Dirichlet series
| L(s) = 1 | − 2·2-s + 8·3-s − 9·4-s + 8·5-s − 16·6-s + 20·8-s + 10·9-s − 16·10-s − 72·12-s + 12·13-s + 64·15-s + 35·16-s + 8·17-s − 20·18-s + 36·19-s − 72·20-s − 12·23-s + 160·24-s − 8·25-s − 24·26-s − 88·27-s + 4·29-s − 128·30-s + 80·31-s − 86·32-s − 16·34-s − 90·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4.61·3-s − 9/2·4-s + 3.57·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s − 5.05·10-s − 20.7·12-s + 3.32·13-s + 16.5·15-s + 35/4·16-s + 1.94·17-s − 4.71·18-s + 8.25·19-s − 16.0·20-s − 2.50·23-s + 32.6·24-s − 8/5·25-s − 4.70·26-s − 16.9·27-s + 0.742·29-s − 23.3·30-s + 14.3·31-s − 15.2·32-s − 2.74·34-s − 15·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{40} \cdot 41^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{40} \cdot 41^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(7^{40} \cdot 41^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.27391\times 10^{24}\) |
| Root analytic conductor: | \(4.00523\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 7^{40} \cdot 41^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(402.7329243\) |
| \(L(\frac12)\) | \(\approx\) | \(402.7329243\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 41 | \( ( 1 - T )^{20} \) | |
| good | 2 | \( 1 + p T + 13 T^{2} + 3 p^{3} T^{3} + 45 p T^{4} + 19 p^{3} T^{5} + 55 p^{3} T^{6} + 343 p T^{7} + 1713 T^{8} + 313 p^{3} T^{9} + 5667 T^{10} + 987 p^{3} T^{11} + 16605 T^{12} + 11135 p T^{13} + 11041 p^{2} T^{14} + 3577 p^{4} T^{15} + 53949 p T^{16} + 33793 p^{2} T^{17} + 243189 T^{18} + 146975 p T^{19} + 506133 T^{20} + 146975 p^{2} T^{21} + 243189 p^{2} T^{22} + 33793 p^{5} T^{23} + 53949 p^{5} T^{24} + 3577 p^{9} T^{25} + 11041 p^{8} T^{26} + 11135 p^{8} T^{27} + 16605 p^{8} T^{28} + 987 p^{12} T^{29} + 5667 p^{10} T^{30} + 313 p^{14} T^{31} + 1713 p^{12} T^{32} + 343 p^{14} T^{33} + 55 p^{17} T^{34} + 19 p^{18} T^{35} + 45 p^{17} T^{36} + 3 p^{20} T^{37} + 13 p^{18} T^{38} + p^{20} T^{39} + p^{20} T^{40} \) |
| 3 | \( 1 - 8 T + 2 p^{3} T^{2} - 88 p T^{3} + 1150 T^{4} - 4324 T^{5} + 14960 T^{6} - 47228 T^{7} + 15533 p^{2} T^{8} - 43028 p^{2} T^{9} + 339190 p T^{10} - 2536820 T^{11} + 6044272 T^{12} - 13775228 T^{13} + 30152662 T^{14} - 63452032 T^{15} + 14295083 p^{2} T^{16} - 27951088 p^{2} T^{17} + 474850744 T^{18} - 865817032 T^{19} + 1525440191 T^{20} - 865817032 p T^{21} + 474850744 p^{2} T^{22} - 27951088 p^{5} T^{23} + 14295083 p^{6} T^{24} - 63452032 p^{5} T^{25} + 30152662 p^{6} T^{26} - 13775228 p^{7} T^{27} + 6044272 p^{8} T^{28} - 2536820 p^{9} T^{29} + 339190 p^{11} T^{30} - 43028 p^{13} T^{31} + 15533 p^{14} T^{32} - 47228 p^{13} T^{33} + 14960 p^{14} T^{34} - 4324 p^{15} T^{35} + 1150 p^{16} T^{36} - 88 p^{18} T^{37} + 2 p^{21} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 5 | \( 1 - 8 T + 72 T^{2} - 384 T^{3} + 17 p^{3} T^{4} - 9004 T^{5} + 39052 T^{6} - 142704 T^{7} + 4246 p^{3} T^{8} - 1742364 T^{9} + 5790782 T^{10} - 3486416 p T^{11} + 52894486 T^{12} - 147702968 T^{13} + 414505318 T^{14} - 1081511548 T^{15} + 2830571057 T^{16} - 6931611904 T^{17} + 17006679256 T^{18} - 39174910428 T^{19} + 90356308826 