Dirichlet series
| L(s) = 1 | + 4·2-s + 2·3-s + 6·4-s − 8·5-s + 8·6-s − 13·7-s + 4·8-s + 3·9-s − 32·10-s − 2·11-s + 12·12-s − 6·13-s − 52·14-s − 16·15-s + 16-s + 6·17-s + 12·18-s + 9·19-s − 48·20-s − 26·21-s − 8·22-s − 3·23-s + 8·24-s + 28·25-s − 24·26-s + 2·27-s − 78·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 3·4-s − 3.57·5-s + 3.26·6-s − 4.91·7-s + 1.41·8-s + 9-s − 10.1·10-s − 0.603·11-s + 3.46·12-s − 1.66·13-s − 13.8·14-s − 4.13·15-s + 1/4·16-s + 1.45·17-s + 2.82·18-s + 2.06·19-s − 10.7·20-s − 5.67·21-s − 1.70·22-s − 0.625·23-s + 1.63·24-s + 28/5·25-s − 4.70·26-s + 0.384·27-s − 14.7·28-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
| \(L(1)\) | \(\approx\) | \(0.02499540460\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02499540460\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) | |
| 5 | \( ( 1 + T + T^{2} )^{8} \) | |
| 31 | \( 1 + 19 T + 143 T^{2} + 304 T^{3} - 2663 T^{4} - 18080 T^{5} + 3652 T^{6} + 549417 T^{7} + 3990975 T^{8} + 549417 p T^{9} + 3652 p^{2} T^{10} - 18080 p^{3} T^{11} - 2663 p^{4} T^{12} + 304 p^{5} T^{13} + 143 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 7 | \( 1 + 13 T + 90 T^{2} + 426 T^{3} + 1524 T^{4} + 4317 T^{5} + 9775 T^{6} + 2616 p T^{7} + 36422 T^{8} + 122912 T^{9} + 538590 T^{10} + 2029570 T^{11} + 5999551 T^{12} + 13558795 T^{13} + 21714359 T^{14} + 18992431 T^{15} + 7214435 T^{16} + 18992431 p T^{17} + 21714359 p^{2} T^{18} + 13558795 p^{3} T^{19} + 5999551 p^{4} T^{20} + 2029570 p^{5} T^{21} + 538590 p^{6} T^{22} + 122912 p^{7} T^{23} + 36422 p^{8} T^{24} + 2616 p^{10} T^{25} + 9775 p^{10} T^{26} + 4317 p^{11} T^{27} + 1524 p^{12} T^{28} + 426 p^{13} T^{29} + 90 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) |
| 11 | \( 1 + 2 T + 28 T^{2} + 131 T^{3} + 564 T^{4} + 2210 T^{5} + 9989 T^{6} + 24098 T^{7} + 103258 T^{8} + 255647 T^{9} + 922792 T^{10} + 2571056 T^{11} + 12933810 T^{12} + 37855991 T^{13} + 193091759 T^{14} + 53569711 p T^{15} + 2409190281 T^{16} + 53569711 p^{2} T^{17} + 193091759 p^{2} T^{18} + 37855991 p^{3} T^{19} + 12933810 p^{4} T^{20} + 2571056 p^{5} T^{21} + 922792 p^{6} T^{22} + 255647 p^{7} T^{23} + 103258 p^{8} T^{24} + 24098 p^{9} T^{25} + 9989 p^{10} T^{26} + 2210 p^{11} T^{27} + 564 p^{12} T^{28} + 131 p^{13} T^{29} + 28 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 6 T + 40 T^{2} + 53 T^{3} + 144 T^{4} - 1186 T^{5} + 2765 T^{6} + 6451 T^{7} + 99252 T^{8} - 43586 T^{9} - 376020 T^{10} - 2724760 T^{11} + 1158816 T^{12} + 39359895 T^{13} - 24186384 T^{14} - 408914687 T^{15} - 4553010175 T^{16} - 408914687 p T^{17} - 24186384 p^{2} T^{18} + 39359895 p^{3} T^{19} + 1158816 p^{4} T^{20} - 2724760 p^{5} T^{21} - 376020 p^{6} T^{22} - 43586 