Dirichlet series
| L(s) = 1 | + 4·2-s + 19·3-s + 5·5-s + 76·6-s + 19·7-s − 21·8-s + 190·9-s + 20·10-s − 9·11-s + 24·13-s + 76·14-s + 95·15-s − 26·16-s + 17·17-s + 760·18-s + 23·19-s + 361·21-s − 36·22-s − 17·23-s − 399·24-s − 16·25-s + 96·26-s + 1.33e3·27-s + 2·29-s + 380·30-s + 16·31-s + 27·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 10.9·3-s + 2.23·5-s + 31.0·6-s + 7.18·7-s − 7.42·8-s + 63.3·9-s + 6.32·10-s − 2.71·11-s + 6.65·13-s + 20.3·14-s + 24.5·15-s − 6.5·16-s + 4.12·17-s + 179.·18-s + 5.27·19-s + 78.7·21-s − 7.67·22-s − 3.54·23-s − 81.4·24-s − 3.19·25-s + 18.8·26-s + 255.·27-s + 0.371·29-s + 69.3·30-s + 2.87·31-s + 4.77·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 7^{19} \cdot 127^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 7^{19} \cdot 127^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(3^{19} \cdot 7^{19} \cdot 127^{19}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72863\times 10^{25}\) |
| Root analytic conductor: | \(4.61477\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((38,\ 3^{19} \cdot 7^{19} \cdot 127^{19} ,\ ( \ : [1/2]^{19} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.050957902\times10^{7}\) |
| \(L(\frac12)\) | \(\approx\) | \(1.050957902\times10^{7}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{19} \) |
| 7 | \( ( 1 - T )^{19} \) | |
| 127 | \( ( 1 - T )^{19} \) | |
| good | 2 | \( 1 - p^{2} T + p^{4} T^{2} - 43 T^{3} + 57 p T^{4} - 251 T^{5} + 541 T^{6} - 65 p^{4} T^{7} + 1959 T^{8} - 851 p^{2} T^{9} + 5807 T^{10} - 73 p^{7} T^{11} + 14813 T^{12} - 2817 p^{3} T^{13} + 16949 p T^{14} - 49667 T^{15} + 72273 T^{16} - 103641 T^{17} + 74019 p T^{18} - 104829 p T^{19} + 74019 p^{2} T^{20} - 103641 p^{2} T^{21} + 72273 p^{3} T^{22} - 49667 p^{4} T^{23} + 16949 p^{6} T^{24} - 2817 p^{9} T^{25} + 14813 p^{7} T^{26} - 73 p^{15} T^{27} + 5807 p^{9} T^{28} - 851 p^{12} T^{29} + 1959 p^{11} T^{30} - 65 p^{16} T^{31} + 541 p^{13} T^{32} - 251 p^{14} T^{33} + 57 p^{16} T^{34} - 43 p^{16} T^{35} + p^{21} T^{36} - p^{20} T^{37} + p^{19} T^{38} \) |
| 5 | \( 1 - p T + 41 T^{2} - 168 T^{3} + 832 T^{4} - 2996 T^{5} + 11691 T^{6} - 37986 T^{7} + 128552 T^{8} - 382033 T^{9} + 1168939 T^{10} - 3222334 T^{11} + 9102821 T^{12} - 4715131 p T^{13} + 12457096 p T^{14} - 152712478 T^{15} + 76177624 p T^{16} - 886961463 T^{17} + 2101604027 T^{18} - 4656991668 T^{19} + 2101604027 p T^{20} - 886961463 p^{2} T^{21} + 76177624 p^{4} T^{22} - 152712478 p^{4} T^{23} + 12457096 p^{6} T^{24} - 4715131 p^{7} T^{25} + 9102821 p^{7} T^{26} - 3222334 p^{8} T^{27} + 1168939 p^{9} T^{28} - 382033 p^{10} T^{29} + 128552 p^{11} T^{30} - 37986 p^{12} T^{31} + 11691 p^{13} T^{32} - 2996 p^{14} T^{33} + 832 p^{15} T^{34} - 168 p^{16} T^{35} + 41 p^{17} T^{36} - p^{19} T^{37} + p^{19} T^{38} \) | |