T^{20} - 39174910428 p T^{21} + 17006679256 p^{2} T^{22} - 6931611904 p^{3} T^{23} + 2830571057 p^{4} T^{24} - 1081511548 p^{5} T^{25} + 414505318 p^{6} T^{26} - 147702968 p^{7} T^{27} + 52894486 p^{8} T^{28} - 3486416 p^{10} T^{29} + 5790782 p^{10} T^{30} - 1742364 p^{11} T^{31} + 4246 p^{15} T^{32} - 142704 p^{13} T^{33} + 39052 p^{14} T^{34} - 9004 p^{15} T^{35} + 17 p^{19} T^{36} - 384 p^{17} T^{37} + 72 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 118 T^{2} - 48 T^{3} + 6940 T^{4} - 4968 T^{5} + 274334 T^{6} - 253504 T^{7} + 8230854 T^{8} - 8623240 T^{9} + 199488002 T^{10} - 221588984 T^{11} + 4048815422 T^{12} - 4586283352 T^{13} + 70400158082 T^{14} - 79217170200 T^{15} + 1064372283529 T^{16} - 1165843530480 T^{17} + 14122131561000 T^{18} - 14793883394904 T^{19} + 165270460487788 T^{20} - 14793883394904 p T^{21} + 14122131561000 p^{2} T^{22} - 1165843530480 p^{3} T^{23} + 1064372283529 p^{4} T^{24} - 79217170200 p^{5} T^{25} + 70400158082 p^{6} T^{26} - 4586283352 p^{7} T^{27} + 4048815422 p^{8} T^{28} - 221588984 p^{9} T^{29} + 199488002 p^{10} T^{30} - 8623240 p^{11} T^{31} + 8230854 p^{12} T^{32} - 253504 p^{13} T^{33} + 274334 p^{14} T^{34} - 4968 p^{15} T^{35} + 6940 p^{16} T^{36} - 48 p^{17} T^{37} + 118 p^{18} T^{38} + p^{20} T^{40} \) | |
| 13 | \( 1 - 12 T + 162 T^{2} - 1348 T^{3} + 11391 T^{4} - 76244 T^{5} + 503238 T^{6} - 2860300 T^{7} + 1220094 p T^{8} - 78595288 T^{9} + 378798880 T^{10} - 1660181216 T^{11} + 7078847534 T^{12} - 27663926544 T^{13} + 105678763228 T^{14} - 371720204456 T^{15} + 1298297800421 T^{16} - 4225917320468 T^{17} + 14249311356054 T^{18} - 46462289478260 T^{19} + 167577058110571 T^{20} - 46462289478260 p T^{21} + 14249311356054 p^{2} T^{22} - 4225917320468 p^{3} T^{23} + 1298297800421 p^{4} T^{24} - 371720204456 p^{5} T^{25} + 105678763228 p^{6} T^{26} - 27663926544 p^{7} T^{27} + 7078847534 p^{8} T^{28} - 1660181216 p^{9} T^{29} + 378798880 p^{10} T^{30} - 78595288 p^{11} T^{31} + 1220094 p^{13} T^{32} - 2860300 p^{13} T^{33} + 503238 p^{14} T^{34} - 76244 p^{15} T^{35} + 11391 p^{16} T^{36} - 1348 p^{17} T^{37} + 162 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 8 T + 188 T^{2} - 1404 T^{3} + 18085 T^{4} - 124924 T^{5} + 1175044 T^{6} - 7503260 T^{7} + 57635920 T^{8} - 341058196 T^{9} + 2264483658 T^{10} - 12456956532 T^{11} + 73852475484 T^{12} - 378800564684 T^{13} + 2045377570258 T^{14} - 9805660532232 T^{15} + 48824768187985 T^{16} - 12889937305720 p T^{17} + 1014001283185446 T^{18} - 250751314639304 p T^{19} + 18417422467269295 T^{20} - 250751314639304 p^{2} T^{21} + 1014001283185446 p^{2} T^{22} - 12889937305720 p^{4} T^{23} + 48824768187985 p^{4} T^{24} - 9805660532232 p^{5} T^{25} + 2045377570258 p^{6} T^{26} - 378800564684 p^{7} T^{27} + 73852475484 p^{8} T^{28} - 12456956532 p^{9} T^{29} + 2264483658 p^{10} T^{30} - 341058196 p^{11} T^{31} + 57635920 p^{12} T^{32} - 7503260 p^{13} T^{33} + 1175044 p^{14} T^{34} - 124924 p^{15} T^{35} + 18085 p^{16} T^{36} - 1404 p^{17} T^{37} + 188 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 - 36 T + 850 T^{2} - 14832 T^{3} + 213196 T^{4} - 2615244 T^{5} + 