Dirichlet series
| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 12·5-s − 4·6-s − 2·7-s + 9-s − 24·10-s − 7·11-s + 2·12-s − 8·13-s + 4·14-s + 24·15-s + 13·17-s − 2·18-s − 8·19-s + 12·20-s − 4·21-s + 14·22-s + 85·25-s + 16·26-s − 2·28-s + 3·29-s − 48·30-s + 9·31-s − 14·33-s − 26·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 5.36·5-s − 1.63·6-s − 0.755·7-s + 1/3·9-s − 7.58·10-s − 2.11·11-s + 0.577·12-s − 2.21·13-s + 1.06·14-s + 6.19·15-s + 3.15·17-s − 0.471·18-s − 1.83·19-s + 2.68·20-s − 0.872·21-s + 2.98·22-s + 17·25-s + 3.13·26-s − 0.377·28-s + 0.557·29-s − 8.76·30-s + 1.61·31-s − 2.43·33-s − 4.45·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 23^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.56012\times 10^{17}\) |
| Root analytic conductor: | \(2.77732\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 23^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(170.0175981\) |
| \(L(\frac12)\) | \(\approx\) | \(170.0175981\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) |
| 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \) | |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \) | |
| 23 | \( 1 - 32 T^{2} - 330 T^{3} + 628 T^{4} + 9471 T^{5} + 37423 T^{6} - 191697 T^{7} - 1119029 T^{8} + 517242 T^{9} + 29715973 T^{10} + 517242 p T^{11} - 1119029 p^{2} T^{12} - 191697 p^{3} T^{13} + 37423 p^{4} T^{14} + 9471 p^{5} T^{15} + 628 p^{6} T^{16} - 330 p^{7} T^{17} - 32 p^{8} T^{18} + p^{10} T^{20} \) | |
| good | 5 | \( 1 - 12 T + 59 T^{2} - 166 T^{3} + 422 T^{4} - 1366 T^{5} + 778 p T^{6} - 8577 T^{7} + 702 p^{2} T^{8} - 28482 T^{9} + 22609 T^{10} - 15897 T^{11} + 5459 T^{12} + 461804 T^{13} - 2100643 T^{14} + 1023478 p T^{15} - 10545058 T^{16} + 19287066 T^{17} - 7258608 p T^{18} + 107673792 T^{19} - 297940529 T^{20} + 107673792 p T^{21} - 7258608 p^{3} T^{22} + 19287066 p^{3} T^{23} - 10545058 p^{4} T^{24} + 1023478 p^{6} T^{25} - 2100643 p^{6} T^{26} + 461804 p^{7} T^{27} + 5459 p^{8} T^{28} - 15897 p^{9} T^{29} + 22609 p^{10} T^{30} - 28482 p^{11} T^{31} + 702 p^{14} T^{32} - 8577 p^{13} T^{33} + 778 p^{15} T^{34} - 1366 p^{15} T^{35} + 422 p^{16} T^{36} - 166 p^{17} T^{37} + 59 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \) |
| 11 | \( 1 + 7 T + 38 T^{2} + 167 T^{3} + 905 T^{4} + 3431 T^{5} + 12148 T^{6} + 34821 T^{7} + 107380 T^{8} + 160916 T^{9} - 45429 T^{10} - 213430 p T^{11} - 1153951 p T^{12} - 5593971 p T^{13} - 19534299 p T^{14} - 54926346 p T^{15} - 12747175 p^{2} T^{16} - 24909102 p^{2} T^{17} - 27620476 p^{2} T^{18} + 108791347 p^{2} T^{19} + 423010325 p^{2} T^{20} + 108791347 p^{3} T^{21} - 27620476 p^{4} T^{22} - 24909102 p^{5} T^{23} - 12747175 p^{6} T^{24} - 54926346 p^{6} T^{25} - 19534299 p^{7} T^{26} - 5593971 p^{8} T^{27} - 1153951 p^{9} T^{28} - 213430 p^{10} T^{29} - 45429 p^{10} T^{30} + 160916 p^{11} T^{31} + 107380 p^{12} T^{32} + 34821 p^{13} T^{33} + 12148 p^{14} T^{34} + 3431 p^{15} T^{35} + 905 p^{16} T^{36} + 167 p^{17} T^{37} + 38 