Dirichlet series
| L(s) = 1 | + 3·2-s + 5·3-s − 4-s + 16·5-s + 15·6-s + 7-s − 11·8-s + 48·10-s + 4·11-s − 5·12-s + 15·13-s + 3·14-s + 80·15-s − 8·16-s + 20·17-s − 4·19-s − 16·20-s + 5·21-s + 12·22-s + 16·23-s − 55·24-s + 103·25-s + 45·26-s − 44·27-s − 28-s − 16·29-s + 240·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2.88·3-s − 1/2·4-s + 7.15·5-s + 6.12·6-s + 0.377·7-s − 3.88·8-s + 15.1·10-s + 1.20·11-s − 1.44·12-s + 4.16·13-s + 0.801·14-s + 20.6·15-s − 2·16-s + 4.85·17-s − 0.917·19-s − 3.57·20-s + 1.09·21-s + 2.55·22-s + 3.33·23-s − 11.2·24-s + 20.5·25-s + 8.82·26-s − 8.46·27-s − 0.188·28-s − 2.97·29-s + 43.8·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(23^{16} \cdot 29^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.19224\times 10^{11}\) |
| Root analytic conductor: | \(2.30781\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 23^{16} \cdot 29^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(402.4994771\) |
| \(L(\frac12)\) | \(\approx\) | \(402.4994771\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 23 | \( ( 1 - T )^{16} \) |
| 29 | \( ( 1 + T )^{16} \) | |
| good | 2 | \( 1 - 3 T + 5 p T^{2} - 11 p T^{3} + 51 T^{4} - 89 T^{5} + 165 T^{6} - 123 p T^{7} + 3 p^{7} T^{8} - 497 T^{9} + 177 p^{2} T^{10} - 413 p T^{11} + 577 p T^{12} - 1343 T^{13} + 1975 T^{14} - 1197 p T^{15} + 963 p^{2} T^{16} - 1197 p^{2} T^{17} + 1975 p^{2} T^{18} - 1343 p^{3} T^{19} + 577 p^{5} T^{20} - 413 p^{6} T^{21} + 177 p^{8} T^{22} - 497 p^{7} T^{23} + 3 p^{15} T^{24} - 123 p^{10} T^{25} + 165 p^{10} T^{26} - 89 p^{11} T^{27} + 51 p^{12} T^{28} - 11 p^{14} T^{29} + 5 p^{15} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 - 5 T + 25 T^{2} - p^{4} T^{3} + 256 T^{4} - 214 p T^{5} + 1565 T^{6} - 3244 T^{7} + 2203 p T^{8} - 11798 T^{9} + 21167 T^{10} - 33862 T^{11} + 57059 T^{12} - 86765 T^{13} + 49445 p T^{14} - 229759 T^{15} + 419134 T^{16} - 229759 p T^{17} + 49445 p^{3} T^{18} - 86765 p^{3} T^{19} + 57059 p^{4} T^{20} - 33862 p^{5} T^{21} + 21167 p^{6} T^{22} - 11798 p^{7} T^{23} + 2203 p^{9} T^{24} - 3244 p^{9} T^{25} + 1565 p^{10} T^{26} - 214 p^{12} T^{27} + 256 p^{12} T^{28} - p^{17} T^{29} + 25 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 5 | \( 1 - 16 T + 153 T^{2} - 1088 T^{3} + 6372 T^{4} - 32211 T^{5} + 144828 T^{6} - 589852 T^{7} + 2205462 T^{8} - 7640406 T^{9} + 24699256 T^{10} - 74890147 T^{11} + 213843063 T^{12} - 115339326 p T^{13} + 1472098531 T^{14} - 3561811102 T^{15} + 8175960316 T^{16} - 3561811102 p T^{17} + 1472098531 p^{2} T^{18} - 115339326 p^{4} T^{19} + 213843063 p^{4} T^{20} - 74890147 p^{5} T^{21} + 24699256 p^{6} T^{22} - 7640406 p^{7} T^{23} + 2205462 p^{8} T^{24} - 589852 p^{9} T^{25} + 144828 p^{10} T^{26} - 32211 p^{11} T^{27} + 6372 p^{12} T^{28} - 1088 p^{13} T^{29} + 153 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 7 | \( 1 - T + 41 T^{2} - 25 T^{3} + 127 p T^{4} - 262 T^{5} + 13422 T^{6} + 1475 T^{7} + 22083 p T^{8} + 97281 T^{9} + 206433 p T^{10} + 1808766 T^{11} + 11521521 T^{12} + 22075619 T^{13} + 83322450 T^{14} + 200046589 T^{15} + 584108128 T^{16} + 200046589 p T^{17} + 