Dirichlet series
| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s + 5-s + 16·6-s − 8·7-s − 30·8-s + 16·9-s − 4·10-s − 32·12-s + 12·13-s + 32·14-s − 4·15-s + 92·16-s − 4·17-s − 64·18-s + 4·19-s + 8·20-s + 32·21-s + 2·23-s + 120·24-s + 4·25-s − 48·26-s − 34·27-s − 64·28-s − 6·29-s + 16·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s + 0.447·5-s + 6.53·6-s − 3.02·7-s − 10.6·8-s + 16/3·9-s − 1.26·10-s − 9.23·12-s + 3.32·13-s + 8.55·14-s − 1.03·15-s + 23·16-s − 0.970·17-s − 15.0·18-s + 0.917·19-s + 1.78·20-s + 6.98·21-s + 0.417·23-s + 24.4·24-s + 4/5·25-s − 9.41·26-s − 6.54·27-s − 12.0·28-s − 1.11·29-s + 2.92·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(67^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(67^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.69010\times 10^{-6}\) |
| Root analytic conductor: | \(0.731435\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 67^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.008741823609\) |
| \(L(\frac12)\) | \(\approx\) | \(0.008741823609\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 67 | \( 1 + 22 T + 331 T^{2} + 2222 T^{3} - 2496 T^{4} - 327800 T^{5} - 4137748 T^{6} - 27737666 T^{7} + 14210054 T^{8} + 2219845276 T^{9} + 26753077319 T^{10} + 2219845276 p T^{11} + 14210054 p^{2} T^{12} - 27737666 p^{3} T^{13} - 4137748 p^{4} T^{14} - 327800 p^{5} T^{15} - 2496 p^{6} T^{16} + 2222 p^{7} T^{17} + 331 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) |
| good | 2 | \( ( 1 + p T + p T^{2} + 11 T^{3} + 9 p T^{4} + 7 p T^{5} + 47 T^{6} + 77 T^{7} + 15 p^{2} T^{8} + 15 p^{3} T^{9} + 241 T^{10} + 15 p^{4} T^{11} + 15 p^{4} T^{12} + 77 p^{3} T^{13} + 47 p^{4} T^{14} + 7 p^{6} T^{15} + 9 p^{7} T^{16} + 11 p^{7} T^{17} + p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} )^{2} \) |
| 3 | \( 1 + 4 T - 10 p T^{3} - 59 T^{4} + 32 T^{5} + 29 p^{2} T^{6} + 68 p T^{7} - 59 p^{2} T^{8} - 718 T^{9} + 584 p T^{10} + 3838 T^{11} - 3163 T^{12} - 15631 T^{13} - 5806 T^{14} + 32263 T^{15} + 18557 T^{16} - 120932 T^{17} - 61673 p T^{18} + 256030 T^{19} + 1051837 T^{20} + 256030 p T^{21} - 61673 p^{3} T^{22} - 120932 p^{3} T^{23} + 18557 p^{4} T^{24} + 32263 p^{5} T^{25} - 5806 p^{6} T^{26} - 15631 p^{7} T^{27} - 3163 p^{8} T^{28} + 3838 p^{9} T^{29} + 584 p^{11} T^{30} - 718 p^{11} T^{31} - 59 p^{14} T^{32} + 68 p^{14} T^{33} + 29 p^{16} T^{34} + 32 p^{15} T^{35} - 59 p^{16} T^{36} - 10 p^{18} T^{37} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 5 | \( 1 - T - 3 T^{2} + 13 T^{3} - 21 T^{4} - 57 T^{5} + 107 T^{6} + 22 T^{7} - 553 T^{8} + 19 p^{2} T^{9} + 1713 T^{10} - 154 T^{11} - 7887 T^{12} + 22097 T^{13} + 39468 T^{14} - 27527 p T^{15} + 21611 p T^{16} + 929329 T^{17} - 1506536 T^{18} - 836493 p T^{19} + 