p^{7} T^{23} + 99252 p^{8} T^{24} + 6451 p^{9} T^{25} + 2765 p^{10} T^{26} - 1186 p^{11} T^{27} + 144 p^{12} T^{28} + 53 p^{13} T^{29} + 40 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 6 T + 40 T^{2} - 280 T^{3} + 1427 T^{4} - 7564 T^{5} + 38455 T^{6} - 8496 p T^{7} + 686810 T^{8} - 2447340 T^{9} + 7168728 T^{10} - 20309080 T^{11} + 22730001 T^{12} + 179716958 T^{13} - 1273745355 T^{14} + 8644601088 T^{15} - 36194380957 T^{16} + 8644601088 p T^{17} - 1273745355 p^{2} T^{18} + 179716958 p^{3} T^{19} + 22730001 p^{4} T^{20} - 20309080 p^{5} T^{21} + 7168728 p^{6} T^{22} - 2447340 p^{7} T^{23} + 686810 p^{8} T^{24} - 8496 p^{10} T^{25} + 38455 p^{10} T^{26} - 7564 p^{11} T^{27} + 1427 p^{12} T^{28} - 280 p^{13} T^{29} + 40 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 9 T + 14 T^{2} + 183 T^{3} - 815 T^{4} - 915 T^{5} + 15601 T^{6} - 26154 T^{7} - 206809 T^{8} + 1426704 T^{9} - 534148 T^{10} - 30367155 T^{11} + 116221541 T^{12} + 210032100 T^{13} - 2078002487 T^{14} - 122288334 T^{15} + 32268782365 T^{16} - 122288334 p T^{17} - 2078002487 p^{2} T^{18} + 210032100 p^{3} T^{19} + 116221541 p^{4} T^{20} - 30367155 p^{5} T^{21} - 534148 p^{6} T^{22} + 1426704 p^{7} T^{23} - 206809 p^{8} T^{24} - 26154 p^{9} T^{25} + 15601 p^{10} T^{26} - 915 p^{11} T^{27} - 815 p^{12} T^{28} + 183 p^{13} T^{29} + 14 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 + 3 T + 24 T^{2} + 17 T^{3} + 1368 T^{4} - 2348 T^{5} + 8549 T^{6} - 119028 T^{7} + 860459 T^{8} - 4217963 T^{9} + 14064572 T^{10} - 77975029 T^{11} + 635714425 T^{12} - 3206744519 T^{13} + 6631202290 T^{14} - 89560153231 T^{15} + 191713228795 T^{16} - 89560153231 p T^{17} + 6631202290 p^{2} T^{18} - 3206744519 p^{3} T^{19} + 635714425 p^{4} T^{20} - 77975029 p^{5} T^{21} + 14064572 p^{6} T^{22} - 4217963 p^{7} T^{23} + 860459 p^{8} T^{24} - 119028 p^{9} T^{25} + 8549 p^{10} T^{26} - 2348 p^{11} T^{27} + 1368 p^{12} T^{28} + 17 p^{13} T^{29} + 24 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + T - 90 T^{2} + 134 T^{3} + 4374 T^{4} - 10091 T^{5} - 86741 T^{6} + 385973 T^{7} + 491092 T^{8} + 22059 T^{9} + 33175575 T^{10} - 206333291 T^{11} - 244969228 T^{12} + 8701119797 T^{13} + 15775302661 T^{14} - 1333607600 p T^{15} - 4577932543 p T^{16} - 1333607600 p^{2} T^{17} + 15775302661 p^{2} T^{18} + 8701119797 p^{3} T^{19} - 244969228 p^{4} T^{20} - 206333291 p^{5} T^{21} + 33175575 p^{6} T^{22} + 22059 p^{7} T^{23} + 491092 p^{8} T^{24} + 385973 p^{9} T^{25} - 86741 p^{10} T^{26} - 10091 p^{11} T^{27} + 4374 p^{12} T^{28} + 134 p^{13} T^{29} - 90 p^{14} T^{30} + p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 3 T - 4 p T^{2} - 131 T^{3} + 11943 T^{4} - 6392 T^{5} - 610469 T^{6} + 695697 T^{7} + 