83322450 p^{2} T^{18} + 22075619 p^{3} T^{19} + 11521521 p^{4} T^{20} + 1808766 p^{5} T^{21} + 206433 p^{7} T^{22} + 97281 p^{7} T^{23} + 22083 p^{9} T^{24} + 1475 p^{9} T^{25} + 13422 p^{10} T^{26} - 262 p^{11} T^{27} + 127 p^{13} T^{28} - 25 p^{13} T^{29} + 41 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 4 T + 87 T^{2} - 305 T^{3} + 3846 T^{4} - 12369 T^{5} + 116999 T^{6} - 351654 T^{7} + 2743519 T^{8} - 7762962 T^{9} + 52405763 T^{10} - 12714274 p T^{11} + 840621025 T^{12} - 2111707908 T^{13} + 11524132159 T^{14} - 27123517842 T^{15} + 136290044578 T^{16} - 27123517842 p T^{17} + 11524132159 p^{2} T^{18} - 2111707908 p^{3} T^{19} + 840621025 p^{4} T^{20} - 12714274 p^{6} T^{21} + 52405763 p^{6} T^{22} - 7762962 p^{7} T^{23} + 2743519 p^{8} T^{24} - 351654 p^{9} T^{25} + 116999 p^{10} T^{26} - 12369 p^{11} T^{27} + 3846 p^{12} T^{28} - 305 p^{13} T^{29} + 87 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 15 T + 193 T^{2} - 1750 T^{3} + 14182 T^{4} - 97594 T^{5} + 618110 T^{6} - 3532117 T^{7} + 18918983 T^{8} - 94054959 T^{9} + 443969101 T^{10} - 1977759271 T^{11} + 8449925802 T^{12} - 34448927420 T^{13} + 135720783671 T^{14} - 513755415830 T^{15} + 1887460003930 T^{16} - 513755415830 p T^{17} + 135720783671 p^{2} T^{18} - 34448927420 p^{3} T^{19} + 8449925802 p^{4} T^{20} - 1977759271 p^{5} T^{21} + 443969101 p^{6} T^{22} - 94054959 p^{7} T^{23} + 18918983 p^{8} T^{24} - 3532117 p^{9} T^{25} + 618110 p^{10} T^{26} - 97594 p^{11} T^{27} + 14182 p^{12} T^{28} - 1750 p^{13} T^{29} + 193 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 20 T + 318 T^{2} - 3634 T^{3} + 35651 T^{4} - 295522 T^{5} + 2190070 T^{6} - 14437961 T^{7} + 87119572 T^{8} - 479762707 T^{9} + 2459921959 T^{10} - 11733318129 T^{11} + 53092304636 T^{12} - 228644231681 T^{13} + 959837015325 T^{14} - 3951257438490 T^{15} + 16310811598792 T^{16} - 3951257438490 p T^{17} + 959837015325 p^{2} T^{18} - 228644231681 p^{3} T^{19} + 53092304636 p^{4} T^{20} - 11733318129 p^{5} T^{21} + 2459921959 p^{6} T^{22} - 479762707 p^{7} T^{23} + 87119572 p^{8} T^{24} - 14437961 p^{9} T^{25} + 2190070 p^{10} T^{26} - 295522 p^{11} T^{27} + 35651 p^{12} T^{28} - 3634 p^{13} T^{29} + 318 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 4 T + 162 T^{2} + 37 p T^{3} + 13635 T^{4} + 60384 T^{5} + 779746 T^{6} + 3423643 T^{7} + 33618775 T^{8} + 144056893 T^{9} + 1156283192 T^{10} + 4778942484 T^{11} + 32844550826 T^{12} + 129483358143 T^{13} + 787683013956 T^{14} + 2924003086674 T^{15} + 16151768751174 T^{16} + 2924003086674 p T^{17} + 787683013956 p^{2} T^{18} + 129483358143 p^{3} T^{19} + 32844550826 p^{4} T^{20} + 4778942484 p^{5} T^{21} + 1156283192 p^{6} T^{22} + 144056893 p^{7} T^{23} + 33618775 p^{8} T^{24} + 3423643 p^{9} T^{25} + 779746 p^{10} T^{26} + 60384 p^{11} T^{27} + 13635 p^{12} T^{28} + 37 p^{14} T^{29} + 162 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 233 T^{2} + 411 T^{3} + 27649 T^{4} + 85522 T^{5} + 2307254 T^{6} + 8925993 T^{7} + 151860395 T^{8} + 635536599 T^{9} + 8212180782 T^{10} + 34789676356 T^{11} + 373071508736 T^{12} + 1540593820289 T^{13} + 