9370961 T^{20} - 836493 p^{2} T^{21} - 1506536 p^{2} T^{22} + 929329 p^{3} T^{23} + 21611 p^{5} T^{24} - 27527 p^{6} T^{25} + 39468 p^{6} T^{26} + 22097 p^{7} T^{27} - 7887 p^{8} T^{28} - 154 p^{9} T^{29} + 1713 p^{10} T^{30} + 19 p^{13} T^{31} - 553 p^{12} T^{32} + 22 p^{13} T^{33} + 107 p^{14} T^{34} - 57 p^{15} T^{35} - 21 p^{16} T^{36} + 13 p^{17} T^{37} - 3 p^{18} T^{38} - p^{19} T^{39} + p^{20} T^{40} \) | |
| 7 | \( 1 + 8 T + 4 T^{2} - 172 T^{3} - 689 T^{4} + 240 T^{5} + 9155 T^{6} + 22359 T^{7} - 26931 T^{8} - 225282 T^{9} - 648 p^{3} T^{10} + 952635 T^{11} + 1861901 T^{12} - 5533803 T^{13} - 18257254 T^{14} + 28130013 T^{15} + 201430179 T^{16} + 227527528 T^{17} - 517064292 T^{18} - 1733215754 T^{19} - 3131732651 T^{20} - 1733215754 p T^{21} - 517064292 p^{2} T^{22} + 227527528 p^{3} T^{23} + 201430179 p^{4} T^{24} + 28130013 p^{5} T^{25} - 18257254 p^{6} T^{26} - 5533803 p^{7} T^{27} + 1861901 p^{8} T^{28} + 952635 p^{9} T^{29} - 648 p^{13} T^{30} - 225282 p^{11} T^{31} - 26931 p^{12} T^{32} + 22359 p^{13} T^{33} + 9155 p^{14} T^{34} + 240 p^{15} T^{35} - 689 p^{16} T^{36} - 172 p^{17} T^{37} + 4 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 - 17 T^{2} - 2 p T^{3} + 410 T^{4} + 124 p T^{5} - 3681 T^{6} - 1744 p T^{7} + 23637 T^{8} + 30233 p T^{9} + 736891 T^{10} - 124771 p T^{11} - 9590697 T^{12} - 111844 p T^{13} + 156096790 T^{14} + 51699567 p T^{15} - 109141160 T^{16} - 386349913 p T^{17} - 3604903678 T^{18} + 328692050 p^{2} T^{19} + 222374833517 T^{20} + 328692050 p^{3} T^{21} - 3604903678 p^{2} T^{22} - 386349913 p^{4} T^{23} - 109141160 p^{4} T^{24} + 51699567 p^{6} T^{25} + 156096790 p^{6} T^{26} - 111844 p^{8} T^{27} - 9590697 p^{8} T^{28} - 124771 p^{10} T^{29} + 736891 p^{10} T^{30} + 30233 p^{12} T^{31} + 23637 p^{12} T^{32} - 1744 p^{14} T^{33} - 3681 p^{14} T^{34} + 124 p^{16} T^{35} + 410 p^{16} T^{36} - 2 p^{18} T^{37} - 17 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 12 T + 47 T^{2} + 59 T^{3} - 1111 T^{4} + 3273 T^{5} + 722 T^{6} - 2893 p T^{7} + 189433 T^{8} - 389264 T^{9} - 1949118 T^{10} + 13660229 T^{11} - 8495966 T^{12} - 155995849 T^{13} + 544850877 T^{14} - 659291528 T^{15} - 69654469 p T^{16} + 18371210775 T^{17} - 79662795190 T^{18} - 77273357650 T^{19} + 1293941144201 T^{20} - 77273357650 p T^{21} - 79662795190 p^{2} T^{22} + 18371210775 p^{3} T^{23} - 69654469 p^{5} T^{24} - 659291528 p^{5} T^{25} + 544850877 p^{6} T^{26} - 155995849 p^{7} T^{27} - 8495966 p^{8} T^{28} + 13660229 p^{9} T^{29} - 1949118 p^{10} T^{30} - 389264 p^{11} T^{31} + 189433 p^{12} T^{32} - 2893 p^{14} T^{33} + 722 p^{14} T^{34} + 3273 p^{15} T^{35} - 1111 p^{16} T^{36} + 59 p^{17} T^{37} + 47 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 + 4 T + 23 