p^{18} T^{38} + 7 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 8 T - 19 T^{2} - 272 T^{3} + 21 p T^{4} + 6098 T^{5} - 272 T^{6} - 85030 T^{7} + 22978 T^{8} + 1124444 T^{9} - 1499 p^{2} T^{10} - 12264736 T^{11} - 1916939 T^{12} + 116736118 T^{13} + 18904052 p T^{14} - 538185210 T^{15} - 5292412874 T^{16} - 1645167846 T^{17} + 82709333520 T^{18} + 24187583364 T^{19} - 1139098218145 T^{20} + 24187583364 p T^{21} + 82709333520 p^{2} T^{22} - 1645167846 p^{3} T^{23} - 5292412874 p^{4} T^{24} - 538185210 p^{5} T^{25} + 18904052 p^{7} T^{26} + 116736118 p^{7} T^{27} - 1916939 p^{8} T^{28} - 12264736 p^{9} T^{29} - 1499 p^{12} T^{30} + 1124444 p^{11} T^{31} + 22978 p^{12} T^{32} - 85030 p^{13} T^{33} - 272 p^{14} T^{34} + 6098 p^{15} T^{35} + 21 p^{17} T^{36} - 272 p^{17} T^{37} - 19 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 13 T + 5 T^{2} + 713 T^{3} - 3600 T^{4} - 5121 T^{5} + 109061 T^{6} - 437283 T^{7} + 7403 T^{8} + 10925563 T^{9} - 55433432 T^{10} - 3789841 T^{11} + 1002413171 T^{12} - 198641614 p T^{13} + 90127932 T^{14} + 40814076258 T^{15} - 216038333791 T^{16} + 323177465207 T^{17} + 1921703104277 T^{18} - 6668285404473 T^{19} + 3781429761099 T^{20} - 6668285404473 p T^{21} + 1921703104277 p^{2} T^{22} + 323177465207 p^{3} T^{23} - 216038333791 p^{4} T^{24} + 40814076258 p^{5} T^{25} + 90127932 p^{6} T^{26} - 198641614 p^{8} T^{27} + 1002413171 p^{8} T^{28} - 3789841 p^{9} T^{29} - 55433432 p^{10} T^{30} + 10925563 p^{11} T^{31} + 7403 p^{12} T^{32} - 437283 p^{13} T^{33} + 109061 p^{14} T^{34} - 5121 p^{15} T^{35} - 3600 p^{16} T^{36} + 713 p^{17} T^{37} + 5 p^{18} T^{38} - 13 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 8 T + 43 T^{2} + 8 p T^{3} + 24 T^{4} + 472 T^{5} - 8023 T^{6} - 95180 T^{7} - 611123 T^{8} - 3212575 T^{9} - 4874189 T^{10} + 15745621 T^{11} + 203164059 T^{12} + 1464486584 T^{13} + 7186795644 T^{14} + 28395095649 T^{15} + 92364811266 T^{16} + 23848321025 T^{17} - 1407219074134 T^{18} - 10882534909834 T^{19} - 65394743766087 T^{20} - 10882534909834 p T^{21} - 1407219074134 p^{2} T^{22} + 23848321025 p^{3} T^{23} + 92364811266 p^{4} T^{24} + 28395095649 p^{5} T^{25} + 7186795644 p^{6} T^{26} + 1464486584 p^{7} T^{27} + 203164059 p^{8} T^{28} + 15745621 p^{9} T^{29} - 4874189 p^{10} T^{30} - 3212575 p^{11} T^{31} - 611123 p^{12} T^{32} - 95180 p^{13} T^{33} - 8023 p^{14} T^{34} + 472 p^{15} T^{35} + 24 p^{16} T^{36} + 8 p^{18} T^{37} + 43 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 - 3 T - p T^{2} + 345 T^{3} - 2240 T^{4} + 4288 T^{5} + 47578 T^{6} - 814246 T^{7} + 4985615 T^{8} - 13818807 T^{9} - 78421044 T^{10} + 1295784767 T^{11} - 7927684427 T^{12} + 25827351561 T^{13} + 69830116319 T^{14} - 1466230038626 T^{15} + 9944600189188 T^{16} - 35011150610692 T^{17} - 22810673620600 T^{18} + 1267110179260777 T^{19} - 9706612518579650 T^{20} + 1267110179260777 p T^{21} - 22810673620600 p^{2} T^{22} - 35011150610692 p^{3} T^{23} + 9944600189188 p^{4} T^{24} - 