| 11 | \( 1 + 9 T + 114 T^{2} + 846 T^{3} + 6834 T^{4} + 43631 T^{5} + 279049 T^{6} + 1570491 T^{7} + 8622246 T^{8} + 43583626 T^{9} + 213037611 T^{10} + 979860254 T^{11} + 4350940113 T^{12} + 18383133454 T^{13} + 75002259850 T^{14} + 26632354399 p T^{15} + 1105867682544 T^{16} + 364512200344 p T^{17} + 14057469809074 T^{18} + 47407746375208 T^{19} + 14057469809074 p T^{20} + 364512200344 p^{3} T^{21} + 1105867682544 p^{3} T^{22} + 26632354399 p^{5} T^{23} + 75002259850 p^{5} T^{24} + 18383133454 p^{6} T^{25} + 4350940113 p^{7} T^{26} + 979860254 p^{8} T^{27} + 213037611 p^{9} T^{28} + 43583626 p^{10} T^{29} + 8622246 p^{11} T^{30} + 1570491 p^{12} T^{31} + 279049 p^{13} T^{32} + 43631 p^{14} T^{33} + 6834 p^{15} T^{34} + 846 p^{16} T^{35} + 114 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 - 24 T + 402 T^{2} - 5026 T^{3} + 52972 T^{4} - 482098 T^{5} + 3925963 T^{6} - 28966932 T^{7} + 196682920 T^{8} - 1238002338 T^{9} + 7283995209 T^{10} - 40243236364 T^{11} + 209809038801 T^{12} - 1035189853758 T^{13} + 4848463373224 T^{14} - 21594807026076 T^{15} + 7048975701384 p T^{16} - 370834674951958 T^{17} + 1432622313733372 T^{18} - 406538880764884 p T^{19} + 1432622313733372 p T^{20} - 370834674951958 p^{2} T^{21} + 7048975701384 p^{4} T^{22} - 21594807026076 p^{4} T^{23} + 4848463373224 p^{5} T^{24} - 1035189853758 p^{6} T^{25} + 209809038801 p^{7} T^{26} - 40243236364 p^{8} T^{27} + 7283995209 p^{9} T^{28} - 1238002338 p^{10} T^{29} + 196682920 p^{11} T^{30} - 28966932 p^{12} T^{31} + 3925963 p^{13} T^{32} - 482098 p^{14} T^{33} + 52972 p^{15} T^{34} - 5026 p^{16} T^{35} + 402 p^{17} T^{36} - 24 p^{18} T^{37} + p^{19} T^{38} \) | |
| 17 | \( 1 - p T + 284 T^{2} - 3184 T^{3} + 32551 T^{4} - 278609 T^{5} + 2179544 T^{6} - 15312493 T^{7} + 99697602 T^{8} - 601566144 T^{9} + 3415708339 T^{10} - 1076653110 p T^{11} + 93623696982 T^{12} - 457896515162 T^{13} + 2163836903127 T^{14} - 9881182119343 T^{15} + 43988887312946 T^{16} - 190801647107236 T^{17} + 811083332879856 T^{18} - 3375158517221308 T^{19} + 811083332879856 p T^{20} - 190801647107236 p^{2} T^{21} + 43988887312946 p^{3} T^{22} - 9881182119343 p^{4} T^{23} + 2163836903127 p^{5} T^{24} - 457896515162 p^{6} T^{25} + 93623696982 p^{7} T^{26} - 1076653110 p^{9} T^{27} + 3415708339 p^{9} T^{28} - 601566144 p^{10} T^{29} + 99697602 p^{11} T^{30} - 15312493 p^{12} T^{31} + 2179544 p^{13} T^{32} - 278609 p^{14} T^{33} + 32551 p^{15} T^{34} - 3184 p^{16} T^{35} + 284 p^{17} T^{36} - p^{19} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 - 23 T + 416 T^{2} - 5333 T^{3} + 59693 T^{4} - 564360 T^{5} + 4868647 T^{6} - 37688990 T^{7} + 272710055 T^{8} - 1823898623 T^{9} + 11572107859 T^{10} - 69036155352 T^{11} + 394624910977 T^{12} - 2144671314045 T^{13} + 11249450770817 T^{14} - 2975422408788 p T^{15} + 275667627613884 T^{16} - 1294507948679753 T^{17} + 5918690016197407 T^{18} - 1375265661118110 p T^{19} + 5918690016197407 p T^{20} - 1294507948679753 p^{2} T^{21} + 275667627613884 p^{3} T^{22} - 2975422408788 p^{5} T^{23} + 11249450770817 p^{5} T^{24} - 2144671314045 p^{6} T^{25} + 394624910977 p^{7} T^{26} - 69036155352 p^{8} T^{27} + 11572107859 p^{9} T^{28} - 1823898623 p^{10} T^{29} + 272710055 p^{11} T^{30} - 37688990 