14475921162419 T^{14} + 56653078471506 T^{15} + 483416812385190 T^{16} + 56653078471506 p T^{17} + 14475921162419 p^{2} T^{18} + 1540593820289 p^{3} T^{19} + 373071508736 p^{4} T^{20} + 34789676356 p^{5} T^{21} + 8212180782 p^{6} T^{22} + 635536599 p^{7} T^{23} + 151860395 p^{8} T^{24} + 8925993 p^{9} T^{25} + 2307254 p^{10} T^{26} + 85522 p^{11} T^{27} + 27649 p^{12} T^{28} + 411 p^{13} T^{29} + 233 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 5 T + 285 T^{2} - 1339 T^{3} + 41237 T^{4} - 188625 T^{5} + 4060511 T^{6} - 501906 p T^{7} + 305421340 T^{8} - 1413924211 T^{9} + 18662771011 T^{10} - 87416923987 T^{11} + 963231740724 T^{12} - 4505250808383 T^{13} + 43078895001993 T^{14} - 196151157407296 T^{15} + 1694967246538164 T^{16} - 196151157407296 p T^{17} + 43078895001993 p^{2} T^{18} - 4505250808383 p^{3} T^{19} + 963231740724 p^{4} T^{20} - 87416923987 p^{5} T^{21} + 18662771011 p^{6} T^{22} - 1413924211 p^{7} T^{23} + 305421340 p^{8} T^{24} - 501906 p^{10} T^{25} + 4060511 p^{10} T^{26} - 188625 p^{11} T^{27} + 41237 p^{12} T^{28} - 1339 p^{13} T^{29} + 285 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 7 T + 479 T^{2} - 3737 T^{3} + 113000 T^{4} - 927971 T^{5} + 17487630 T^{6} - 143932917 T^{7} + 1982002515 T^{8} - 15741652212 T^{9} + 173398279305 T^{10} - 1294524515988 T^{11} + 12037605284290 T^{12} - 83034933842372 T^{13} + 673331302488874 T^{14} - 2521807540268 p^{2} T^{15} + 30592154579437316 T^{16} - 2521807540268 p^{3} T^{17} + 673331302488874 p^{2} T^{18} - 83034933842372 p^{3} T^{19} + 12037605284290 p^{4} T^{20} - 1294524515988 p^{5} T^{21} + 173398279305 p^{6} T^{22} - 15741652212 p^{7} T^{23} + 1982002515 p^{8} T^{24} - 143932917 p^{9} T^{25} + 17487630 p^{10} T^{26} - 927971 p^{11} T^{27} + 113000 p^{12} T^{28} - 3737 p^{13} T^{29} + 479 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 17 T + 513 T^{2} + 7205 T^{3} + 124688 T^{4} + 1493488 T^{5} + 19330948 T^{6} + 201987772 T^{7} + 2160610148 T^{8} + 20032919829 T^{9} + 185964243015 T^{10} + 1549827842217 T^{11} + 12822536040455 T^{12} + 96941169343511 T^{13} + 725554514257835 T^{14} + 5003884842566475 T^{15} + 34140214704561310 T^{16} + 5003884842566475 p T^{17} + 725554514257835 p^{2} T^{18} + 96941169343511 p^{3} T^{19} + 12822536040455 p^{4} T^{20} + 1549827842217 p^{5} T^{21} + 185964243015 p^{6} T^{22} + 20032919829 p^{7} T^{23} + 2160610148 p^{8} T^{24} + 201987772 p^{9} T^{25} + 19330948 p^{10} T^{26} + 1493488 p^{11} T^{27} + 124688 p^{12} T^{28} + 7205 p^{13} T^{29} + 513 p^{14} T^{30} + 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 29 T + 866 T^{2} - 16104 T^{3} + 288964 T^{4} - 4059574 T^{5} + 54737743 T^{6} - 625314880 T^{7} + 6889440501 T^{8} - 66797235645 T^{9} + 629927859687 T^{10} - 5348449236143 T^{11} + 44672448605843 T^{12} - 341591995236157 T^{13} + 2602589459913176 T^{14} - 18379631794667350 T^{15} + 130553878485010118 T^{16} - 18379631794667350 p T^{17} + 2602589459913176 p^{2} T^{18} - 341591995236157 p^{3} T^{19} + 44672448605843 p^{4} T^{20} - 5348449236143 p^{5} T^{21} + 629927859687 p^{6} T^{22} - 66797235645 p^{7} T^{23} + 6889440501 