28297470 T^{6} - 274680524 T^{7} + 2429084823 T^{8} - 19761847584 T^{9} + 149213238548 T^{10} - 55378217648 p T^{11} + 6968334560366 T^{12} - 43526384683348 T^{13} + 257403895691492 T^{14} - 1445357420049560 T^{15} + 7725859686222233 T^{16} - 39386485891561008 T^{17} + 191806819405103800 T^{18} - 893159691307611332 T^{19} + 3979674328886919883 T^{20} - 893159691307611332 p T^{21} + 191806819405103800 p^{2} T^{22} - 39386485891561008 p^{3} T^{23} + 7725859686222233 p^{4} T^{24} - 1445357420049560 p^{5} T^{25} + 257403895691492 p^{6} T^{26} - 43526384683348 p^{7} T^{27} + 6968334560366 p^{8} T^{28} - 55378217648 p^{10} T^{29} + 149213238548 p^{10} T^{30} - 19761847584 p^{11} T^{31} + 2429084823 p^{12} T^{32} - 274680524 p^{13} T^{33} + 28297470 p^{14} T^{34} - 2615244 p^{15} T^{35} + 213196 p^{16} T^{36} - 14832 p^{17} T^{37} + 850 p^{18} T^{38} - 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + 12 T + 264 T^{2} + 2536 T^{3} + 33322 T^{4} + 269640 T^{5} + 2694120 T^{6} + 822212 p T^{7} + 6838029 p T^{8} + 975374992 T^{9} + 7077829886 T^{10} + 39272677016 T^{11} + 256551997944 T^{12} + 1286998279004 T^{13} + 7771548029638 T^{14} + 35698353619756 T^{15} + 204851440961633 T^{16} + 879534542310616 T^{17} + 4938010042319026 T^{18} + 20433799859147148 T^{19} + 114483924767861047 T^{20} + 20433799859147148 p T^{21} + 4938010042319026 p^{2} T^{22} + 879534542310616 p^{3} T^{23} + 204851440961633 p^{4} T^{24} + 35698353619756 p^{5} T^{25} + 7771548029638 p^{6} T^{26} + 1286998279004 p^{7} T^{27} + 256551997944 p^{8} T^{28} + 39272677016 p^{9} T^{29} + 7077829886 p^{10} T^{30} + 975374992 p^{11} T^{31} + 6838029 p^{13} T^{32} + 822212 p^{14} T^{33} + 2694120 p^{14} T^{34} + 269640 p^{15} T^{35} + 33322 p^{16} T^{36} + 2536 p^{17} T^{37} + 264 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 4 T + 220 T^{2} - 244 T^{3} + 22487 T^{4} + 32268 T^{5} + 1632064 T^{6} + 5509328 T^{7} + 102545570 T^{8} + 438777020 T^{9} + 5759544558 T^{10} + 25922431552 T^{11} + 281410604106 T^{12} + 1289331313248 T^{13} + 11973920371798 T^{14} + 55252548869188 T^{15} + 453676297490541 T^{16} + 2046976880657540 T^{17} + 15492456976010384 T^{18} + 2294694751456120 p T^{19} + 475112880598256414 T^{20} + 2294694751456120 p^{2} T^{21} + 15492456976010384 p^{2} T^{22} + 2046976880657540 p^{3} T^{23} + 453676297490541 p^{4} T^{24} + 55252548869188 p^{5} T^{25} + 11973920371798 p^{6} T^{26} + 1289331313248 p^{7} T^{27} + 281410604106 p^{8} T^{28} + 25922431552 p^{9} T^{29} + 5759544558 p^{10} T^{30} + 438777020 p^{11} T^{31} + 102545570 p^{12} T^{32} + 5509328 p^{13} T^{33} + 1632064 p^{14} T^{34} + 32268 p^{15} T^{35} + 22487 p^{16} T^{36} - 244 p^{17} T^{37} + 220 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 80 T + 3462 T^{2} - 105312 T^{3} + 80747 p T^{4} - 49192900 T^{5} + 26709040 p T^{6} - 12224645744 T^{7} + 161090092794 T^{8} - 1919482258212 T^{9} + 20892465795374 T^{10} - 209404899706928 T^{11} + 1945347509950614 T^{12} - 16838784245902600 T^{13} + 136391722177006866 T^{14} - 1037364865659009948 T^{15} + 7429138295090538405 T^{16} - 50204137294293347096 T^{17} + \)\(32\!