T^{2} - 98 T^{3} - 369 T^{4} - 3151 T^{5} + 4007 T^{6} - 1319 p T^{7} - 103 p T^{8} - 1085227 T^{9} + 1234321 T^{10} + 6875733 T^{11} + 110389896 T^{12} + 174608135 T^{13} + 929928804 T^{14} + 880136991 T^{15} + 7529772399 T^{16} - 27910442404 T^{17} - 212436030472 T^{18} - 1266205108674 T^{19} - 4984110309553 T^{20} - 1266205108674 p T^{21} - 212436030472 p^{2} T^{22} - 27910442404 p^{3} T^{23} + 7529772399 p^{4} T^{24} + 880136991 p^{5} T^{25} + 929928804 p^{6} T^{26} + 174608135 p^{7} T^{27} + 110389896 p^{8} T^{28} + 6875733 p^{9} T^{29} + 1234321 p^{10} T^{30} - 1085227 p^{11} T^{31} - 103 p^{13} T^{32} - 1319 p^{14} T^{33} + 4007 p^{14} T^{34} - 3151 p^{15} T^{35} - 369 p^{16} T^{36} - 98 p^{17} T^{37} + 23 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 4 T - 99 T^{2} + 450 T^{3} + 4930 T^{4} - 25630 T^{5} - 154704 T^{6} + 992871 T^{7} + 3102422 T^{8} - 30108226 T^{9} - 24373382 T^{10} + 774037077 T^{11} - 946766279 T^{12} - 17403723491 T^{13} + 52366874138 T^{14} + 332911659990 T^{15} - 1601291439259 T^{16} - 4871699159314 T^{17} + 38249643922226 T^{18} + 35195485874571 T^{19} - 778641426100757 T^{20} + 35195485874571 p T^{21} + 38249643922226 p^{2} T^{22} - 4871699159314 p^{3} T^{23} - 1601291439259 p^{4} T^{24} + 332911659990 p^{5} T^{25} + 52366874138 p^{6} T^{26} - 17403723491 p^{7} T^{27} - 946766279 p^{8} T^{28} + 774037077 p^{9} T^{29} - 24373382 p^{10} T^{30} - 30108226 p^{11} T^{31} + 3102422 p^{12} T^{32} + 992871 p^{13} T^{33} - 154704 p^{14} T^{34} - 25630 p^{15} T^{35} + 4930 p^{16} T^{36} + 450 p^{17} T^{37} - 99 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 - 2 T - 31 T^{2} - 2 T^{3} - 316 T^{4} - 730 T^{5} + 23885 T^{6} + 56030 T^{7} + 111765 T^{8} + 1555174 T^{9} - 7697464 T^{10} - 90791540 T^{11} - 166339093 T^{12} - 710988326 T^{13} + 1466397079 T^{14} + 54555820790 T^{15} + 187699252302 T^{16} + 363677360704 T^{17} + 265513112603 T^{18} - 780113930112 p T^{19} - 6333603331551 p T^{20} - 780113930112 p^{2} T^{21} + 265513112603 p^{2} T^{22} + 363677360704 p^{3} T^{23} + 187699252302 p^{4} T^{24} + 54555820790 p^{5} T^{25} + 1466397079 p^{6} T^{26} - 710988326 p^{7} T^{27} - 166339093 p^{8} T^{28} - 90791540 p^{9} T^{29} - 7697464 p^{10} T^{30} + 1555174 p^{11} T^{31} + 111765 p^{12} T^{32} + 56030 p^{13} T^{33} + 23885 p^{14} T^{34} - 730 p^{15} T^{35} - 316 p^{16} T^{36} - 2 p^{17} T^{37} - 31 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( ( 1 + 3 T + 199 T^{2} + 713 T^{3} + 18802 T^{4} + 75629 T^{5} + 1132092 T^{6} + 4785416 T^{7} + 48975416 T^{8} + 200939092 T^{9} + 1613044537 T^{10} + 200939092 p T^{11} + 48975416 p^{2} T^{12} + 4785416 p^{3} T^{13} + 1132092 p^{4} T^{14} + 75629 p^{5} T^{15} + 18802 p^{6} T^{16} + 713 p^{7} T^{17} + 199 