7} + 1026886 T^{8} - 3443002 T^{9} - 14501021 T^{10} + 66441302 T^{11} + 14537836 T^{12} - 1425152082 T^{13} + 1919499764 T^{14} + 14005679392 T^{15} - 28241219869 T^{16} + 14005679392 p T^{17} + 1919499764 p^{2} T^{18} - 1425152082 p^{3} T^{19} + 14537836 p^{4} T^{20} + 66441302 p^{5} T^{21} - 14501021 p^{6} T^{22} - 3443002 p^{7} T^{23} + 1026886 p^{8} T^{24} + 131820 p^{9} T^{25} - 40522 p^{10} T^{26} - 788 p^{11} T^{27} + 1613 p^{12} T^{28} - 104 p^{13} T^{29} - 3 p^{15} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 4 T - 91 T^{2} - 321 T^{3} + 3688 T^{4} + 15551 T^{5} - 59230 T^{6} - 582117 T^{7} - 653313 T^{8} + 19023023 T^{9} + 61480485 T^{10} - 478486734 T^{11} - 2058403222 T^{12} + 8513750864 T^{13} + 60166778338 T^{14} - 70325300504 T^{15} - 1518443132555 T^{16} - 70325300504 p T^{17} + 60166778338 p^{2} T^{18} + 8513750864 p^{3} T^{19} - 2058403222 p^{4} T^{20} - 478486734 p^{5} T^{21} + 61480485 p^{6} T^{22} + 19023023 p^{7} T^{23} - 653313 p^{8} T^{24} - 582117 p^{9} T^{25} - 59230 p^{10} T^{26} + 15551 p^{11} T^{27} + 3688 p^{12} T^{28} - 321 p^{13} T^{29} - 91 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 5 T - 90 T^{2} + 253 T^{3} + 4625 T^{4} - 9873 T^{5} - 3754 p T^{6} + 558635 T^{7} + 485456 T^{8} - 36171623 T^{9} + 65241644 T^{10} + 1280605273 T^{11} - 3131271334 T^{12} - 27731175618 T^{13} + 164646624844 T^{14} + 237208671818 T^{15} - 5840728889199 T^{16} + 237208671818 p T^{17} + 164646624844 p^{2} T^{18} - 27731175618 p^{3} T^{19} - 3131271334 p^{4} T^{20} + 1280605273 p^{5} T^{21} + 65241644 p^{6} T^{22} - 36171623 p^{7} T^{23} + 485456 p^{8} T^{24} + 558635 p^{9} T^{25} - 3754 p^{11} T^{26} - 9873 p^{11} T^{27} + 4625 p^{12} T^{28} + 253 p^{13} T^{29} - 90 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 10 T + 178 T^{2} + 1268 T^{3} + 12461 T^{4} + 69358 T^{5} + 493598 T^{6} + 2427584 T^{7} + 16468759 T^{8} + 2427584 p T^{9} + 493598 p^{2} T^{10} + 69358 p^{3} T^{11} + 12461 p^{4} T^{12} + 1268 p^{5} T^{13} + 178 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 + 17 T + 97 T^{2} - 140 T^{3} - 5126 T^{4} - 173 p T^{5} + 350235 T^{6} + 2797559 T^{7} + 1508479 T^{8} - 102557662 T^{9} - 627481465 T^{10} + 105589435 T^{11} + 18566319432 T^{12} + 64622225638 T^{13} - 301044527788 T^{14} - 4712665734834 T^{15} - 36029402124689 T^{16} - 4712665734834 p T^{17} - 301044527788 p^{2} T^{18} + 64622225638 p^{3} T^{19} + 18566319432 p^{4} T^{20} + 105589435 p^{5} T^{21} - 627481465 p^{6} T^{22} - 102557662 p^{7} T^{23} + 1508479 p^{8} T^{24} + 2797559 p^{9} T^{25} + 350235 p^{10} T^{26} - 173 p^{12} T^{27} - 5126 p^{12} T^{28} - 140 p^{13} T^{29} + 97 p^{14} T^{30} + 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 4 T - 192 T^{2} - 1050 T^{3} + 12660 T^{4} + 1844 p T^{5} - 118552 T^{6} + 310726 T^{7} - 16347685 T^{8} - 311449595 T^{9} - 117808468 T^{10} + 10105899058 T^{11} + 39173780611 T^{12} + 334286193795 T^{13} + 1726804835745 T^{14} - 15557427188974 T^{15} - 