p^{8} T^{24} - 625314880 p^{9} T^{25} + 54737743 p^{10} T^{26} - 4059574 p^{11} T^{27} + 288964 p^{12} T^{28} - 16104 p^{13} T^{29} + 866 p^{14} T^{30} - 29 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 63 T + 2345 T^{2} - 63312 T^{3} + 1370906 T^{4} - 25004561 T^{5} + 397150870 T^{6} - 5614072755 T^{7} + 71799158661 T^{8} - 840804285427 T^{9} + 9101064515451 T^{10} - 91708149305311 T^{11} + 865122714883027 T^{12} - 7672397016244700 T^{13} + 64172863595393290 T^{14} - 507317180372376107 T^{15} + 3795670584889688578 T^{16} - 507317180372376107 p T^{17} + 64172863595393290 p^{2} T^{18} - 7672397016244700 p^{3} T^{19} + 865122714883027 p^{4} T^{20} - 91708149305311 p^{5} T^{21} + 9101064515451 p^{6} T^{22} - 840804285427 p^{7} T^{23} + 71799158661 p^{8} T^{24} - 5614072755 p^{9} T^{25} + 397150870 p^{10} T^{26} - 25004561 p^{11} T^{27} + 1370906 p^{12} T^{28} - 63312 p^{13} T^{29} + 2345 p^{14} T^{30} - 63 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 11 T + 581 T^{2} - 5608 T^{3} + 170450 T^{4} - 1484025 T^{5} + 33371964 T^{6} - 265309873 T^{7} + 4861313076 T^{8} - 35523364556 T^{9} + 557181999478 T^{10} - 3753509547737 T^{11} + 51882494368759 T^{12} - 322292125090558 T^{13} + 3998789482813805 T^{14} - 22855714893812772 T^{15} + 257642804366027452 T^{16} - 22855714893812772 p T^{17} + 3998789482813805 p^{2} T^{18} - 322292125090558 p^{3} T^{19} + 51882494368759 p^{4} T^{20} - 3753509547737 p^{5} T^{21} + 557181999478 p^{6} T^{22} - 35523364556 p^{7} T^{23} + 4861313076 p^{8} T^{24} - 265309873 p^{9} T^{25} + 33371964 p^{10} T^{26} - 1484025 p^{11} T^{27} + 170450 p^{12} T^{28} - 5608 p^{13} T^{29} + 581 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 430 T^{2} - 442 T^{3} + 96022 T^{4} - 176950 T^{5} + 14719103 T^{6} - 37933840 T^{7} + 1731467400 T^{8} - 5670296486 T^{9} + 166556630888 T^{10} - 649179906236 T^{11} + 13666457049488 T^{12} - 59356051906612 T^{13} + 984542318044971 T^{14} - 4416129220363530 T^{15} + 63361352311918146 T^{16} - 4416129220363530 p T^{17} + 984542318044971 p^{2} T^{18} - 59356051906612 p^{3} T^{19} + 13666457049488 p^{4} T^{20} - 649179906236 p^{5} T^{21} + 166556630888 p^{6} T^{22} - 5670296486 p^{7} T^{23} + 1731467400 p^{8} T^{24} - 37933840 p^{9} T^{25} + 14719103 p^{10} T^{26} - 176950 p^{11} T^{27} + 96022 p^{12} T^{28} - 442 p^{13} T^{29} + 430 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 + 13 T + 10 p T^{2} + 8188 T^{3} + 230543 T^{4} + 2619614 T^{5} + 53087488 T^{6} + 558208387 T^{7} + 9073012510 T^{8} + 88068094121 T^{9} + 1213486439819 T^{10} + 10855099028016 T^{11} + 130938836056731 T^{12} + 1077211910225838 T^{13} + 173186268929509 p T^{14} + 87507315070039515 T^{15} + 852538297434471918 T^{16} + 87507315070039515 p T^{17} + 173186268929509 p^{3} T^{18} + 1077211910225838 p^{3} T^{19} + 130938836056731 p^{4} T^{20} + 10855099028016 p^{5} T^{21} + 1213486439819 p^{6} T^{22} + 88068094121 p^{7} T^{23} + 9073012510 p^{8} T^{24} + 558208387 p^{9} T^{25} + 53087488 p^{10} T^{26} + 2619614 p^{11} T^{27} + 230543 p^{12} T^{28} + 8188 p^{13} T^{29} + 10 p^{15} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 + 23 T + 884 T^{2} + 15988 T^{3} + 350501 T^{4} + 5252152 T^{5} + 84915421 T^{6} + 1089093351 T^{7} + 14332646888 T^{8} + 161103070680 T^{9} + 1821849114561 T^{10} + 18326234398836 T^{11} + 185017677286106 T^{12} + 1701877538128570 T^{13} + 15819528055733742 T^{14} + 135879776864301866 T^{15} + 1188115050713486000 T^{16} + 135879776864301866 p T^{17} + 15819528055733742 p^{2} T^{18} + 1701877538128570 p^{3} T^{19} + 185017677286106 p^{4} T^{20} + 18326234398836 p^{5} T^{21} + 1821849114561 p^{6} T^{22} + 161103070680 p^{7} T^{23} + 14332646888 p^{8} T^{24} + 1089093351 p^{9} T^{25} + 84915421 p^{10} T^{26} + 5252152 p^{11} T^{27} + 350501 p^{12} T^{28} + 15988 p^{13} T^{29} + 884 p^{14} T^{30} + 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 38 T + 1102 T^{2} + 23637 T^{3} + 449118 T^{4} + 7416121 T^{5} + 112854931 T^{6} + 1568043333 T^{7} + 20466071212 T^{8} + 249509182915 T^{9} + 2887586787526 T^{10} + 31589362290113 T^{11} + 330110919046364 T^{12} + 3282626306498577 T^{13} + 31291315331137657 T^{14} + 284805720605396682 T^{15} + 2488970007910991602 T^{16} + 284805720605396682 p T^{17} + 31291315331137657 p^{2} T^{18} + 3282626306498577 p^{3} T^{19} + 330110919046364 p^{4} T^{20} + 31589362290113 p^{5} T^{21} + 2887586787526 p^{6} T^{22} + 249509182915 p^{7} T^{23} + 20466071212 p^{8} T^{24} + 1568043333 p^{9} T^{25} + 112854931 p^{10} T^{26} + 7416121 p^{11} T^{27} + 449118 p^{12} T^{28} + 23637 p^{13} T^{29} + 1102 p^{14} T^{30} + 38 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 27 T + 693 T^{2} + 9018 T^{3} + 123643 T^{4} + 857696 T^{5} + 9649395 T^{6} + 34236492 T^{7} + 874865634 T^{8} + 3707513929 T^{9} + 117694754759 T^{10} + 461194101439 T^{11} + 11627414069339 T^{12} + 25230497718801 T^{13} + 855497369602862 T^{14} + 175508400238644 T^{15} + 60554252891018408 T^{16} + 175508400238644 p T^{17} + 855497369602862 p^{2} T^{18} + 25230497718801 p^{3} T^{19} + 11627414069339 p^{4} T^{20} + 461194101439 p^{5} T^{21} + 117694754759 p^{6} T^{22} + 3707513929 p^{7} T^{23} + 874865634 p^{8} T^{24} + 34236492 p^{9} T^{25} + 9649395 p^{10} T^{26} + 857696 p^{11} T^{27} + 123643 p^{12} T^{28} + 9018 p^{13} T^{29} + 693 p^{14} T^{30} + 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 36 T + 1196 T^{2} - 27517 T^{3} + 571881 T^{4} - 9920614 T^{5} + 158245423 T^{6} - 2231747673 T^{7} + 29379555190 T^{8} - 351597127364 T^{9} + 3980053468120 T^{10} - 41759240431813 T^{11} + 421455095839285 T^{12} - 4024465951535984 T^{13} + 37861643012502909 T^{14} - 345281369780647867 T^{15} + 3174230346311553110 T^{16} - 345281369780647867 p T^{17} + 37861643012502909 p^{2} T^{18} - 4024465951535984 p^{3} T^{19} + 421455095839285 p^{4} T^{20} - 41759240431813 p^{5} T^{21} + 3980053468120 p^{6} T^{22} - 351597127364 p^{7} T^{23} + 29379555190 p^{8} T^{24} - 2231747673 p^{9} T^{25} + 158245423 p^{10} T^{26} - 9920614 p^{11} T^{27} + 571881 p^{12} T^{28} - 27517 p^{13} T^{29} + 1196 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 16 T + 811 T^{2} + 11248 T^{3} + 323872 T^{4} + 4023539 T^{5} + 86234371 T^{6} + 978851675 T^{7} + 