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( 1 - 83 T^{2} - 242 T^{3} + 3167 T^{4} + 10076 T^{5} - 49294 T^{6} - 9174 T^{7} + 2662671 T^{8} - 10579954 T^{9} - 190623718 T^{10} - 110524964 T^{11} + 7676497944 T^{12} + 16556908742 T^{13} - 142991373724 T^{14} - 233035298221 T^{15} + 3484843896102 T^{16} - 18012821522905 T^{17} - 198649592552807 T^{18} + 361913915279730 T^{19} + 9209224689631077 T^{20} + 361913915279730 p T^{21} - 198649592552807 p^{2} T^{22} - 18012821522905 p^{3} T^{23} + 3484843896102 p^{4} T^{24} - 233035298221 p^{5} T^{25} - 142991373724 p^{6} T^{26} + 16556908742 p^{7} T^{27} + 7676497944 p^{8} T^{28} - 110524964 p^{9} T^{29} - 190623718 p^{10} T^{30} - 10579954 p^{11} T^{31} + 2662671 p^{12} T^{32} - 9174 p^{13} T^{33} - 49294 p^{14} T^{34} + 10076 p^{15} T^{35} + 3167 p^{16} T^{36} - 242 p^{17} T^{37} - 83 p^{18} T^{38} + p^{20} T^{40} \) | |
| 37 | \( ( 1 - 12 T + 316 T^{2} - 3084 T^{3} + 46667 T^{4} - 381194 T^{5} + 4228303 T^{6} - 29329830 T^{7} + 260171950 T^{8} - 1538744490 T^{9} + 11357216397 T^{10} - 1538744490 p T^{11} + 260171950 p^{2} T^{12} - 29329830 p^{3} T^{13} + 4228303 p^{4} T^{14} - 381194 p^{5} T^{15} + 46667 p^{6} T^{16} - 3084 p^{7} T^{17} + 316 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 41 | \( 1 + 6 T + 53 T^{2} - 126 T^{3} + 3438 T^{4} + 274 T^{5} + 5267 p T^{6} - 1167958 T^{7} + 9266235 T^{8} - 68497462 T^{9} + 17055122 p T^{10} - 3803191788 T^{11} + 34429742397 T^{12} - 233434296990 T^{13} + 1872904670541 T^{14} - 9842397577324 T^{15} + 95704533752026 T^{16} - 511660284853930 T^{17} + 3962217445172503 T^{18} - 20610298943986994 T^{19} + 191429154826750259 T^{20} - 20610298943986994 p T^{21} + 3962217445172503 p^{2} T^{22} - 511660284853930 p^{3} T^{23} + 95704533752026 p^{4} T^{24} - 9842397577324 p^{5} T^{25} + 1872904670541 p^{6} T^{26} - 233434296990 p^{7} T^{27} + 34429742397 p^{8} T^{28} - 3803191788 p^{9} T^{29} + 17055122 p^{11} T^{30} - 68497462 p^{11} T^{31} + 9266235 p^{12} T^{32} - 1167958 p^{13} T^{33} + 5267 p^{15} T^{34} + 274 p^{15} T^{35} + 3438 p^{16} T^{36} - 126 p^{17} T^{37} + 53 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 22 T + 204 T^{2} + 957 T^{3} - 3562 T^{4} - 117106 T^{5} - 1000576 T^{6} - 3545509 T^{7} + 16767103 T^{8} + 350694146 T^{9} + 2580775230 T^{10} + 9153007589 T^{11} - 16913942607 T^{12} - 457112932034 T^{13} - 3482790949105 T^{14} - 19230997072529 T^{15} - 73828075545149 T^{16} - 79165101297348 T^{17} + 2068199829600536 T^{18} + 31166898033135766 T^{19} + 260743546082835243 T^{20} + 31166898033135766 p T^{21} + 2068199829600536 p^{2} T^{22} - 79165101297348 p^{3} T^{23} - 73828075545149 p^{4} T^{24} - 19230997072529 p^{5} T^{25} - 3482790949105 p^{6} T^{26} - 457112932034 