\cdots\!46\)\( T^{18} - \)\(19\!\cdots\!32\)\( T^{19} + \)\(11\!\cdots\!02\)\( T^{20} - \)\(19\!\cdots\!32\)\( p T^{21} + \)\(32\!\cdots\!46\)\( p^{2} T^{22} - 50204137294293347096 p^{3} T^{23} + 7429138295090538405 p^{4} T^{24} - 1037364865659009948 p^{5} T^{25} + 136391722177006866 p^{6} T^{26} - 16838784245902600 p^{7} T^{27} + 1945347509950614 p^{8} T^{28} - 209404899706928 p^{9} T^{29} + 20892465795374 p^{10} T^{30} - 1919482258212 p^{11} T^{31} + 161090092794 p^{12} T^{32} - 12224645744 p^{13} T^{33} + 26709040 p^{15} T^{34} - 49192900 p^{15} T^{35} + 80747 p^{17} T^{36} - 105312 p^{17} T^{37} + 3462 p^{18} T^{38} - 80 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 - 4 T + 374 T^{2} - 1660 T^{3} + 70523 T^{4} - 338700 T^{5} + 8965224 T^{6} - 45621500 T^{7} + 865632083 T^{8} - 4582276244 T^{9} + 67710044698 T^{10} - 366407864592 T^{11} + 4461496150465 T^{12} - 24259552100896 T^{13} + 253783168848574 T^{14} - 1363596231230644 T^{15} + 12654287455652152 T^{16} - 66128635855706968 T^{17} + 558189785908848346 T^{18} - 2794489891421497944 T^{19} + 21888197222810126128 T^{20} - 2794489891421497944 p T^{21} + 558189785908848346 p^{2} T^{22} - 66128635855706968 p^{3} T^{23} + 12654287455652152 p^{4} T^{24} - 1363596231230644 p^{5} T^{25} + 253783168848574 p^{6} T^{26} - 24259552100896 p^{7} T^{27} + 4461496150465 p^{8} T^{28} - 366407864592 p^{9} T^{29} + 67710044698 p^{10} T^{30} - 4582276244 p^{11} T^{31} + 865632083 p^{12} T^{32} - 45621500 p^{13} T^{33} + 8965224 p^{14} T^{34} - 338700 p^{15} T^{35} + 70523 p^{16} T^{36} - 1660 p^{17} T^{37} + 374 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 506 T^{2} - 260 T^{3} + 2883 p T^{4} - 108184 T^{5} + 19695068 T^{6} - 20707044 T^{7} + 2291957422 T^{8} - 2384282952 T^{9} + 209137290724 T^{10} - 179028757692 T^{11} + 15642979002248 T^{12} - 8541669931816 T^{13} + 991388610021174 T^{14} - 184064643392124 T^{15} + 54749735683103433 T^{16} + 7151756916697124 T^{17} + 2700979966466963536 T^{18} + 835585910411141060 T^{19} + \)\(12\!\cdots\!59\)\( T^{20} + 835585910411141060 p T^{21} + 2700979966466963536 p^{2} T^{22} + 7151756916697124 p^{3} T^{23} + 54749735683103433 p^{4} T^{24} - 184064643392124 p^{5} T^{25} + 991388610021174 p^{6} T^{26} - 8541669931816 p^{7} T^{27} + 15642979002248 p^{8} T^{28} - 179028757692 p^{9} T^{29} + 209137290724 p^{10} T^{30} - 2384282952 p^{11} T^{31} + 2291957422 p^{12} T^{32} - 20707044 p^{13} T^{33} + 19695068 p^{14} T^{34} - 108184 p^{15} T^{35} + 2883 p^{17} T^{36} - 260 p^{17} T^{37} + 506 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 - 32 T + 1054 T^{2} - 22228 T^{3} + 446949 T^{4} - 7249148 T^{5} + 111313470 T^{6} - 1492042972 T^{7} + 19005714531 T^{8} - 219398395704 T^{9} + 2420133399230 T^{10} - 24711993291428 T^{11} + 242401505158173 T^{12} - 2230767047947508 T^{13} + 19813104531813714 T^{14} - 166592131922405944 T^{15} + 1357040640856503412 T^{16} - 10528006969596451932 T^{17} + 79348272356280509140 T^{18} - \)\(57\!\cdots\!80\)\( T^{19} + \)\(40\!\cdots\!48\)\( T^{20} - \)\(57\!