1466230038626 p^{5} T^{25} + 69830116319 p^{6} T^{26} + 25827351561 p^{7} T^{27} - 7927684427 p^{8} T^{28} + 1295784767 p^{9} T^{29} - 78421044 p^{10} T^{30} - 13818807 p^{11} T^{31} + 4985615 p^{12} T^{32} - 814246 p^{13} T^{33} + 47578 p^{14} T^{34} + 4288 p^{15} T^{35} - 2240 p^{16} T^{36} + 345 p^{17} T^{37} - p^{19} T^{38} - 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 - 9 T + 33 T^{2} + 87 T^{3} - 938 T^{4} + 870 T^{5} + 3318 T^{6} - 49364 T^{7} + 862608 T^{8} - 6909788 T^{9} + 5587475 T^{10} + 19138740 T^{11} + 1246930139 T^{12} - 8766452816 T^{13} + 25857740025 T^{14} - 24097186289 T^{15} + 234812624684 T^{16} + 2580448924289 T^{17} - 24751899944130 T^{18} - 247632415803180 T^{19} + 2664124346076111 T^{20} - 247632415803180 p T^{21} - 24751899944130 p^{2} T^{22} + 2580448924289 p^{3} T^{23} + 234812624684 p^{4} T^{24} - 24097186289 p^{5} T^{25} + 25857740025 p^{6} T^{26} - 8766452816 p^{7} T^{27} + 1246930139 p^{8} T^{28} + 19138740 p^{9} T^{29} + 5587475 p^{10} T^{30} - 6909788 p^{11} T^{31} + 862608 p^{12} T^{32} - 49364 p^{13} T^{33} + 3318 p^{14} T^{34} + 870 p^{15} T^{35} - 938 p^{16} T^{36} + 87 p^{17} T^{37} + 33 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + T - 65 T^{2} - 72 T^{3} + 1390 T^{4} - 4848 T^{5} - 1731 T^{6} + 244572 T^{7} + 802011 T^{8} + 6245385 T^{9} - 63121692 T^{10} - 3105766 p T^{11} + 3175659841 T^{12} - 13839831684 T^{13} - 94599339377 T^{14} + 544834379602 T^{15} - 9417576247 T^{16} - 16542540109249 T^{17} - 1233006585711 p T^{18} + 363077924130993 T^{19} + 7086777833125181 T^{20} + 363077924130993 p T^{21} - 1233006585711 p^{3} T^{22} - 16542540109249 p^{3} T^{23} - 9417576247 p^{4} T^{24} + 544834379602 p^{5} T^{25} - 94599339377 p^{6} T^{26} - 13839831684 p^{7} T^{27} + 3175659841 p^{8} T^{28} - 3105766 p^{10} T^{29} - 63121692 p^{10} T^{30} + 6245385 p^{11} T^{31} + 802011 p^{12} T^{32} + 244572 p^{13} T^{33} - 1731 p^{14} T^{34} - 4848 p^{15} T^{35} + 1390 p^{16} T^{36} - 72 p^{17} T^{37} - 65 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 - 10 T + 16 T^{2} + 490 T^{3} - 481 T^{4} - 30676 T^{5} + 97192 T^{6} + 1398606 T^{7} - 8792297 T^{8} - 17970734 T^{9} + 51378616 T^{10} + 2511477694 T^{11} - 13506047897 T^{12} - 11278216192 T^{13} + 168369853928 T^{14} + 1273724136322 T^{15} - 1242883941833 T^{16} - 68453876636882 T^{17} + 540186243344726 T^{18} - 4262709741831820 T^{19} + 28875496281911263 T^{20} - 4262709741831820 p T^{21} + 540186243344726 p^{2} T^{22} - 68453876636882 p^{3} T^{23} - 1242883941833 p^{4} T^{24} + 1273724136322 p^{5} T^{25} + 168369853928 p^{6} T^{26} - 11278216192 p^{7} T^{27} - 13506047897 p^{8} T^{28} + 2511477694 p^{9} T^{29} + 51378616 p^{10} T^{30} - 17970734 p^{11} T^{31} - 8792297 p^{12} T^{32} + 1398606 p^{13} T^{33} + 97192 p^{14} T^{34} - 30676 p^{15} T^{35} - 481 p^{16} T^{36} + 490 p^{17} T^{37} + 16 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 15 T + 72 T^{2} + 1069 T^{3} + 14110 T^{4} + 59450 T^{5} + 387823 