p^{7} T^{27} - 16913942607 p^{8} T^{28} + 9153007589 p^{9} T^{29} + 2580775230 p^{10} T^{30} + 350694146 p^{11} T^{31} + 16767103 p^{12} T^{32} - 3545509 p^{13} T^{33} - 1000576 p^{14} T^{34} - 117106 p^{15} T^{35} - 3562 p^{16} T^{36} + 957 p^{17} T^{37} + 204 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 - 16 T + 59 T^{2} + 763 T^{3} - 7287 T^{4} + 37722 T^{5} - 266690 T^{6} + 1038263 T^{7} + 15788946 T^{8} - 149727516 T^{9} + 990376049 T^{10} - 8555207838 T^{11} + 52049130822 T^{12} + 227201674743 T^{13} - 4593526238851 T^{14} + 36157276755294 T^{15} - 157905500270851 T^{16} + 7123295951920 p T^{17} + 4941730216479663 T^{18} - 57589493622900977 T^{19} + 488235542071738743 T^{20} - 57589493622900977 p T^{21} + 4941730216479663 p^{2} T^{22} + 7123295951920 p^{4} T^{23} - 157905500270851 p^{4} T^{24} + 36157276755294 p^{5} T^{25} - 4593526238851 p^{6} T^{26} + 227201674743 p^{7} T^{27} + 52049130822 p^{8} T^{28} - 8555207838 p^{9} T^{29} + 990376049 p^{10} T^{30} - 149727516 p^{11} T^{31} + 15788946 p^{12} T^{32} + 1038263 p^{13} T^{33} - 266690 p^{14} T^{34} + 37722 p^{15} T^{35} - 7287 p^{16} T^{36} + 763 p^{17} T^{37} + 59 p^{18} T^{38} - 16 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + T - 4 T^{2} - 451 T^{3} - 6289 T^{4} - 35468 T^{5} + 241403 T^{6} + 1516339 T^{7} + 26632309 T^{8} + 86512690 T^{9} - 444748842 T^{10} - 6283052633 T^{11} + 12365442854 T^{12} - 151186372850 T^{13} + 2134131841350 T^{14} - 15188997760031 T^{15} - 214981349448667 T^{16} - 1545221951437271 T^{17} + 1811553372758976 T^{18} + 58837901363967088 T^{19} + 1227375230596548727 T^{20} + 58837901363967088 p T^{21} + 1811553372758976 p^{2} T^{22} - 1545221951437271 p^{3} T^{23} - 214981349448667 p^{4} T^{24} - 15188997760031 p^{5} T^{25} + 2134131841350 p^{6} T^{26} - 151186372850 p^{7} T^{27} + 12365442854 p^{8} T^{28} - 6283052633 p^{9} T^{29} - 444748842 p^{10} T^{30} + 86512690 p^{11} T^{31} + 26632309 p^{12} T^{32} + 1516339 p^{13} T^{33} + 241403 p^{14} T^{34} - 35468 p^{15} T^{35} - 6289 p^{16} T^{36} - 451 p^{17} T^{37} - 4 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 26 T + 262 T^{2} + 1729 T^{3} + 15987 T^{4} + 135668 T^{5} + 489974 T^{6} - 2707517 T^{7} - 78190979 T^{8} - 1038026272 T^{9} - 9367978369 T^{10} - 76760184199 T^{11} - 667447801311 T^{12} - 4561850018135 T^{13} - 17941380028635 T^{14} - 7843869743551 T^{15} + 725247366801604 T^{16} + 11377947375984990 T^{17} + 125535850806252507 T^{18} + 1263117394649158289 T^{19} + 10992092800686404599 T^{20} + 1263117394649158289 p T^{21} + 125535850806252507 p^{2} T^{22} + 11377947375984990 p^{3} T^{23} + 725247366801604 p^{4} T^{24} - 7843869743551 p^{5} T^{25} - 17941380028635 p^{6} T^{26} - 4561850018135 p^{7} T^{27} - 667447801311 