\cdots\!80\)\( p T^{21} + 79348272356280509140 p^{2} T^{22} - 10528006969596451932 p^{3} T^{23} + 1357040640856503412 p^{4} T^{24} - 166592131922405944 p^{5} T^{25} + 19813104531813714 p^{6} T^{26} - 2230767047947508 p^{7} T^{27} + 242401505158173 p^{8} T^{28} - 24711993291428 p^{9} T^{29} + 2420133399230 p^{10} T^{30} - 219398395704 p^{11} T^{31} + 19005714531 p^{12} T^{32} - 1492042972 p^{13} T^{33} + 111313470 p^{14} T^{34} - 7249148 p^{15} T^{35} + 446949 p^{16} T^{36} - 22228 p^{17} T^{37} + 1054 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 - 4 T + 584 T^{2} - 1300 T^{3} + 169427 T^{4} - 155500 T^{5} + 33254640 T^{6} + 990320 T^{7} + 4979714798 T^{8} + 3500809188 T^{9} + 602915899774 T^{10} + 702863596368 T^{11} + 60989810945046 T^{12} + 89141676459680 T^{13} + 5259984505732254 T^{14} + 8541232596335756 T^{15} + 391701482247910545 T^{16} + 657449612733733044 T^{17} + 25386378216055146108 T^{18} + 41761549760320485088 T^{19} + \)\(14\!\cdots\!86\)\( T^{20} + 41761549760320485088 p T^{21} + 25386378216055146108 p^{2} T^{22} + 657449612733733044 p^{3} T^{23} + 391701482247910545 p^{4} T^{24} + 8541232596335756 p^{5} T^{25} + 5259984505732254 p^{6} T^{26} + 89141676459680 p^{7} T^{27} + 60989810945046 p^{8} T^{28} + 702863596368 p^{9} T^{29} + 602915899774 p^{10} T^{30} + 3500809188 p^{11} T^{31} + 4979714798 p^{12} T^{32} + 990320 p^{13} T^{33} + 33254640 p^{14} T^{34} - 155500 p^{15} T^{35} + 169427 p^{16} T^{36} - 1300 p^{17} T^{37} + 584 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 - 32 T + 1152 T^{2} - 25224 T^{3} + 554909 T^{4} - 9586340 T^{5} + 162283028 T^{6} - 2365075960 T^{7} + 33525032782 T^{8} - 427758810284 T^{9} + 5295067780982 T^{10} - 60491889842272 T^{11} + 669839688701614 T^{12} - 6949954494440600 T^{13} + 69873828708462014 T^{14} - 664531361577479268 T^{15} + 6123656874409519873 T^{16} - 53692784608363601920 T^{17} + \)\(45\!\cdots\!08\)\( T^{18} - \)\(36\!\cdots\!52\)\( T^{19} + \)\(29\!\cdots\!98\)\( T^{20} - \)\(36\!\cdots\!52\)\( p T^{21} + \)\(45\!\cdots\!08\)\( p^{2} T^{22} - 53692784608363601920 p^{3} T^{23} + 6123656874409519873 p^{4} T^{24} - 664531361577479268 p^{5} T^{25} + 69873828708462014 p^{6} T^{26} - 6949954494440600 p^{7} T^{27} + 669839688701614 p^{8} T^{28} - 60491889842272 p^{9} T^{29} + 5295067780982 p^{10} T^{30} - 427758810284 p^{11} T^{31} + 33525032782 p^{12} T^{32} - 2365075960 p^{13} T^{33} + 162283028 p^{14} T^{34} - 9586340 p^{15} T^{35} + 554909 p^{16} T^{36} - 25224 p^{17} T^{37} + 1152 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 44 T + 1650 T^{2} - 43108 T^{3} + 1001356 T^{4} - 19510876 T^{5} + 347743822 T^{6} - 5531538940 T^{7} + 81951261982 T^{8} - 1116220971516 T^{9} + 14331179981302 T^{10} - 172041573544228 T^{11} + 1962985818315698 T^{12} - 21163810901955204 T^{13} + 218124016507465870 T^{14} - 2138725833819101548 T^{15} + 20124979775029100817 T^{16} - \)\(18\!\cdots\!44\)\( T^{17} + \)\(15\!\cdots\!08\)\( T^{18} - \)\(12\!\cdots\!40\)\( T^{19} + \)\(10\!\cdots\!48\)\( T^{20} - \)\(12\!\cdots\!40\)\( p T^{21} + \)\(15\!\cdots\!08\)\( p^{2} T^{22} - \)\(18\!