p^{12} T^{31} + 4868647 p^{13} T^{32} - 564360 p^{14} T^{33} + 59693 p^{15} T^{34} - 5333 p^{16} T^{35} + 416 p^{17} T^{36} - 23 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 + 17 T + 17 p T^{2} + 4769 T^{3} + 65448 T^{4} + 27849 p T^{5} + 6681855 T^{6} + 55642304 T^{7} + 483186514 T^{8} + 3544873790 T^{9} + 26828147273 T^{10} + 177193952328 T^{11} + 1199885605017 T^{12} + 7232532287266 T^{13} + 44528435308294 T^{14} + 247034068307600 T^{15} + 1396581507639034 T^{16} + 7165658273752720 T^{17} + 37417693376472169 T^{18} + 177916417007843598 T^{19} + 37417693376472169 p T^{20} + 7165658273752720 p^{2} T^{21} + 1396581507639034 p^{3} T^{22} + 247034068307600 p^{4} T^{23} + 44528435308294 p^{5} T^{24} + 7232532287266 p^{6} T^{25} + 1199885605017 p^{7} T^{26} + 177193952328 p^{8} T^{27} + 26828147273 p^{9} T^{28} + 3544873790 p^{10} T^{29} + 483186514 p^{11} T^{30} + 55642304 p^{12} T^{31} + 6681855 p^{13} T^{32} + 27849 p^{15} T^{33} + 65448 p^{15} T^{34} + 4769 p^{16} T^{35} + 17 p^{18} T^{36} + 17 p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 - 2 T + 250 T^{2} - 416 T^{3} + 31743 T^{4} - 40300 T^{5} + 2734376 T^{6} - 2380316 T^{7} + 180736146 T^{8} - 93979076 T^{9} + 9822415154 T^{10} - 2467083804 T^{11} + 457248535436 T^{12} - 31843829608 T^{13} + 18656669806136 T^{14} + 735202286540 T^{15} + 675508971482301 T^{16} + 63722501247098 T^{17} + 21852698540725853 T^{18} + 2340535489419288 T^{19} + 21852698540725853 p T^{20} + 63722501247098 p^{2} T^{21} + 675508971482301 p^{3} T^{22} + 735202286540 p^{4} T^{23} + 18656669806136 p^{5} T^{24} - 31843829608 p^{6} T^{25} + 457248535436 p^{7} T^{26} - 2467083804 p^{8} T^{27} + 9822415154 p^{9} T^{28} - 93979076 p^{10} T^{29} + 180736146 p^{11} T^{30} - 2380316 p^{12} T^{31} + 2734376 p^{13} T^{32} - 40300 p^{14} T^{33} + 31743 p^{15} T^{34} - 416 p^{16} T^{35} + 250 p^{17} T^{36} - 2 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 - 16 T + 444 T^{2} - 5891 T^{3} + 94707 T^{4} - 1068715 T^{5} + 12949771 T^{6} - 127334359 T^{7} + 1280574803 T^{8} - 11191113269 T^{9} + 97892201385 T^{10} - 771667588986 T^{11} + 6026996383307 T^{12} - 43307805557421 T^{13} + 306799204066761 T^{14} - 2023856746847967 T^{15} + 13129737685672700 T^{16} - 79857016141786523 T^{17} + 15390896619446123 p T^{18} - 2680713867518439722 T^{19} + 15390896619446123 p^{2} T^{20} - 79857016141786523 p^{2} T^{21} + 13129737685672700 p^{3} T^{22} - 2023856746847967 p^{4} T^{23} + 306799204066761 p^{5} T^{24} - 43307805557421 p^{6} T^{25} + 6026996383307 p^{7} T^{26} - 771667588986 p^{8} T^{27} + 97892201385 p^{9} T^{28} - 11191113269 p^{10} T^{29} + 1280574803 p^{11} T^{30} - 127334359 p^{12} T^{31} + 12949771 p^{13} T^{32} - 1068715 p^{14} T^{33} + 94707 p^{15} T^{34} - 5891 p^{16} T^{35} + 444 p^{17} T^{36} - 16 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 - 56 T + 1919 T^{2} - 48113 T^{3} + 978205 T^{4} - 16860046 T^{5} + 254586879 T^{6} - 3434890242 T^{7} + 42066352834 T^{8} - 472720007895 T^{9} + 4918349057024 T^{10} - 47690421274916 T^{11} + 433348998146773 T^{12} - 3705349253519629 T^{13} + 29916574318889081 T^{14} - 228662300563992558 T^{15} + 1658031228676445269 T^{16} - 11421713005692758978 T^{17} + 2022493037507193579 p T^{18} - \)\(46\!