p^{8} T^{28} - 76760184199 p^{9} T^{29} - 9367978369 p^{10} T^{30} - 1038026272 p^{11} T^{31} - 78190979 p^{12} T^{32} - 2707517 p^{13} T^{33} + 489974 p^{14} T^{34} + 135668 p^{15} T^{35} + 15987 p^{16} T^{36} + 1729 p^{17} T^{37} + 262 p^{18} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 26 T + 192 T^{2} - 638 T^{3} - 11413 T^{4} + 140488 T^{5} + 3130580 T^{6} + 12804194 T^{7} - 117960453 T^{8} - 862837786 T^{9} + 9387017308 T^{10} + 128742346578 T^{11} + 25079137059 T^{12} - 7555730738060 T^{13} - 28728084291316 T^{14} + 299543866740754 T^{15} + 1920548972915287 T^{16} - 20711803769520246 T^{17} - 259975687678518810 T^{18} - 314130001690793428 T^{19} + 7331189391251145535 T^{20} - 314130001690793428 p T^{21} - 259975687678518810 p^{2} T^{22} - 20711803769520246 p^{3} T^{23} + 1920548972915287 p^{4} T^{24} + 299543866740754 p^{5} T^{25} - 28728084291316 p^{6} T^{26} - 7555730738060 p^{7} T^{27} + 25079137059 p^{8} T^{28} + 128742346578 p^{9} T^{29} + 9387017308 p^{10} T^{30} - 862837786 p^{11} T^{31} - 117960453 p^{12} T^{32} + 12804194 p^{13} T^{33} + 3130580 p^{14} T^{34} + 140488 p^{15} T^{35} - 11413 p^{16} T^{36} - 638 p^{17} T^{37} + 192 p^{18} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 20 T + 137 T^{2} + 814 T^{3} - 13486 T^{4} + 37235 T^{5} + 427973 T^{6} + 2877680 T^{7} - 77077218 T^{8} + 680575148 T^{9} + 1078134739 T^{10} - 31821019452 T^{11} + 235221271022 T^{12} + 904417697667 T^{13} + 7754653459435 T^{14} - 133474532939748 T^{15} + 1735912848222366 T^{16} - 53336103370168 T^{17} - 4362461753893346 T^{18} + 828068207173433314 T^{19} - 4583461853239821025 T^{20} + 828068207173433314 p T^{21} - 4362461753893346 p^{2} T^{22} - 53336103370168 p^{3} T^{23} + 1735912848222366 p^{4} T^{24} - 133474532939748 p^{5} T^{25} + 7754653459435 p^{6} T^{26} + 904417697667 p^{7} T^{27} + 235221271022 p^{8} T^{28} - 31821019452 p^{9} T^{29} + 1078134739 p^{10} T^{30} + 680575148 p^{11} T^{31} - 77077218 p^{12} T^{32} + 2877680 p^{13} T^{33} + 427973 p^{14} T^{34} + 37235 p^{15} T^{35} - 13486 p^{16} T^{36} + 814 p^{17} T^{37} + 137 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 + 55 T + 1520 T^{2} + 30822 T^{3} + 551396 T^{4} + 9113962 T^{5} + 138325395 T^{6} + 1949657688 T^{7} + 25981829118 T^{8} + 329580553005 T^{9} + 3989772510394 T^{10} + 46307341590169 T^{11} + 517230812918486 T^{12} + 5570455518168376 T^{13} + 794049606507005 p T^{14} + 583915144576721991 T^{15} + 5699573104386656653 T^{16} + 53951638137532009457 T^{17} + \)\(49\!\cdots\!69\)\( T^{18} + \)\(44\!\cdots\!11\)\( T^{19} + \)\(38\!\cdots\!39\)\( T^{20} + \)\(44\!\cdots\!11\)\( p T^{21} + \)\(49\!