T^{6} + 4804129 T^{7} + 13666287 T^{8} + 8202467 T^{9} + 673759310 T^{10} + 280921597 T^{11} - 31612516173 T^{12} + 55428284892 T^{13} + 739562320230 T^{14} - 158533151025 p T^{15} + 27182113422301 T^{16} + 701167023570863 T^{17} + 614466293402544 T^{18} + 11425359426187101 T^{19} + 257150015533444279 T^{20} + 11425359426187101 p T^{21} + 614466293402544 p^{2} T^{22} + 701167023570863 p^{3} T^{23} + 27182113422301 p^{4} T^{24} - 158533151025 p^{6} T^{25} + 739562320230 p^{6} T^{26} + 55428284892 p^{7} T^{27} - 31612516173 p^{8} T^{28} + 280921597 p^{9} T^{29} + 673759310 p^{10} T^{30} + 8202467 p^{11} T^{31} + 13666287 p^{12} T^{32} + 4804129 p^{13} T^{33} + 387823 p^{14} T^{34} + 59450 p^{15} T^{35} + 14110 p^{16} T^{36} + 1069 p^{17} T^{37} + 72 p^{18} T^{38} + 15 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( ( 1 + 14 T + 281 T^{2} + 2858 T^{3} + 32976 T^{4} + 265969 T^{5} + 2302053 T^{6} + 16210597 T^{7} + 120471453 T^{8} + 812821019 T^{9} + 5719516109 T^{10} + 812821019 p T^{11} + 120471453 p^{2} T^{12} + 16210597 p^{3} T^{13} + 2302053 p^{4} T^{14} + 265969 p^{5} T^{15} + 32976 p^{6} T^{16} + 2858 p^{7} T^{17} + 281 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 - 38 T + 546 T^{2} - 2201 T^{3} - 33483 T^{4} + 478977 T^{5} - 1583482 T^{6} - 11238642 T^{7} + 98019617 T^{8} - 226237515 T^{9} + 3942442455 T^{10} - 47098113479 T^{11} - 186521546734 T^{12} + 6578196218801 T^{13} - 30060087327914 T^{14} - 271001285771005 T^{15} + 3464350149588826 T^{16} - 6830002897785093 T^{17} - 90452162530345980 T^{18} + 519526335797023643 T^{19} - 1111782128740224503 T^{20} + 519526335797023643 p T^{21} - 90452162530345980 p^{2} T^{22} - 6830002897785093 p^{3} T^{23} + 3464350149588826 p^{4} T^{24} - 271001285771005 p^{5} T^{25} - 30060087327914 p^{6} T^{26} + 6578196218801 p^{7} T^{27} - 186521546734 p^{8} T^{28} - 47098113479 p^{9} T^{29} + 3942442455 p^{10} T^{30} - 226237515 p^{11} T^{31} + 98019617 p^{12} T^{32} - 11238642 p^{13} T^{33} - 1583482 p^{14} T^{34} + 478977 p^{15} T^{35} - 33483 p^{16} T^{36} - 2201 p^{17} T^{37} + 546 p^{18} T^{38} - 38 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 10 T - 218 T^{2} - 3131 T^{3} + 9633 T^{4} + 353005 T^{5} + 1356212 T^{6} - 11287720 T^{7} - 135484885 T^{8} - 858152597 T^{9} - 3355492963 T^{10} + 53680263287 T^{11} + 971533327154 T^{12} + 3556792341501 T^{13} - 40235567128916 T^{14} - 388589666986928 T^{15} - 98664384746299 T^{16} + 7197040321784822 T^{17} - 13154141015722551 T^{18} + 131331497170919753 T^{19} + 4625749023695232321 T^{20} + 131331497170919753 p T^{21} - 13154141015722551 p^{2} T^{22} + 7197040321784822 p^{3} T^{23} - 98664384746299 p^{4} T^{24} - 388589666986928 p^{5} T^{25} - 40235567128916 p^{6} T^{26} + 3556792341501 p^{7} T^{27} + 971533327154 p^{8} T^{28} + 53680263287 p^{9} T^{29} - 3355492963 p^{10} T^{30} - 858152597 p^{11} T^{31} - 135484885 p^{12} T^{32} - 11287720 p^{13} T^{33} + 1356212 p^{14} T^{34} + 353005 p^{15} T^{35} + 