\cdots\!78\)\( T^{19} + 2022493037507193579 p^{2} T^{20} - 11421713005692758978 p^{2} T^{21} + 1658031228676445269 p^{3} T^{22} - 228662300563992558 p^{4} T^{23} + 29916574318889081 p^{5} T^{24} - 3705349253519629 p^{6} T^{25} + 433348998146773 p^{7} T^{26} - 47690421274916 p^{8} T^{27} + 4918349057024 p^{9} T^{28} - 472720007895 p^{10} T^{29} + 42066352834 p^{11} T^{30} - 3434890242 p^{12} T^{31} + 254586879 p^{13} T^{32} - 16860046 p^{14} T^{33} + 978205 p^{15} T^{34} - 48113 p^{16} T^{35} + 1919 p^{17} T^{36} - 56 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 + 7 T + 372 T^{2} + 2494 T^{3} + 72192 T^{4} + 464718 T^{5} + 9659139 T^{6} + 59703885 T^{7} + 994893253 T^{8} + 5895564041 T^{9} + 83515099021 T^{10} + 473396571593 T^{11} + 5907087984900 T^{12} + 31939933276623 T^{13} + 359380353673970 T^{14} + 1847303925885007 T^{15} + 19050148050782657 T^{16} + 92701601847103523 T^{17} + 886536429602488599 T^{18} + 4062791660052265530 T^{19} + 886536429602488599 p T^{20} + 92701601847103523 p^{2} T^{21} + 19050148050782657 p^{3} T^{22} + 1847303925885007 p^{4} T^{23} + 359380353673970 p^{5} T^{24} + 31939933276623 p^{6} T^{25} + 5907087984900 p^{7} T^{26} + 473396571593 p^{8} T^{27} + 83515099021 p^{9} T^{28} + 5895564041 p^{10} T^{29} + 994893253 p^{11} T^{30} + 59703885 p^{12} T^{31} + 9659139 p^{13} T^{32} + 464718 p^{14} T^{33} + 72192 p^{15} T^{34} + 2494 p^{16} T^{35} + 372 p^{17} T^{36} + 7 p^{18} T^{37} + p^{19} T^{38} \) | |
| 43 | \( 1 - 19 T + 541 T^{2} - 8177 T^{3} + 138902 T^{4} - 40845 p T^{5} + 22810067 T^{6} - 249749424 T^{7} + 2714308406 T^{8} - 26376020952 T^{9} + 251013493425 T^{10} - 2205434613904 T^{11} + 18891895734477 T^{12} - 152330616533600 T^{13} + 1197167503618210 T^{14} - 8968364455463072 T^{15} + 65582001060951154 T^{16} - 460827672059565390 T^{17} + 3166245237871279593 T^{18} - 488121458168209114 p T^{19} + 3166245237871279593 p T^{20} - 460827672059565390 p^{2} T^{21} + 65582001060951154 p^{3} T^{22} - 8968364455463072 p^{4} T^{23} + 1197167503618210 p^{5} T^{24} - 152330616533600 p^{6} T^{25} + 18891895734477 p^{7} T^{26} - 2205434613904 p^{8} T^{27} + 251013493425 p^{9} T^{28} - 26376020952 p^{10} T^{29} + 2714308406 p^{11} T^{30} - 249749424 p^{12} T^{31} + 22810067 p^{13} T^{32} - 40845 p^{15} T^{33} + 138902 p^{15} T^{34} - 8177 p^{16} T^{35} + 541 p^{17} T^{36} - 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 - 25 T + 904 T^{2} - 16894 T^{3} + 361649 T^{4} - 5490414 T^{5} + 88619801 T^{6} - 1143888312 T^{7} + 15210985459 T^{8} - 171652595794 T^{9} + 1965291833203 T^{10} - 19743875550578 T^{11} + 199637602810875 T^{12} - 1807202412082554 T^{13} + 16393786057089957 T^{14} - 134786478179220970 T^{15} + 1107872880178406762 T^{16} - 8312551665909679693 T^{17} + 62279110124776561297 T^{18} - \)\(42\!