205862902762616 T^{16} - 15557427188974 p T^{17} + 1726804835745 p^{2} T^{18} + 334286193795 p^{3} T^{19} + 39173780611 p^{4} T^{20} + 10105899058 p^{5} T^{21} - 117808468 p^{6} T^{22} - 311449595 p^{7} T^{23} - 16347685 p^{8} T^{24} + 310726 p^{9} T^{25} - 118552 p^{10} T^{26} + 1844 p^{12} T^{27} + 12660 p^{12} T^{28} - 1050 p^{13} T^{29} - 192 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 2 T - 114 T^{2} + 462 T^{3} + 2458 T^{4} - 23488 T^{5} + 352230 T^{6} - 1566326 T^{7} - 21757843 T^{8} + 162442451 T^{9} - 302621900 T^{10} + 1452957778 T^{11} + 53371954223 T^{12} - 675639116707 T^{13} + 7322083811 p T^{14} + 20464654342628 T^{15} - 131150172814280 T^{16} + 20464654342628 p T^{17} + 7322083811 p^{3} T^{18} - 675639116707 p^{3} T^{19} + 53371954223 p^{4} T^{20} + 1452957778 p^{5} T^{21} - 302621900 p^{6} T^{22} + 162442451 p^{7} T^{23} - 21757843 p^{8} T^{24} - 1566326 p^{9} T^{25} + 352230 p^{10} T^{26} - 23488 p^{11} T^{27} + 2458 p^{12} T^{28} + 462 p^{13} T^{29} - 114 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 9 T - 197 T^{2} + 1585 T^{3} + 21340 T^{4} - 134687 T^{5} - 1636537 T^{6} + 8146039 T^{7} + 79074605 T^{8} - 387441825 T^{9} - 877237823 T^{10} + 14080839417 T^{11} - 233186144144 T^{12} - 360899754460 T^{13} + 25717302015820 T^{14} + 4640903085204 T^{15} - 1639783640239701 T^{16} + 4640903085204 p T^{17} + 25717302015820 p^{2} T^{18} - 360899754460 p^{3} T^{19} - 233186144144 p^{4} T^{20} + 14080839417 p^{5} T^{21} - 877237823 p^{6} T^{22} - 387441825 p^{7} T^{23} + 79074605 p^{8} T^{24} + 8146039 p^{9} T^{25} - 1636537 p^{10} T^{26} - 134687 p^{11} T^{27} + 21340 p^{12} T^{28} + 1585 p^{13} T^{29} - 197 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 3 T - 222 T^{2} - 1268 T^{3} + 21125 T^{4} + 202864 T^{5} - 641368 T^{6} - 14780424 T^{7} - 69947300 T^{8} + 166824588 T^{9} + 8908745816 T^{10} + 66564926521 T^{11} - 338435358500 T^{12} - 103225453856 p T^{13} - 10818592018550 T^{14} + 174540100849772 T^{15} + 1637097984762411 T^{16} + 174540100849772 p T^{17} - 10818592018550 p^{2} T^{18} - 103225453856 p^{4} T^{19} - 338435358500 p^{4} T^{20} + 66564926521 p^{5} T^{21} + 8908745816 p^{6} T^{22} + 166824588 p^{7} T^{23} - 69947300 p^{8} T^{24} - 14780424 p^{9} T^{25} - 641368 p^{10} T^{26} + 202864 p^{11} T^{27} + 21125 p^{12} T^{28} - 1268 p^{13} T^{29} - 222 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 35 T + 818 T^{2} - 13540 T^{3} + 184878 T^{4} - 2099485 T^{5} + 21125671 T^{6} - 188371500 T^{7} + 1545475420 T^{8} - 188371500 p T^{9} + 21125671 p^{2} T^{10} - 2099485 p^{3} T^{11} + 184878 p^{4} T^{12} - 13540 p^{5} T^{13} + 818 p^{6} T^{14} - 35 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 33 T + 731 T^{2} + 11028 T^{3} + 140012 T^{4} + 1505616 T^{5} + 15273270 T^{6} + 140650929 T^{7} + 1218557692 T^{8} + 140650929 p T^{9} + 15273270 p^{2} T^{10} + 1505616 p^{3} T^{11} + 140012 p^{4} T^{12} + 11028 p^{5} T^{13} + 731 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 