17292857403 T^{8} + 181368267110 T^{9} + 2777395005049 T^{10} + 27066173146294 T^{11} + 369771559980595 T^{12} + 3354466560687775 T^{13} + 41625490530195845 T^{14} + 351042231905929943 T^{15} + 4002398398064518426 T^{16} + 351042231905929943 p T^{17} + 41625490530195845 p^{2} T^{18} + 3354466560687775 p^{3} T^{19} + 369771559980595 p^{4} T^{20} + 27066173146294 p^{5} T^{21} + 2777395005049 p^{6} T^{22} + 181368267110 p^{7} T^{23} + 17292857403 p^{8} T^{24} + 978851675 p^{9} T^{25} + 86234371 p^{10} T^{26} + 4023539 p^{11} T^{27} + 323872 p^{12} T^{28} + 11248 p^{13} T^{29} + 811 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 + 30 T + 926 T^{2} + 17818 T^{3} + 357637 T^{4} + 5630907 T^{5} + 91483144 T^{6} + 1261788379 T^{7} + 17841643467 T^{8} + 223742731957 T^{9} + 2856843348275 T^{10} + 33124989986201 T^{11} + 388835507548649 T^{12} + 4209189925697002 T^{13} + 45935998553919015 T^{14} + 465832374979483898 T^{15} + 4752194657058211340 T^{16} + 465832374979483898 p T^{17} + 45935998553919015 p^{2} T^{18} + 4209189925697002 p^{3} T^{19} + 388835507548649 p^{4} T^{20} + 33124989986201 p^{5} T^{21} + 2856843348275 p^{6} T^{22} + 223742731957 p^{7} T^{23} + 17841643467 p^{8} T^{24} + 1261788379 p^{9} T^{25} + 91483144 p^{10} T^{26} + 5630907 p^{11} T^{27} + 357637 p^{12} T^{28} + 17818 p^{13} T^{29} + 926 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.86511940333301402770566206980, −2.76870276003720750198423075013, −2.70674057497773048026797269784, −2.60435680906888525694255054089, −2.46059151784143982711822032450, −2.45660221968565723091574690408, −2.31133758067115764767952379297, −2.16706170665421223258469598115, −2.11763972102688109809777699315, −2.09220105089759713345930171201, −2.06365290810351837726457092287, −1.95710475289815049042525174035, −1.86579903455996398479064390585, −1.57195574705848677486262265441, −1.51752874629228380928264464171, −1.49717067104105362621042922726, −1.43753758837644928812941616585, −1.39822304900822278672771261132, −1.36375103132713772730030699039, −1.27211289770000172727214958157, −1.13491634484006525420018702370, −0.887933692681204713308516191209, −0.67690630770963988258318948973, −0.44665184182056498038474904150, −0.26837458967246695331760195596, 0.26837458967246695331760195596, 0.44665184182056498038474904150, 0.67690630770963988258318948973, 0.887933692681204713308516191209, 1.13491634484006525420018702370, 1.27211289770000172727214958157, 1.36375103132713772730030699039, 1.39822304900822278672771261132, 1.43753758837644928812941616585, 1.49717067104105362621042922726, 1.51752874629228380928264464171, 1.57195574705848677486262265441, 1.86579903455996398479064390585, 1.95710475289815049042525174035, 2.06365290810351837726457092287, 2.09220105089759713345930171201, 2.11763972102688109809777699315, 2.16706170665421223258469598115, 2.31133758067115764767952379297, 2.45660221968565723091574690408, 2.46059151784143982711822032450, 2.60435680906888525694255054089, 2.70674057497773048026797269784, 2.76870276003720750198423075013, 2.86511940333301402770566206980
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.