\cdots\!44\)\( p^{3} T^{23} + 20124979775029100817 p^{4} T^{24} - 2138725833819101548 p^{5} T^{25} + 218124016507465870 p^{6} T^{26} - 21163810901955204 p^{7} T^{27} + 1962985818315698 p^{8} T^{28} - 172041573544228 p^{9} T^{29} + 14331179981302 p^{10} T^{30} - 1116220971516 p^{11} T^{31} + 81951261982 p^{12} T^{32} - 5531538940 p^{13} T^{33} + 347743822 p^{14} T^{34} - 19510876 p^{15} T^{35} + 1001356 p^{16} T^{36} - 43108 p^{17} T^{37} + 1650 p^{18} T^{38} - 44 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 + 4 T + 508 T^{2} + 2676 T^{3} + 119783 T^{4} + 785404 T^{5} + 17867516 T^{6} + 138787264 T^{7} + 1993805958 T^{8} + 17121061356 T^{9} + 191693784190 T^{10} + 1640777965272 T^{11} + 17373926578870 T^{12} + 134842203434952 T^{13} + 1474761640508702 T^{14} + 10353104142184508 T^{15} + 112106275921829321 T^{16} + 773704682842873564 T^{17} + 7683496896669878492 T^{18} + 55515909626186811336 T^{19} + \)\(51\!\cdots\!50\)\( T^{20} + 55515909626186811336 p T^{21} + 7683496896669878492 p^{2} T^{22} + 773704682842873564 p^{3} T^{23} + 112106275921829321 p^{4} T^{24} + 10353104142184508 p^{5} T^{25} + 1474761640508702 p^{6} T^{26} + 134842203434952 p^{7} T^{27} + 17373926578870 p^{8} T^{28} + 1640777965272 p^{9} T^{29} + 191693784190 p^{10} T^{30} + 17121061356 p^{11} T^{31} + 1993805958 p^{12} T^{32} + 138787264 p^{13} T^{33} + 17867516 p^{14} T^{34} + 785404 p^{15} T^{35} + 119783 p^{16} T^{36} + 2676 p^{17} T^{37} + 508 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 - 8 T + 762 T^{2} - 3576 T^{3} + 275000 T^{4} - 597800 T^{5} + 66381154 T^{6} - 16110776 T^{7} + 12386993870 T^{8} + 14552435320 T^{9} + 1904761586734 T^{10} + 4109636225576 T^{11} + 248644265143658 T^{12} + 692801064948840 T^{13} + 28008670199006558 T^{14} + 87750614257749208 T^{15} + 2749348361125969377 T^{16} + 8953364907147647152 T^{17} + \)\(23\!\cdots\!96\)\( T^{18} + \)\(75\!\cdots\!44\)\( T^{19} + \)\(17\!\cdots\!88\)\( T^{20} + \)\(75\!\cdots\!44\)\( p T^{21} + \)\(23\!\cdots\!96\)\( p^{2} T^{22} + 8953364907147647152 p^{3} T^{23} + 2749348361125969377 p^{4} T^{24} + 87750614257749208 p^{5} T^{25} + 28008670199006558 p^{6} T^{26} + 692801064948840 p^{7} T^{27} + 248644265143658 p^{8} T^{28} + 4109636225576 p^{9} T^{29} + 1904761586734 p^{10} T^{30} + 14552435320 p^{11} T^{31} + 12386993870 p^{12} T^{32} - 16110776 p^{13} T^{33} + 66381154 p^{14} T^{34} - 597800 p^{15} T^{35} + 275000 p^{16} T^{36} - 3576 p^{17} T^{37} + 762 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 48 T + 1942 T^{2} - 56264 T^{3} + 1433780 T^{4} - 31189032 T^{5} + 615525402 T^{6} - 10961436744 T^{7} + 180602859582 T^{8} - 2754197341656 T^{9} + 39361752586386 T^{10} - 528064119106464 T^{11} + 6697259691407002 T^{12} - 80430473082955408 T^{13} + 918686945721583994 T^{14} - 9992605985967244920 T^{15} + \)\(10\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!04\)\( T^{17} + \)\(97\!\cdots\!60\)\( T^{18} - \)\(89\!\cdots\!64\)\( T^{19} + \)\(77\!\cdots\!20\)\( T^{20} - \)\(89\!\cdots\!64\)\( p T^{21} + \)\(97\!\cdots\!60\)\( p^{2} T^{22} - \)\(10\!\cdots\!04\)\( p^{3} T^{23} + \)\(10\!