\cdots\!32\)\( T^{19} + 62279110124776561297 p T^{20} - 8312551665909679693 p^{2} T^{21} + 1107872880178406762 p^{3} T^{22} - 134786478179220970 p^{4} T^{23} + 16393786057089957 p^{5} T^{24} - 1807202412082554 p^{6} T^{25} + 199637602810875 p^{7} T^{26} - 19743875550578 p^{8} T^{27} + 1965291833203 p^{9} T^{28} - 171652595794 p^{10} T^{29} + 15210985459 p^{11} T^{30} - 1143888312 p^{12} T^{31} + 88619801 p^{13} T^{32} - 5490414 p^{14} T^{33} + 361649 p^{15} T^{34} - 16894 p^{16} T^{35} + 904 p^{17} T^{36} - 25 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 - 18 T + 729 T^{2} - 10786 T^{3} + 248005 T^{4} - 3156553 T^{5} + 53632959 T^{6} - 604817569 T^{7} + 8403382051 T^{8} - 85661456697 T^{9} + 1025414824379 T^{10} - 9578389113218 T^{11} + 101873882940499 T^{12} - 879848025198707 T^{13} + 8477677851613975 T^{14} - 68061668751127647 T^{15} + 601679391573782590 T^{16} - 4501409880480265273 T^{17} + 694527557822202388 p T^{18} - \)\(25\!\cdots\!92\)\( T^{19} + 694527557822202388 p^{2} T^{20} - 4501409880480265273 p^{2} T^{21} + 601679391573782590 p^{3} T^{22} - 68061668751127647 p^{4} T^{23} + 8477677851613975 p^{5} T^{24} - 879848025198707 p^{6} T^{25} + 101873882940499 p^{7} T^{26} - 9578389113218 p^{8} T^{27} + 1025414824379 p^{9} T^{28} - 85661456697 p^{10} T^{29} + 8403382051 p^{11} T^{30} - 604817569 p^{12} T^{31} + 53632959 p^{13} T^{32} - 3156553 p^{14} T^{33} + 248005 p^{15} T^{34} - 10786 p^{16} T^{35} + 729 p^{17} T^{36} - 18 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 - 11 T + 625 T^{2} - 6723 T^{3} + 195828 T^{4} - 2071013 T^{5} + 41248883 T^{6} - 425750946 T^{7} + 6571435112 T^{8} - 65378594338 T^{9} + 841495712425 T^{10} - 7973031579512 T^{11} + 89681549643081 T^{12} - 802152520437678 T^{13} + 8123426402651024 T^{14} - 68246189918775774 T^{15} + 633654102053130070 T^{16} - 4987112049848041672 T^{17} + 42904033250390068767 T^{18} - \)\(31\!\cdots\!46\)\( T^{19} + 42904033250390068767 p T^{20} - 4987112049848041672 p^{2} T^{21} + 633654102053130070 p^{3} T^{22} - 68246189918775774 p^{4} T^{23} + 8123426402651024 p^{5} T^{24} - 802152520437678 p^{6} T^{25} + 89681549643081 p^{7} T^{26} - 7973031579512 p^{8} T^{27} + 841495712425 p^{9} T^{28} - 65378594338 p^{10} T^{29} + 6571435112 p^{11} T^{30} - 425750946 p^{12} T^{31} + 41248883 p^{13} T^{32} - 2071013 p^{14} T^{33} + 195828 p^{15} T^{34} - 6723 p^{16} T^{35} + 625 p^{17} T^{36} - 11 p^{18} T^{37} + p^{19} T^{38} \) | |
| 61 | \( 1 - 26 T + 838 T^{2} - 15329 T^{3} + 298583 T^{4} - 4294165 T^{5} + 63615119 T^{6} - 761644687 T^{7} + 9351456495 T^{8} - 96581854497 T^{9} + 1031428248461 T^{10} - 9443877771964 T^{11} + 90986491118717 T^{12} - 758071858895611 T^{13} + 6812650245376485 T^{14} - 53082743698333913 T^{15} + 458408958260406348 T^{16} - 3427992803436643349 T^{17} + 29015749356968724265 T^{18} - \)\(21\!