25 T + 271 T^{2} + 1875 T^{3} + 13250 T^{4} + 237965 T^{5} + 3923885 T^{6} + 38046565 T^{7} + 280829375 T^{8} + 2408499025 T^{9} + 29229043953 T^{10} + 333874473375 T^{11} + 2740063410138 T^{12} + 19427096939650 T^{13} + 170155340195350 T^{14} + 1854559470150980 T^{15} + 18049813134587545 T^{16} + 1854559470150980 p T^{17} + 170155340195350 p^{2} T^{18} + 19427096939650 p^{3} T^{19} + 2740063410138 p^{4} T^{20} + 333874473375 p^{5} T^{21} + 29229043953 p^{6} T^{22} + 2408499025 p^{7} T^{23} + 280829375 p^{8} T^{24} + 38046565 p^{9} T^{25} + 3923885 p^{10} T^{26} + 237965 p^{11} T^{27} + 13250 p^{12} T^{28} + 1875 p^{13} T^{29} + 271 p^{14} T^{30} + 25 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 10 T - 33 T^{2} - 834 T^{3} + 2787 T^{4} - 51878 T^{5} - 1145126 T^{6} + 2213260 T^{7} + 88356034 T^{8} - 311014596 T^{9} - 1998502047 T^{10} + 54232354964 T^{11} - 49756974248 T^{12} - 3252591841782 T^{13} + 16575722380732 T^{14} + 116315383864870 T^{15} - 2060507451506835 T^{16} + 116315383864870 p T^{17} + 16575722380732 p^{2} T^{18} - 3252591841782 p^{3} T^{19} - 49756974248 p^{4} T^{20} + 54232354964 p^{5} T^{21} - 1998502047 p^{6} T^{22} - 311014596 p^{7} T^{23} + 88356034 p^{8} T^{24} + 2213260 p^{9} T^{25} - 1145126 p^{10} T^{26} - 51878 p^{11} T^{27} + 2787 p^{12} T^{28} - 834 p^{13} T^{29} - 33 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 14 T + 80 T^{2} - 2433 T^{3} + 31885 T^{4} - 197806 T^{5} + 3193816 T^{6} - 36774450 T^{7} + 226108448 T^{8} - 2631012362 T^{9} + 27060349107 T^{10} - 161818055056 T^{11} + 1519564961146 T^{12} - 14702482063651 T^{13} + 90696981876897 T^{14} - 787177755772854 T^{15} + 8457104799952696 T^{16} - 787177755772854 p T^{17} + 90696981876897 p^{2} T^{18} - 14702482063651 p^{3} T^{19} + 1519564961146 p^{4} T^{20} - 161818055056 p^{5} T^{21} + 27060349107 p^{6} T^{22} - 2631012362 p^{7} T^{23} + 226108448 p^{8} T^{24} - 36774450 p^{9} T^{25} + 3193816 p^{10} T^{26} - 197806 p^{11} T^{27} + 31885 p^{12} T^{28} - 2433 p^{13} T^{29} + 80 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 20 T - 53 T^{2} + 3866 T^{3} - 128 T^{4} - 685658 T^{5} + 2359924 T^{6} + 87662450 T^{7} - 533060956 T^{8} - 9176740306 T^{9} + 84036159393 T^{10} + 767435033324 T^{11} - 10506213535413 T^{12} - 48490109956932 T^{13} + 1110080659587932 T^{14} + 1397516285150740 T^{15} - 97743848084615880 T^{16} + 1397516285150740 p T^{17} + 1110080659587932 p^{2} T^{18} - 48490109956932 p^{3} T^{19} - 10506213535413 p^{4} T^{20} + 767435033324 p^{5} T^{21} + 84036159393 p^{6} T^{22} - 9176740306 p^{7} T^{23} - 533060956 p^{8} T^{24} + 87662450 p^{9} T^{25} + 2359924 p^{10} T^{26} - 685658 p^{11} T^{27} - 128 p^{12} T^{28} + 3866 p^{13} T^{29} - 53 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 13 T + 100 T^{2} + 1099 T^{3} + 2907 T^{4} - 52473 T^{5} - 394240 T^{6} - 10474443 T^{7} - 164830802 T^{8} - 1689712741 T^{9} - 20879194316 T^{10} - 131376419745 T^{11} + 139482521804 T^{12} + 