\cdots\!45\)\( p^{4} T^{24} - 9992605985967244920 p^{5} T^{25} + 918686945721583994 p^{6} T^{26} - 80430473082955408 p^{7} T^{27} + 6697259691407002 p^{8} T^{28} - 528064119106464 p^{9} T^{29} + 39361752586386 p^{10} T^{30} - 2754197341656 p^{11} T^{31} + 180602859582 p^{12} T^{32} - 10961436744 p^{13} T^{33} + 615525402 p^{14} T^{34} - 31189032 p^{15} T^{35} + 1433780 p^{16} T^{36} - 56264 p^{17} T^{37} + 1942 p^{18} T^{38} - 48 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 4 T + 744 T^{2} + 3204 T^{3} + 274302 T^{4} + 1114476 T^{5} + 66854640 T^{6} + 228599980 T^{7} + 12124345485 T^{8} + 30973250912 T^{9} + 1759640243608 T^{10} + 2880535376928 T^{11} + 216553343817768 T^{12} + 181564983044224 T^{13} + 23661783887002376 T^{14} + 7171924176109248 T^{15} + 2357550074150624818 T^{16} + 127612800881544008 T^{17} + \)\(21\!\cdots\!68\)\( T^{18} - 3227028660750646520 T^{19} + \)\(17\!\cdots\!04\)\( T^{20} - 3227028660750646520 p T^{21} + \)\(21\!\cdots\!68\)\( p^{2} T^{22} + 127612800881544008 p^{3} T^{23} + 2357550074150624818 p^{4} T^{24} + 7171924176109248 p^{5} T^{25} + 23661783887002376 p^{6} T^{26} + 181564983044224 p^{7} T^{27} + 216553343817768 p^{8} T^{28} + 2880535376928 p^{9} T^{29} + 1759640243608 p^{10} T^{30} + 30973250912 p^{11} T^{31} + 12124345485 p^{12} T^{32} + 228599980 p^{13} T^{33} + 66854640 p^{14} T^{34} + 1114476 p^{15} T^{35} + 274302 p^{16} T^{36} + 3204 p^{17} T^{37} + 744 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 - 8 T + 652 T^{2} - 3792 T^{3} + 203652 T^{4} - 810608 T^{5} + 41396172 T^{6} - 94217504 T^{7} + 6243872430 T^{8} - 3834384688 T^{9} + 752626097124 T^{10} + 814590982952 T^{11} + 75624310995034 T^{12} + 224318035845656 T^{13} + 6485612334199396 T^{14} + 34384736717270752 T^{15} + 488098252382532241 T^{16} + 4060655772846815224 T^{17} + 34603086671340481632 T^{18} + \)\(39\!\cdots\!64\)\( T^{19} + \)\(26\!\cdots\!36\)\( T^{20} + \)\(39\!\cdots\!64\)\( p T^{21} + 34603086671340481632 p^{2} T^{22} + 4060655772846815224 p^{3} T^{23} + 488098252382532241 p^{4} T^{24} + 34384736717270752 p^{5} T^{25} + 6485612334199396 p^{6} T^{26} + 224318035845656 p^{7} T^{27} + 75624310995034 p^{8} T^{28} + 814590982952 p^{9} T^{29} + 752626097124 p^{10} T^{30} - 3834384688 p^{11} T^{31} + 6243872430 p^{12} T^{32} - 94217504 p^{13} T^{33} + 41396172 p^{14} T^{34} - 810608 p^{15} T^{35} + 203652 p^{16} T^{36} - 3792 p^{17} T^{37} + 652 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 - 20 T + 808 T^{2} - 12856 T^{3} + 314542 T^{4} - 4326688 T^{5} + 81767262 T^{6} - 1024025692 T^{7} + 16342815399 T^{8} - 192379723056 T^{9} + 2700529362264 T^{10} - 30408160260956 T^{11} + 383587331961062 T^{12} - 4167192887975976 T^{13} + 47960522080828044 T^{14} - 504816645425875312 T^{15} + 5374084024329144099 T^{16} - 54821471985120211132 T^{17} + \)\(54\!\cdots\!98\)\( T^{18} - \)\(53\!\cdots\!00\)\( T^{19} + \)\(50\!\cdots\!95\)\( T^{20} - \)\(53\!\cdots\!00\)\( p T^{21} + \)\(54\!