\cdots\!10\)\( T^{19} + 29015749356968724265 p T^{20} - 3427992803436643349 p^{2} T^{21} + 458408958260406348 p^{3} T^{22} - 53082743698333913 p^{4} T^{23} + 6812650245376485 p^{5} T^{24} - 758071858895611 p^{6} T^{25} + 90986491118717 p^{7} T^{26} - 9443877771964 p^{8} T^{27} + 1031428248461 p^{9} T^{28} - 96581854497 p^{10} T^{29} + 9351456495 p^{11} T^{30} - 761644687 p^{12} T^{31} + 63615119 p^{13} T^{32} - 4294165 p^{14} T^{33} + 298583 p^{15} T^{34} - 15329 p^{16} T^{35} + 838 p^{17} T^{36} - 26 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 - 24 T + 1065 T^{2} - 20598 T^{3} + 528809 T^{4} - 8644973 T^{5} + 165769208 T^{6} - 2361709360 T^{7} + 37198542309 T^{8} - 471412532131 T^{9} + 6396164360286 T^{10} - 73119929927616 T^{11} + 878356408508454 T^{12} - 9144679664174609 T^{13} + 98919019711057509 T^{14} - 943722960976640048 T^{15} + 9291719328933613139 T^{16} - 81516068920894131031 T^{17} + \)\(73\!\cdots\!96\)\( T^{18} - \)\(59\!\cdots\!16\)\( T^{19} + \)\(73\!\cdots\!96\)\( p T^{20} - 81516068920894131031 p^{2} T^{21} + 9291719328933613139 p^{3} T^{22} - 943722960976640048 p^{4} T^{23} + 98919019711057509 p^{5} T^{24} - 9144679664174609 p^{6} T^{25} + 878356408508454 p^{7} T^{26} - 73119929927616 p^{8} T^{27} + 6396164360286 p^{9} T^{28} - 471412532131 p^{10} T^{29} + 37198542309 p^{11} T^{30} - 2361709360 p^{12} T^{31} + 165769208 p^{13} T^{32} - 8644973 p^{14} T^{33} + 528809 p^{15} T^{34} - 20598 p^{16} T^{35} + 1065 p^{17} T^{36} - 24 p^{18} T^{37} + p^{19} T^{38} \) | |
| 71 | \( 1 + 32 T + 1227 T^{2} + 28835 T^{3} + 680427 T^{4} + 12842100 T^{5} + 234406211 T^{6} + 3740306330 T^{7} + 57238298625 T^{8} + 796894059719 T^{9} + 10634242395334 T^{10} + 131843183828886 T^{11} + 1569149036104108 T^{12} + 17561592195883763 T^{13} + 189022343080292513 T^{14} + 1927041274911330686 T^{15} + 18922276766488552162 T^{16} + \)\(17\!\cdots\!86\)\( T^{17} + \)\(15\!\cdots\!38\)\( T^{18} + \)\(13\!\cdots\!34\)\( T^{19} + \)\(15\!\cdots\!38\)\( p T^{20} + \)\(17\!\cdots\!86\)\( p^{2} T^{21} + 18922276766488552162 p^{3} T^{22} + 1927041274911330686 p^{4} T^{23} + 189022343080292513 p^{5} T^{24} + 17561592195883763 p^{6} T^{25} + 1569149036104108 p^{7} T^{26} + 131843183828886 p^{8} T^{27} + 10634242395334 p^{9} T^{28} + 796894059719 p^{10} T^{29} + 57238298625 p^{11} T^{30} + 3740306330 p^{12} T^{31} + 234406211 p^{13} T^{32} + 12842100 p^{14} T^{33} + 680427 p^{15} T^{34} + 28835 p^{16} T^{35} + 1227 p^{17} T^{36} + 32 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 - 51 T + 2082 T^{2} - 60320 T^{3} + 1520884 T^{4} - 32377455 T^{5} + 624053883 T^{6} - 10764805837 T^{7} + 171927543584 T^{8} - 2527267835368 T^{9} + 34877971935503 T^{10} - 450100748228450 T^{11} + 5502830968614309 T^{12} - 63523695181462276 T^{13} + 698837312980657430 T^{14} - 7303189643834057099 T^{15} + 73007994854064934486 T^{16} - \)\(69\!\cdots\!98\)\( T^{17} + \)\(63\!\cdots\!58\)\( T^{18} - \)\(55\!\cdots\!84\)\( T^{19} + \)\(63\!\cdots\!58\)\( p T^{20} - \)\(69\!