9633 p^{16} T^{36} - 3131 p^{17} T^{37} - 218 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 + 6 T - 14 T^{2} - 1668 T^{3} - 1076 T^{4} + 25195 T^{5} + 765100 T^{6} - 7250771 T^{7} + 22467044 T^{8} + 67277920 T^{9} + 5031992420 T^{10} - 59478655170 T^{11} - 3470762891 T^{12} - 293065871941 T^{13} + 39692667115917 T^{14} - 217934506894716 T^{15} - 228796915375119 T^{16} - 8246024005362609 T^{17} + 172630942365736459 T^{18} - 706752850171108364 T^{19} - 507509963037681457 T^{20} - 706752850171108364 p T^{21} + 172630942365736459 p^{2} T^{22} - 8246024005362609 p^{3} T^{23} - 228796915375119 p^{4} T^{24} - 217934506894716 p^{5} T^{25} + 39692667115917 p^{6} T^{26} - 293065871941 p^{7} T^{27} - 3470762891 p^{8} T^{28} - 59478655170 p^{9} T^{29} + 5031992420 p^{10} T^{30} + 67277920 p^{11} T^{31} + 22467044 p^{12} T^{32} - 7250771 p^{13} T^{33} + 765100 p^{14} T^{34} + 25195 p^{15} T^{35} - 1076 p^{16} T^{36} - 1668 p^{17} T^{37} - 14 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 36 T + 750 T^{2} - 10735 T^{3} + 122421 T^{4} - 1177049 T^{5} + 9630210 T^{6} - 60460294 T^{7} + 186862718 T^{8} + 1720176102 T^{9} - 40355019157 T^{10} + 501275747514 T^{11} - 4793912068685 T^{12} + 36654084074001 T^{13} - 214851275764160 T^{14} + 759845861999679 T^{15} + 2814917924432911 T^{16} - 95736803824170254 T^{17} + 1269270147580762322 T^{18} - 12823899999930315354 T^{19} + \)\(11\!\cdots\!95\)\( T^{20} - 12823899999930315354 p T^{21} + 1269270147580762322 p^{2} T^{22} - 95736803824170254 p^{3} T^{23} + 2814917924432911 p^{4} T^{24} + 759845861999679 p^{5} T^{25} - 214851275764160 p^{6} T^{26} + 36654084074001 p^{7} T^{27} - 4793912068685 p^{8} T^{28} + 501275747514 p^{9} T^{29} - 40355019157 p^{10} T^{30} + 1720176102 p^{11} T^{31} + 186862718 p^{12} T^{32} - 60460294 p^{13} T^{33} + 9630210 p^{14} T^{34} - 1177049 p^{15} T^{35} + 122421 p^{16} T^{36} - 10735 p^{17} T^{37} + 750 p^{18} T^{38} - 36 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + T + 8 T^{2} + 1725 T^{3} + 1706 T^{4} + 65023 T^{5} + 1417424 T^{6} + 126555 T^{7} + 121874365 T^{8} + 646066727 T^{9} - 341811066 T^{10} + 93927869266 T^{11} + 28041785937 T^{12} + 1541368535775 T^{13} + 26894101261544 T^{14} - 216455340058526 T^{15} + 1879558322992044 T^{16} - 10498830963113806 T^{17} - 124564368864991493 T^{18} + 327452786852004012 T^{19} - 13165210482011996985 T^{20} + 327452786852004012 p T^{21} - 124564368864991493 p^{2} T^{22} - 10498830963113806 p^{3} T^{23} + 1879558322992044 p^{4} T^{24} - 216455340058526 p^{5} T^{25} + 26894101261544 p^{6} T^{26} + 1541368535775 p^{7} T^{27} + 28041785937 p^{8} T^{28} + 93927869266 p^{9} T^{29} - 341811066 p^{10} T^{30} + 646066727 p^{11} T^{31} + 121874365 p^{12} T^{32} + 126555 p^{13} T^{33} + 1417424 p^{14} T^{34} + 65023 p^{15} T^{35} + 1706 p^{16} T^{36} + 1725 p^{17} T^{37} + 8 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 65 T + 1734 T^{2} - 20803 T^{3} - 9132 T^{4} + 3678914 T^{5} - 42671880 T^{6} + 1776304 T^{7} + 4197970945 T^{8} - 28330418864 T^{9} - 195167467341 T^{10} + 3235273639806 T^{11} + 3041126547124 T^{12} - 291625899968854 T^{13} + 1020266363519674 T^{14} + 19250030848627523 T^{15} - 165671944579630148 T^{16} - 644204553347583427 T^{17} + 12945024207967779511 T^{18} + 4430926296150848582 T^{19} - \)\(80\!