7567117614646 T^{13} + 134920030604908 T^{14} + 1486507825420932 T^{15} + 12726722196195757 T^{16} + 1486507825420932 p T^{17} + 134920030604908 p^{2} T^{18} + 7567117614646 p^{3} T^{19} + 139482521804 p^{4} T^{20} - 131376419745 p^{5} T^{21} - 20879194316 p^{6} T^{22} - 1689712741 p^{7} T^{23} - 164830802 p^{8} T^{24} - 10474443 p^{9} T^{25} - 394240 p^{10} T^{26} - 52473 p^{11} T^{27} + 2907 p^{12} T^{28} + 1099 p^{13} T^{29} + 100 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 55 T + 1401 T^{2} - 23569 T^{3} + 339826 T^{4} - 4864391 T^{5} + 65721936 T^{6} - 785650508 T^{7} + 8645087150 T^{8} - 94110792121 T^{9} + 1007041929642 T^{10} - 10090583770539 T^{11} + 96419882430448 T^{12} - 933436409094428 T^{13} + 9122454303430546 T^{14} - 87079630003431973 T^{15} + 838150395971853668 T^{16} - 87079630003431973 p T^{17} + 9122454303430546 p^{2} T^{18} - 933436409094428 p^{3} T^{19} + 96419882430448 p^{4} T^{20} - 10090583770539 p^{5} T^{21} + 1007041929642 p^{6} T^{22} - 94110792121 p^{7} T^{23} + 8645087150 p^{8} T^{24} - 785650508 p^{9} T^{25} + 65721936 p^{10} T^{26} - 4864391 p^{11} T^{27} + 339826 p^{12} T^{28} - 23569 p^{13} T^{29} + 1401 p^{14} T^{30} - 55 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.76384205216231523444698089496, −2.69536029065175775997768109512, −2.63174345676919822035924269928, −2.60183369480164029411219538680, −2.57719907961154536352758954199, −2.28521077572466050961451454007, −1.97992598202064107958823241318, −1.97687841928968996172198760551, −1.92646033575689965792127264468, −1.90158343235274390724055643910, −1.88148623598252119048478545664, −1.87975261618902981261494191972, −1.55839751716899089825485065804, −1.49962371620769368390457206544, −1.32687436625787635886375345746, −1.21638555852789845083297949446, −1.08887669300955302406309330287, −0.879910214041222725193366570695, −0.67018704066761935837509059283, −0.65539653024130172079224266086, −0.58953110164667405379417074222, −0.55342813394187682421325555469, −0.50860081584230384522968096457, −0.23914802263668298805643916396, −0.00655431017867297376219462176, 0.00655431017867297376219462176, 0.23914802263668298805643916396, 0.50860081584230384522968096457, 0.55342813394187682421325555469, 0.58953110164667405379417074222, 0.65539653024130172079224266086, 0.67018704066761935837509059283, 0.879910214041222725193366570695, 1.08887669300955302406309330287, 1.21638555852789845083297949446, 1.32687436625787635886375345746, 1.49962371620769368390457206544, 1.55839751716899089825485065804, 1.87975261618902981261494191972, 1.88148623598252119048478545664, 1.90158343235274390724055643910, 1.92646033575689965792127264468, 1.97687841928968996172198760551, 1.97992598202064107958823241318, 2.28521077572466050961451454007, 2.57719907961154536352758954199, 2.60183369480164029411219538680, 2.63174345676919822035924269928, 2.69536029065175775997768109512, 2.76384205216231523444698089496
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.