19873877 T^{8} - 26620548 T^{9} - 343521381 T^{10} + 562058794 T^{11} - 7112822500 T^{12} - 10871933970 T^{13} + 960250029465 T^{14} + 166885607535 T^{15} - 46752458223399 T^{16} + 166885607535 p T^{17} + 960250029465 p^{2} T^{18} - 10871933970 p^{3} T^{19} - 7112822500 p^{4} T^{20} + 562058794 p^{5} T^{21} - 343521381 p^{6} T^{22} - 26620548 p^{7} T^{23} + 19873877 p^{8} T^{24} + 695697 p^{9} T^{25} - 610469 p^{10} T^{26} - 6392 p^{11} T^{27} + 11943 p^{12} T^{28} - 131 p^{13} T^{29} - 4 p^{15} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - T^{2} + 330 T^{3} + 4541 T^{4} - 3135 T^{5} + 96533 T^{6} + 1357845 T^{7} + 6253882 T^{8} + 4101165 T^{9} + 361975904 T^{10} + 1606456470 T^{11} + 3622705672 T^{12} + 31460783445 T^{13} + 479430133113 T^{14} + 1098233895510 T^{15} + 3705225378291 T^{16} + 1098233895510 p T^{17} + 479430133113 p^{2} T^{18} + 31460783445 p^{3} T^{19} + 3622705672 p^{4} T^{20} + 1606456470 p^{5} T^{21} + 361975904 p^{6} T^{22} + 4101165 p^{7} T^{23} + 6253882 p^{8} T^{24} + 1357845 p^{9} T^{25} + 96533 p^{10} T^{26} - 3135 p^{11} T^{27} + 4541 p^{12} T^{28} + 330 p^{13} T^{29} - p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 2 T - 103 T^{2} + 303 T^{3} + 2356 T^{4} - 14205 T^{5} - 26255 T^{6} + 117704 T^{7} + 8624922 T^{8} - 28211453 T^{9} - 321321381 T^{10} + 2825343295 T^{11} - 4298927143 T^{12} - 18934508797 T^{13} - 207874167321 T^{14} - 1873007329483 T^{15} + 36749415687329 T^{16} - 1873007329483 p T^{17} - 207874167321 p^{2} T^{18} - 18934508797 p^{3} T^{19} - 4298927143 p^{4} T^{20} + 2825343295 p^{5} T^{21} - 321321381 p^{6} T^{22} - 28211453 p^{7} T^{23} + 8624922 p^{8} T^{24} + 117704 p^{9} T^{25} - 26255 p^{10} T^{26} - 14205 p^{11} T^{27} + 2356 p^{12} T^{28} + 303 p^{13} T^{29} - 103 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + T - 87 T^{2} - 196 T^{3} + 3336 T^{4} + 8485 T^{5} - 14435 T^{6} + 452372 T^{7} - 2061623 T^{8} - 76411641 T^{9} + 346390941 T^{10} + 5555400385 T^{11} - 22009837783 T^{12} - 268503511534 T^{13} + 647143746676 T^{14} + 5581509846536 T^{15} - 510876154181 T^{16} + 5581509846536 p T^{17} + 647143746676 p^{2} T^{18} - 268503511534 p^{3} T^{19} - 22009837783 p^{4} T^{20} + 5555400385 p^{5} T^{21} + 346390941 p^{6} T^{22} - 76411641 p^{7} T^{23} - 2061623 p^{8} T^{24} + 452372 p^{9} T^{25} - 14435 p^{10} T^{26} + 8485 p^{11} T^{27} + 3336 p^{12} T^{28} - 196 p^{13} T^{29} - 87 p^{14} T^{30} + p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 14 T + 110 T^{2} - 288 T^{3} - 17651 T^{4} - 179019 T^{5} - 893480 T^{6} + 6038754 T^{7} + 132794767 T^{8} + 1057703851 T^{9} + 3067642280 T^{10} - 43671117665 T^{11} - 600257077184 T^{12} - 3882106187375 T^{13} - 3883313101959 T^{14} + 