\cdots\!47\)\( T^{20} + 4430926296150848582 p T^{21} + 12945024207967779511 p^{2} T^{22} - 644204553347583427 p^{3} T^{23} - 165671944579630148 p^{4} T^{24} + 19250030848627523 p^{5} T^{25} + 1020266363519674 p^{6} T^{26} - 291625899968854 p^{7} T^{27} + 3041126547124 p^{8} T^{28} + 3235273639806 p^{9} T^{29} - 195167467341 p^{10} T^{30} - 28330418864 p^{11} T^{31} + 4197970945 p^{12} T^{32} + 1776304 p^{13} T^{33} - 42671880 p^{14} T^{34} + 3678914 p^{15} T^{35} - 9132 p^{16} T^{36} - 20803 p^{17} T^{37} + 1734 p^{18} T^{38} - 65 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 - 10 T + 52 T^{2} - 148 T^{3} + 10675 T^{4} - 14890 T^{5} - 232235 T^{6} + 65274 p T^{7} + 4150355 T^{8} + 312254910 T^{9} + 762691986 T^{10} - 1090282798 T^{11} + 251273087589 T^{12} - 2428327156411 T^{13} + 22726272682750 T^{14} - 90500690433385 T^{15} + 39974825308301 T^{16} - 352094282120248 T^{17} - 150763402222271421 T^{18} + 12417192188120104 p T^{19} - 15575249815442409179 T^{20} + 12417192188120104 p^{2} T^{21} - 150763402222271421 p^{2} T^{22} - 352094282120248 p^{3} T^{23} + 39974825308301 p^{4} T^{24} - 90500690433385 p^{5} T^{25} + 22726272682750 p^{6} T^{26} - 2428327156411 p^{7} T^{27} + 251273087589 p^{8} T^{28} - 1090282798 p^{9} T^{29} + 762691986 p^{10} T^{30} + 312254910 p^{11} T^{31} + 4150355 p^{12} T^{32} + 65274 p^{14} T^{33} - 232235 p^{14} T^{34} - 14890 p^{15} T^{35} + 10675 p^{16} T^{36} - 148 p^{17} T^{37} + 52 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 14 T + 188 T^{2} + 49 p T^{3} + 38028 T^{4} + 431885 T^{5} + 4807604 T^{6} + 22325193 T^{7} + 157406286 T^{8} - 778466440 T^{9} - 34156227740 T^{10} - 417047795351 T^{11} - 5783112460222 T^{12} - 51367542683583 T^{13} - 416338904649796 T^{14} - 2869276219010234 T^{15} - 2956944863905248 T^{16} + 105200095422204508 T^{17} + 2653188533795058138 T^{18} + 35586208083891419125 T^{19} + \)\(31\!\cdots\!65\)\( T^{20} + 35586208083891419125 p T^{21} + 2653188533795058138 p^{2} T^{22} + 105200095422204508 p^{3} T^{23} - 2956944863905248 p^{4} T^{24} - 2869276219010234 p^{5} T^{25} - 416338904649796 p^{6} T^{26} - 51367542683583 p^{7} T^{27} - 5783112460222 p^{8} T^{28} - 417047795351 p^{9} T^{29} - 34156227740 p^{10} T^{30} - 778466440 p^{11} T^{31} + 157406286 p^{12} T^{32} + 22325193 p^{13} T^{33} + 4807604 p^{14} T^{34} + 431885 p^{15} T^{35} + 38028 p^{16} T^{36} + 49 p^{18} T^{37} + 188 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 18 T - 73 T^{2} - 2029 T^{3} - 2444 T^{4} - 55186 T^{5} + 809482 T^{6} + 17864275 T^{7} - 170854634 T^{8} - 743517086 T^{9} + 13522355425 T^{10} - 47029499187 T^{11} - 245256132844 T^{12} + 1014127569804 T^{13} + 12571021650618 T^{14} + 828950188081053 T^{15} - 783782984306918 T^{16} - 82070339467596823 T^{17} - 929009538108952158 T^{18} + 2398106682688105409 T^{19} + \)\(14\!