\cdots\!69\)\( p^{2} T^{22} + 53951638137532009457 p^{3} T^{23} + 5699573104386656653 p^{4} T^{24} + 583915144576721991 p^{5} T^{25} + 794049606507005 p^{7} T^{26} + 5570455518168376 p^{7} T^{27} + 517230812918486 p^{8} T^{28} + 46307341590169 p^{9} T^{29} + 3989772510394 p^{10} T^{30} + 329580553005 p^{11} T^{31} + 25981829118 p^{12} T^{32} + 1949657688 p^{13} T^{33} + 138325395 p^{14} T^{34} + 9113962 p^{15} T^{35} + 551396 p^{16} T^{36} + 30822 p^{17} T^{37} + 1520 p^{18} T^{38} + 55 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 34 T + 443 T^{2} + 1283 T^{3} - 36256 T^{4} - 333724 T^{5} + 4009518 T^{6} + 103111316 T^{7} + 695197626 T^{8} - 2910495614 T^{9} - 67290369735 T^{10} + 43098932712 T^{11} + 9130956035844 T^{12} + 83147191333094 T^{13} - 14672120223476 T^{14} - 4725697482183805 T^{15} - 4729285251766448 T^{16} + 558635335034362625 T^{17} + 5270951227654411805 T^{18} + 810877419188693719 T^{19} - \)\(25\!\cdots\!73\)\( T^{20} + 810877419188693719 p T^{21} + 5270951227654411805 p^{2} T^{22} + 558635335034362625 p^{3} T^{23} - 4729285251766448 p^{4} T^{24} - 4725697482183805 p^{5} T^{25} - 14672120223476 p^{6} T^{26} + 83147191333094 p^{7} T^{27} + 9130956035844 p^{8} T^{28} + 43098932712 p^{9} T^{29} - 67290369735 p^{10} T^{30} - 2910495614 p^{11} T^{31} + 695197626 p^{12} T^{32} + 103111316 p^{13} T^{33} + 4009518 p^{14} T^{34} - 333724 p^{15} T^{35} - 36256 p^{16} T^{36} + 1283 p^{17} T^{37} + 443 p^{18} T^{38} + 34 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 56 T + 1351 T^{2} - 19174 T^{3} + 194930 T^{4} - 1648232 T^{5} + 12343514 T^{6} - 109590041 T^{7} + 1491415464 T^{8} - 19679219978 T^{9} + 207931314174 T^{10} - 1749484467225 T^{11} + 10150571641911 T^{12} - 33756963735443 T^{13} + 158657766881040 T^{14} - 3123650093370206 T^{15} + 49426847666163323 T^{16} - 456214166302240196 T^{17} + 1136208274133542676 T^{18} + 22438148133658431185 T^{19} - \)\(31\!\cdots\!21\)\( T^{20} + 22438148133658431185 p T^{21} + 1136208274133542676 p^{2} T^{22} - 456214166302240196 p^{3} T^{23} + 49426847666163323 p^{4} T^{24} - 3123650093370206 p^{5} T^{25} + 158657766881040 p^{6} T^{26} - 33756963735443 p^{7} T^{27} + 10150571641911 p^{8} T^{28} - 1749484467225 p^{9} T^{29} + 207931314174 p^{10} T^{30} - 19679219978 p^{11} T^{31} + 1491415464 p^{12} T^{32} - 109590041 p^{13} T^{33} + 12343514 p^{14} T^{34} - 1648232 p^{15} T^{35} + 194930 p^{16} T^{36} - 19174 p^{17} T^{37} + 1351 p^{18} T^{38} - 56 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 3 T - 163 T^{2} - 3081 T^{3} + 8175 T^{4} + 544912 T^{5} + 3201711 T^{6} - 52658195 T^{7} - 788152620 T^{8} + 605047414 T^{9} + 99048742483 T^{10} + 583390192437 T^{11} - 6402830234474 T^{12} - 100364322717580 T^{13} + 12119515390464 T^{14} + 9868922137180465 T^{15} + 49748030020733902 T^{16} - 612933280717367425 T^{17} - 7719970238091240596 T^{18} + 19747075809070909368 T^{19} + \)\(76\!