189951813184482 T^{15} + 1948455101460445 T^{16} + 189951813184482 p T^{17} - 3883313101959 p^{2} T^{18} - 3882106187375 p^{3} T^{19} - 600257077184 p^{4} T^{20} - 43671117665 p^{5} T^{21} + 3067642280 p^{6} T^{22} + 1057703851 p^{7} T^{23} + 132794767 p^{8} T^{24} + 6038754 p^{9} T^{25} - 893480 p^{10} T^{26} - 179019 p^{11} T^{27} - 17651 p^{12} T^{28} - 288 p^{13} T^{29} + 110 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 16 T + 212 T^{2} - 2582 T^{3} + 31029 T^{4} - 324160 T^{5} + 2994941 T^{6} - 25926336 T^{7} + 199820848 T^{8} - 1388796206 T^{9} + 7964002603 T^{10} - 31488379537 T^{11} - 62104814195 T^{12} + 2752733661082 T^{13} - 34869141612979 T^{14} + 353003417494083 T^{15} - 2928815300989099 T^{16} + 353003417494083 p T^{17} - 34869141612979 p^{2} T^{18} + 2752733661082 p^{3} T^{19} - 62104814195 p^{4} T^{20} - 31488379537 p^{5} T^{21} + 7964002603 p^{6} T^{22} - 1388796206 p^{7} T^{23} + 199820848 p^{8} T^{24} - 25926336 p^{9} T^{25} + 2994941 p^{10} T^{26} - 324160 p^{11} T^{27} + 31029 p^{12} T^{28} - 2582 p^{13} T^{29} + 212 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 14 T + 413 T^{2} + 4604 T^{3} + 76267 T^{4} + 694205 T^{5} + 8413162 T^{6} + 63584712 T^{7} + 618636975 T^{8} + 63584712 p T^{9} + 8413162 p^{2} T^{10} + 694205 p^{3} T^{11} + 76267 p^{4} T^{12} + 4604 p^{5} T^{13} + 413 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 47 T + 922 T^{2} - 9681 T^{3} + 66503 T^{4} - 611832 T^{5} + 8858926 T^{6} - 90690738 T^{7} + 468410737 T^{8} - 551178043 T^{9} + 1043724424 T^{10} - 74459299606 T^{11} - 672564337350 T^{12} + 29552182528030 T^{13} - 344235310532445 T^{14} + 2261699978951655 T^{15} - 14094412873142809 T^{16} + 2261699978951655 p T^{17} - 344235310532445 p^{2} T^{18} + 29552182528030 p^{3} T^{19} - 672564337350 p^{4} T^{20} - 74459299606 p^{5} T^{21} + 1043724424 p^{6} T^{22} - 551178043 p^{7} T^{23} + 468410737 p^{8} T^{24} - 90690738 p^{9} T^{25} + 8858926 p^{10} T^{26} - 611832 p^{11} T^{27} + 66503 p^{12} T^{28} - 9681 p^{13} T^{29} + 922 p^{14} T^{30} - 47 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 3 T + 286 T^{2} - 646 T^{3} + 29945 T^{4} - 46015 T^{5} + 774124 T^{6} + 3734493 T^{7} - 131323174 T^{8} + 1224631473 T^{9} - 14828677197 T^{10} + 97422990635 T^{11} - 412761091629 T^{12} - 2248890111925 T^{13} + 40926134912667 T^{14} - 960954418679112 T^{15} + 5071441106594915 T^{16} - 960954418679112 p T^{17} + 40926134912667 p^{2} T^{18} - 2248890111925 p^{3} T^{19} - 412761091629 p^{4} T^{20} + 97422990635 p^{5} T^{21} - 14828677197 p^{6} T^{22} + 1224631473 p^{7} T^{23} - 131323174 p^{8} T^{24} + 3734493 p^{9} T^{25} + 774124 p^{10} T^{26} - 46015 p^{11} T^{27} + 29945 p^{12} T^{28} - 646 