\cdots\!98\)\( p^{2} T^{21} + 73007994854064934486 p^{3} T^{22} - 7303189643834057099 p^{4} T^{23} + 698837312980657430 p^{5} T^{24} - 63523695181462276 p^{6} T^{25} + 5502830968614309 p^{7} T^{26} - 450100748228450 p^{8} T^{27} + 34877971935503 p^{9} T^{28} - 2527267835368 p^{10} T^{29} + 171927543584 p^{11} T^{30} - 10764805837 p^{12} T^{31} + 624053883 p^{13} T^{32} - 32377455 p^{14} T^{33} + 1520884 p^{15} T^{34} - 60320 p^{16} T^{35} + 2082 p^{17} T^{36} - 51 p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 - 30 T + 1323 T^{2} - 30260 T^{3} + 793055 T^{4} - 14916571 T^{5} + 296531429 T^{6} - 4783765086 T^{7} + 78863642538 T^{8} - 1120203367049 T^{9} + 16007575831074 T^{10} - 203725377221417 T^{11} + 2588519002910803 T^{12} - 29871844374454987 T^{13} + 342857430889264467 T^{14} - 3616829807433295478 T^{15} + 37877700456738271513 T^{16} - \)\(36\!\cdots\!51\)\( T^{17} + \)\(35\!\cdots\!65\)\( T^{18} - \)\(31\!\cdots\!06\)\( T^{19} + \)\(35\!\cdots\!65\)\( p T^{20} - \)\(36\!\cdots\!51\)\( p^{2} T^{21} + 37877700456738271513 p^{3} T^{22} - 3616829807433295478 p^{4} T^{23} + 342857430889264467 p^{5} T^{24} - 29871844374454987 p^{6} T^{25} + 2588519002910803 p^{7} T^{26} - 203725377221417 p^{8} T^{27} + 16007575831074 p^{9} T^{28} - 1120203367049 p^{10} T^{29} + 78863642538 p^{11} T^{30} - 4783765086 p^{12} T^{31} + 296531429 p^{13} T^{32} - 14916571 p^{14} T^{33} + 793055 p^{15} T^{34} - 30260 p^{16} T^{35} + 1323 p^{17} T^{36} - 30 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 - T + 11 p T^{2} - 180 T^{3} + 412053 T^{4} + 180103 T^{5} + 122830140 T^{6} + 112698348 T^{7} + 27225502887 T^{8} + 34293478900 T^{9} + 4787033795008 T^{10} + 7085502677768 T^{11} + 695564964961644 T^{12} + 1111539579594292 T^{13} + 85892305345946283 T^{14} + 140345510451137812 T^{15} + 9192190781252108163 T^{16} + 14778538914349895882 T^{17} + \)\(86\!\cdots\!08\)\( T^{18} + \)\(13\!\cdots\!36\)\( T^{19} + \)\(86\!\cdots\!08\)\( p T^{20} + 14778538914349895882 p^{2} T^{21} + 9192190781252108163 p^{3} T^{22} + 140345510451137812 p^{4} T^{23} + 85892305345946283 p^{5} T^{24} + 1111539579594292 p^{6} T^{25} + 695564964961644 p^{7} T^{26} + 7085502677768 p^{8} T^{27} + 4787033795008 p^{9} T^{28} + 34293478900 p^{10} T^{29} + 27225502887 p^{11} T^{30} + 112698348 p^{12} T^{31} + 122830140 p^{13} T^{32} + 180103 p^{14} T^{33} + 412053 p^{15} T^{34} - 180 p^{16} T^{35} + 11 p^{18} T^{36} - p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 - 5 T + 900 T^{2} - 5208 T^{3} + 408753 T^{4} - 2600183 T^{5} + 124544985 T^{6} - 845207354 T^{7} + 28548725824 T^{8} - 202656062858 T^{9} + 5238694079121 T^{10} - 38292815208174 T^{11} + 800260312469475 T^{12} - 5932265418022598 T^{13} + 104523943291455492 T^{14} - 772530166737831198 T^{15} + 11892279508870100778 T^{16} - 85898464458081369316 T^{17} + \)\(11\!\cdots\!55\)\( T^{18} - \)\(82\!\cdots\!80\)\( T^{19} + \)\(11\!\cdots\!55\)\( p T^{20} - 85898464458081369316 p^{2} T^{21} + 11892279508870100778 p^{3} T^{22} - 772530166737831198 p^{4} T^{23} + 104523943291455492 p^{5} T^{24} - 5932265418022598 p^{6} T^{25} + 800260312469475 p^{7} T^{26} - 38292815208174 p^{8} T^{27} + 5238694079121 p^{9} T^{28} - 202656062858 p^{10} T^{29} + 28548725824 p^{11} T^{30} - 845207354 p^{12} T^{31} + 124544985 p^{13} T^{32} - 2600183 p^{14} T^{33} + 408753 p^{15} T^{34} - 5208 p^{16} T^{35} + 900 p^{17} T^{36} - 5 p^{18} T^{37} + p^{19} T^{38} \) | |