\cdots\!51\)\( T^{20} + 19747075809070909368 p T^{21} - 7719970238091240596 p^{2} T^{22} - 612933280717367425 p^{3} T^{23} + 49748030020733902 p^{4} T^{24} + 9868922137180465 p^{5} T^{25} + 12119515390464 p^{6} T^{26} - 100364322717580 p^{7} T^{27} - 6402830234474 p^{8} T^{28} + 583390192437 p^{9} T^{29} + 99048742483 p^{10} T^{30} + 605047414 p^{11} T^{31} - 788152620 p^{12} T^{32} - 52658195 p^{13} T^{33} + 3201711 p^{14} T^{34} + 544912 p^{15} T^{35} + 8175 p^{16} T^{36} - 3081 p^{17} T^{37} - 163 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( ( 1 + 7 T + 565 T^{2} + 3641 T^{3} + 159998 T^{4} + 939501 T^{5} + 29800356 T^{6} + 158828950 T^{7} + 4105222848 T^{8} + 19795183760 T^{9} + 444524390339 T^{10} + 19795183760 p T^{11} + 4105222848 p^{2} T^{12} + 158828950 p^{3} T^{13} + 29800356 p^{4} T^{14} + 939501 p^{5} T^{15} + 159998 p^{6} T^{16} + 3641 p^{7} T^{17} + 565 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.29818899617109570060674391276, −4.13957100561784371729757549608, −4.10514717379487590331036301718, −3.85789049749485178226937806643, −3.82282857429771028782155094776, −3.64797269983527650018884757634, −3.64124888433491507815836276525, −3.47385461564117731945823248618, −3.34308881727280948830198198981, −3.30427847986417632034369316241, −3.29184118294998280142534453394, −3.23241870794106484310981839179, −3.06291897480775723430916687992, −3.06195137789712560775571887286, −2.94286208412495745392370679657, −2.63819765820075461372038159812, −2.62458566673581514342219196113, −2.47088745914474995044291560311, −2.37419043264165879908510815889, −2.32548765358519133328041546853, −1.83313592674506673222455824036, −1.59301527467922879782683336589, −1.44204479357260207131182942422, −1.02168903398244016322867281292, −0.910365863101594388795462763221, 0.910365863101594388795462763221, 1.02168903398244016322867281292, 1.44204479357260207131182942422, 1.59301527467922879782683336589, 1.83313592674506673222455824036, 2.32548765358519133328041546853, 2.37419043264165879908510815889, 2.47088745914474995044291560311, 2.62458566673581514342219196113, 2.63819765820075461372038159812, 2.94286208412495745392370679657, 3.06195137789712560775571887286, 3.06291897480775723430916687992, 3.23241870794106484310981839179, 3.29184118294998280142534453394, 3.30427847986417632034369316241, 3.34308881727280948830198198981, 3.47385461564117731945823248618, 3.64124888433491507815836276525, 3.64797269983527650018884757634, 3.82282857429771028782155094776, 3.85789049749485178226937806643, 4.10514717379487590331036301718, 4.13957100561784371729757549608, 4.29818899617109570060674391276
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.