p^{13} T^{29} + 286 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 27 T + 215 T^{2} + 975 T^{3} + 31512 T^{4} + 435953 T^{5} + 1423270 T^{6} + 7408836 T^{7} + 342374500 T^{8} + 2907158035 T^{9} + 820814802 T^{10} + 59447677815 T^{11} + 2513969015071 T^{12} + 14052675337334 T^{13} - 28577765752440 T^{14} + 654927537373441 T^{15} + 15610915258902883 T^{16} + 654927537373441 p T^{17} - 28577765752440 p^{2} T^{18} + 14052675337334 p^{3} T^{19} + 2513969015071 p^{4} T^{20} + 59447677815 p^{5} T^{21} + 820814802 p^{6} T^{22} + 2907158035 p^{7} T^{23} + 342374500 p^{8} T^{24} + 7408836 p^{9} T^{25} + 1423270 p^{10} T^{26} + 435953 p^{11} T^{27} + 31512 p^{12} T^{28} + 975 p^{13} T^{29} + 215 p^{14} T^{30} + 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 4 T + 122 T^{2} - 1323 T^{3} + 8794 T^{4} - 186735 T^{5} + 1339156 T^{6} - 12747029 T^{7} + 175459938 T^{8} - 1011323314 T^{9} + 13967610333 T^{10} - 130958235868 T^{11} + 834142462565 T^{12} - 10931775902867 T^{13} + 74275165684626 T^{14} - 616404933009968 T^{15} + 7462925071998941 T^{16} - 616404933009968 p T^{17} + 74275165684626 p^{2} T^{18} - 10931775902867 p^{3} T^{19} + 834142462565 p^{4} T^{20} - 130958235868 p^{5} T^{21} + 13967610333 p^{6} T^{22} - 1011323314 p^{7} T^{23} + 175459938 p^{8} T^{24} - 12747029 p^{9} T^{25} + 1339156 p^{10} T^{26} - 186735 p^{11} T^{27} + 8794 p^{12} T^{28} - 1323 p^{13} T^{29} + 122 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 66 T + 2227 T^{2} - 50589 T^{3} + 859690 T^{4} - 11469396 T^{5} + 121538813 T^{6} - 985009623 T^{7} + 5024528033 T^{8} + 6775537419 T^{9} - 515222986646 T^{10} + 6831932768019 T^{11} - 53920312090912 T^{12} + 179388092033028 T^{13} + 2254088567524271 T^{14} - 50724378664925232 T^{15} + 569775766642506967 T^{16} - 50724378664925232 p T^{17} + 2254088567524271 p^{2} T^{18} + 179388092033028 p^{3} T^{19} - 53920312090912 p^{4} T^{20} + 6831932768019 p^{5} T^{21} - 515222986646 p^{6} T^{22} + 6775537419 p^{7} T^{23} + 5024528033 p^{8} T^{24} - 985009623 p^{9} T^{25} + 121538813 p^{10} T^{26} - 11469396 p^{11} T^{27} + 859690 p^{12} T^{28} - 50589 p^{13} T^{29} + 2227 p^{14} T^{30} - 66 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 + 45 T + 864 T^{2} + 8470 T^{3} + 15135 T^{4} - 1057360 T^{5} - 22321570 T^{6} - 249163815 T^{7} - 1366994065 T^{8} + 6313401155 T^{9} + 251496956072 T^{10} + 3371807556715 T^{11} + 27342976010008 T^{12} + 98096537843660 T^{13} - 994176595604870 T^{14} - 24346296336943430 T^{15} - 285441917637819485 T^{16} - 24346296336943430 p T^{17} - 994176595604870 p^{2} T^{18} + 98096537843660 p^{3} T^{19} + 27342976010008 p^{4} T^{20} + 3371807556715 p^{5} T^{21} + 251496956072 p^{6} T^{22} + 6313401155 p^{7} T^{23} - 