\cdots\!11\)\( T^{20} + 2398106682688105409 p T^{21} - 929009538108952158 p^{2} T^{22} - 82070339467596823 p^{3} T^{23} - 783782984306918 p^{4} T^{24} + 828950188081053 p^{5} T^{25} + 12571021650618 p^{6} T^{26} + 1014127569804 p^{7} T^{27} - 245256132844 p^{8} T^{28} - 47029499187 p^{9} T^{29} + 13522355425 p^{10} T^{30} - 743517086 p^{11} T^{31} - 170854634 p^{12} T^{32} + 17864275 p^{13} T^{33} + 809482 p^{14} T^{34} - 55186 p^{15} T^{35} - 2444 p^{16} T^{36} - 2029 p^{17} T^{37} - 73 p^{18} T^{38} + 18 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( ( 1 + 10 T + 146 T^{2} + 2085 T^{3} + 26807 T^{4} + 238283 T^{5} + 2827021 T^{6} + 35309464 T^{7} + 384885023 T^{8} + 3934413428 T^{9} + 45025869681 T^{10} + 3934413428 p T^{11} + 384885023 p^{2} T^{12} + 35309464 p^{3} T^{13} + 2827021 p^{4} T^{14} + 238283 p^{5} T^{15} + 26807 p^{6} T^{16} + 2085 p^{7} T^{17} + 146 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
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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
−2.19833320788941959185341443441, −2.13998064083922070159792548712, −2.04188230509789387345828437869, −1.98745199559896672521757805574, −1.96656971544481158872363844355, −1.85447371442099374573442293995, −1.80867504348453957777223971644, −1.73644398039030216561864472970, −1.69801279365533643473816628007, −1.68071744888507953010920539500, −1.67293559579068566780791625564, −1.54767810867716490664490538096, −1.53755135671199180438339533100, −1.29630732988455481290383862139, −1.15839549610016741628158784790, −1.07513878897180501383302220567, −0.991005363331114507171312770226, −0.801007951336540636095381043301, −0.73319652183511758161702249222, −0.72495475443258319772013014377, −0.64865147950710043763825083231, −0.59845285615773817453336473839, −0.52273496737283478435208488346, −0.43213085349459053680955285820, −0.28417626254980046804154503114, 0.28417626254980046804154503114, 0.43213085349459053680955285820, 0.52273496737283478435208488346, 0.59845285615773817453336473839, 0.64865147950710043763825083231, 0.72495475443258319772013014377, 0.73319652183511758161702249222, 0.801007951336540636095381043301, 0.991005363331114507171312770226, 1.07513878897180501383302220567, 1.15839549610016741628158784790, 1.29630732988455481290383862139, 1.53755135671199180438339533100, 1.54767810867716490664490538096, 1.67293559579068566780791625564, 1.68071744888507953010920539500, 1.69801279365533643473816628007, 1.73644398039030216561864472970, 1.80867504348453957777223971644, 1.85447371442099374573442293995, 1.96656971544481158872363844355, 1.98745199559896672521757805574, 2.04188230509789387345828437869, 2.13998064083922070159792548712, 2.19833320788941959185341443441
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.