\cdots\!98\)\( p^{2} T^{22} - 54821471985120211132 p^{3} T^{23} + 5374084024329144099 p^{4} T^{24} - 504816645425875312 p^{5} T^{25} + 47960522080828044 p^{6} T^{26} - 4167192887975976 p^{7} T^{27} + 383587331961062 p^{8} T^{28} - 30408160260956 p^{9} T^{29} + 2700529362264 p^{10} T^{30} - 192379723056 p^{11} T^{31} + 16342815399 p^{12} T^{32} - 1024025692 p^{13} T^{33} + 81767262 p^{14} T^{34} - 4326688 p^{15} T^{35} + 314542 p^{16} T^{36} - 12856 p^{17} T^{37} + 808 p^{18} T^{38} - 20 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 8 T + 1066 T^{2} - 8140 T^{3} + 5926 p T^{4} - 4231648 T^{5} + 208790556 T^{6} - 1486980364 T^{7} + 57301056433 T^{8} - 394664140872 T^{9} + 12625178523118 T^{10} - 83880214388348 T^{11} + 2315754575630556 T^{12} - 14783337728939600 T^{13} + 361857543145360578 T^{14} - 2208911135139695232 T^{15} + 48893254201549069627 T^{16} - \)\(28\!\cdots\!52\)\( T^{17} + \)\(57\!\cdots\!24\)\( T^{18} - \)\(31\!\cdots\!20\)\( T^{19} + \)\(59\!\cdots\!31\)\( T^{20} - \)\(31\!\cdots\!20\)\( p T^{21} + \)\(57\!\cdots\!24\)\( p^{2} T^{22} - \)\(28\!\cdots\!52\)\( p^{3} T^{23} + 48893254201549069627 p^{4} T^{24} - 2208911135139695232 p^{5} T^{25} + 361857543145360578 p^{6} T^{26} - 14783337728939600 p^{7} T^{27} + 2315754575630556 p^{8} T^{28} - 83880214388348 p^{9} T^{29} + 12625178523118 p^{10} T^{30} - 394664140872 p^{11} T^{31} + 57301056433 p^{12} T^{32} - 1486980364 p^{13} T^{33} + 208790556 p^{14} T^{34} - 4231648 p^{15} T^{35} + 5926 p^{17} T^{36} - 8140 p^{17} T^{37} + 1066 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \) | |
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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
−2.02825535827764658523529704803, −1.86451764332391420747867132226, −1.83532163463131712098403171849, −1.73950763073300363825343033730, −1.72955542198260597214900021996, −1.58588473719268579176336150445, −1.50024235156982695324664027191, −1.46255660918106752998301671600, −1.44392665618216687971335689650, −1.29281163434479496528229935676, −1.04907856994561331756444437989, −1.02558613129325114581125068904, −0.992043231654877471115082128693, −0.969255191205720454899997895051, −0.886925668032828437054197239680, −0.835574606554170552846985394419, −0.822591067014361507677100222026, −0.74115785585808448543716389865, −0.71340986674138112214969769026, −0.67755812195179243951787549314, −0.67240517512390670654392130257, −0.55295984613275047272067603804, −0.54568538641980600916578717235, −0.27355404721401864843912463704, −0.20361237151937738823767285487, 0.20361237151937738823767285487, 0.27355404721401864843912463704, 0.54568538641980600916578717235, 0.55295984613275047272067603804, 0.67240517512390670654392130257, 0.67755812195179243951787549314, 0.71340986674138112214969769026, 0.74115785585808448543716389865, 0.822591067014361507677100222026, 0.835574606554170552846985394419, 0.886925668032828437054197239680, 0.969255191205720454899997895051, 0.992043231654877471115082128693, 1.02558613129325114581125068904, 1.04907856994561331756444437989, 1.29281163434479496528229935676, 1.44392665618216687971335689650, 1.46255660918106752998301671600, 1.50024235156982695324664027191, 1.58588473719268579176336150445, 1.72955542198260597214900021996, 1.73950763073300363825343033730, 1.83532163463131712098403171849, 1.86451764332391420747867132226, 2.02825535827764658523529704803
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.