| 97 | \( 1 - 5 T + 1119 T^{2} - 7425 T^{3} + 626161 T^{4} - 4997793 T^{5} + 234060233 T^{6} - 2106203122 T^{7} + 65583827219 T^{8} - 634772102388 T^{9} + 14616979308761 T^{10} - 146747567731490 T^{11} + 2681647163351003 T^{12} - 27114376204648986 T^{13} + 413553226986122049 T^{14} - 4106187033289173770 T^{15} + 54307299971976751138 T^{16} - \)\(51\!\cdots\!28\)\( T^{17} + \)\(61\!\cdots\!00\)\( T^{18} - \)\(54\!\cdots\!70\)\( T^{19} + \)\(61\!\cdots\!00\)\( p T^{20} - \)\(51\!\cdots\!28\)\( p^{2} T^{21} + 54307299971976751138 p^{3} T^{22} - 4106187033289173770 p^{4} T^{23} + 413553226986122049 p^{5} T^{24} - 27114376204648986 p^{6} T^{25} + 2681647163351003 p^{7} T^{26} - 146747567731490 p^{8} T^{27} + 14616979308761 p^{9} T^{28} - 634772102388 p^{10} T^{29} + 65583827219 p^{11} T^{30} - 2106203122 p^{12} T^{31} + 234060233 p^{13} T^{32} - 4997793 p^{14} T^{33} + 626161 p^{15} T^{34} - 7425 p^{16} T^{35} + 1119 p^{17} T^{36} - 5 p^{18} T^{37} + p^{19} T^{38} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.00130046211259233211472265590, −1.99539866956247881102050978215, −1.98882658606298297176312518928, −1.78329576736817936165546317700, −1.73849914760989173261888802361, −1.73357953656782122150390657563, −1.69245357136407958663347309432, −1.52021006722839376544459731608, −1.51863790533376504267106231484, −1.43021273435326009039713983583, −1.40577398808723919189932774009, −1.18471649642025002564746640216, −1.17021255706326669069144228432, −1.09857898132201767569363705977, −1.04233699149003389835389818196, −1.02594787257228366011201918454, −0.992112357710300089760527685805, −0.991008627957024292521159421979, −0.896253888835693461900649800402, −0.834988727199391607897201394491, −0.816720887692633796458877383915, −0.796134474680772152522261446384, −0.68043822646494679380232906887, −0.52457773651797861372085042937, −0.39955471558835738304640418982, 0.39955471558835738304640418982, 0.52457773651797861372085042937, 0.68043822646494679380232906887, 0.796134474680772152522261446384, 0.816720887692633796458877383915, 0.834988727199391607897201394491, 0.896253888835693461900649800402, 0.991008627957024292521159421979, 0.992112357710300089760527685805, 1.02594787257228366011201918454, 1.04233699149003389835389818196, 1.09857898132201767569363705977, 1.17021255706326669069144228432, 1.18471649642025002564746640216, 1.40577398808723919189932774009, 1.43021273435326009039713983583, 1.51863790533376504267106231484, 1.52021006722839376544459731608, 1.69245357136407958663347309432, 1.73357953656782122150390657563, 1.73849914760989173261888802361, 1.78329576736817936165546317700, 1.98882658606298297176312518928, 1.99539866956247881102050978215, 2.00130046211259233211472265590
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.