1366994065 p^{8} T^{24} - 249163815 p^{9} T^{25} - 22321570 p^{10} T^{26} - 1057360 p^{11} T^{27} + 15135 p^{12} T^{28} + 8470 p^{13} T^{29} + 864 p^{14} T^{30} + 45 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 5 T - 297 T^{2} + 3915 T^{3} + 30975 T^{4} - 873760 T^{5} + 2533715 T^{6} + 97281780 T^{7} - 1017280740 T^{8} - 2645039005 T^{9} + 127021064094 T^{10} - 7863541070 p T^{11} - 5212704210393 T^{12} + 126516387613215 T^{13} - 663916370798780 T^{14} - 5955264258124710 T^{15} + 125986934913464315 T^{16} - 5955264258124710 p T^{17} - 663916370798780 p^{2} T^{18} + 126516387613215 p^{3} T^{19} - 5212704210393 p^{4} T^{20} - 7863541070 p^{6} T^{21} + 127021064094 p^{6} T^{22} - 2645039005 p^{7} T^{23} - 1017280740 p^{8} T^{24} + 97281780 p^{9} T^{25} + 2533715 p^{10} T^{26} - 873760 p^{11} T^{27} + 30975 p^{12} T^{28} + 3915 p^{13} T^{29} - 297 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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
−2.69988074696866844961712285448, −2.66672017636104544404219765578, −2.48729434900947735039691823113, −2.44274334360022943562812110169, −2.38426378214641498003581281461, −2.27317585200507516249295615883, −2.22589262246928209144270900964, −2.18240810359331951146458374570, −2.09673296273316611756928129822, −2.00256552012782416260181688157, −1.96070892687470429992660681327, −1.94178903419024536809384449000, −1.83224188239811593877112208207, −1.42027112666139585942505497094, −1.28299778104705154950325850173, −1.15839211941855911541231710396, −1.14716060792361022326586376726, −1.11704862267825330930854308277, −1.04808766042099698830353131760, −1.01895255368290405248482382395, −0.70997346990553416014128117349, −0.32784402573034384239821893521, −0.19227561527600727662373566229, −0.11971637001640281560677048688, −0.06569444075054337971695682638, 0.06569444075054337971695682638, 0.11971637001640281560677048688, 0.19227561527600727662373566229, 0.32784402573034384239821893521, 0.70997346990553416014128117349, 1.01895255368290405248482382395, 1.04808766042099698830353131760, 1.11704862267825330930854308277, 1.14716060792361022326586376726, 1.15839211941855911541231710396, 1.28299778104705154950325850173, 1.42027112666139585942505497094, 1.83224188239811593877112208207, 1.94178903419024536809384449000, 1.96070892687470429992660681327, 2.00256552012782416260181688157, 2.09673296273316611756928129822, 2.18240810359331951146458374570, 2.22589262246928209144270900964, 2.27317585200507516249295615883, 2.38426378214641498003581281461, 2.44274334360022943562812110169, 2.48729434900947735039691823113, 2.66672017636104544404219765578, 2.69988074696866844961712285448
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.