| L(s) = 1 | − 32·4-s − 376·13-s + 640·16-s + 1.12e3·19-s + 2.89e3·25-s − 880·31-s + 1.57e3·37-s − 5.76e3·43-s + 1.37e3·49-s + 1.20e4·52-s − 2.56e3·61-s − 1.02e4·64-s + 2.37e4·67-s − 1.31e4·73-s − 3.58e4·76-s − 9.59e3·79-s − 1.40e4·97-s − 9.26e4·100-s − 5.74e3·103-s + 7.61e4·109-s + 4.53e4·121-s + 2.81e4·124-s + 127-s + 131-s + 137-s + 139-s − 5.04e4·148-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2.22·13-s + 5/2·16-s + 3.10·19-s + 4.63·25-s − 0.915·31-s + 1.15·37-s − 3.11·43-s + 4/7·49-s + 4.44·52-s − 0.687·61-s − 5/2·64-s + 5.29·67-s − 2.47·73-s − 6.20·76-s − 1.53·79-s − 1.48·97-s − 9.26·100-s − 0.541·103-s + 6.41·109-s + 3.09·121-s + 1.83·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 2.30·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.141054502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141054502\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| good | 5 | \( 1 - 2896 T^{2} + 3638054 T^{4} - 2780750784 T^{6} + 1742925696491 T^{8} - 2780750784 p^{8} T^{10} + 3638054 p^{16} T^{12} - 2896 p^{24} T^{14} + p^{32} T^{16} \) |
| 11 | \( 1 - 45376 T^{2} + 1125096230 T^{4} - 1811353663872 p T^{6} + 310095746834536235 T^{8} - 1811353663872 p^{9} T^{10} + 1125096230 p^{16} T^{12} - 45376 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 188 T + 74156 T^{2} + 8050092 T^{3} + 2287381274 T^{4} + 8050092 p^{4} T^{5} + 74156 p^{8} T^{6} + 188 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 158704 T^{2} + 29778019076 T^{4} - 3081760811859408 T^{6} + \)\(31\!\cdots\!86\)\( T^{8} - 3081760811859408 p^{8} T^{10} + 29778019076 p^{16} T^{12} - 158704 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 - 560 T + 377398 T^{2} - 156336320 T^{3} + 75881622811 T^{4} - 156336320 p^{4} T^{5} + 377398 p^{8} T^{6} - 560 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1281040 T^{2} + 888819235094 T^{4} - 405553787863355712 T^{6} + \)\(13\!\cdots\!15\)\( T^{8} - 405553787863355712 p^{8} T^{10} + 888819235094 p^{16} T^{12} - 1281040 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 2268016 T^{2} + 3273567985028 T^{4} - 3388632307546760784 T^{6} + \)\(27\!\cdots\!98\)\( T^{8} - 3388632307546760784 p^{8} T^{10} + 3273567985028 p^{16} T^{12} - 2268016 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 440 T + 2095934 T^{2} + 1329203904 T^{3} + 2254233834035 T^{4} + 1329203904 p^{4} T^{5} + 2095934 p^{8} T^{6} + 440 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 788 T + 2100034 T^{2} - 2401740416 T^{3} + 2338205615659 T^{4} - 2401740416 p^{4} T^{5} + 2100034 p^{8} T^{6} - 788 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 6887696 T^{2} + 28777618813030 T^{4} - 91098876450878501312 T^{6} + \)\(23\!\cdots\!95\)\( T^{8} - 91098876450878501312 p^{8} T^{10} + 28777618813030 p^{16} T^{12} - 6887696 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 2884 T + 7001404 T^{2} + 5531256052 T^{3} + 7968256367770 T^{4} + 5531256052 p^{4} T^{5} + 7001404 p^{8} T^{6} + 2884 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 25995728 T^{2} + 331798032373252 T^{4} - \)\(27\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!00\)\( p^{8} T^{10} + 331798032373252 p^{16} T^{12} - 25995728 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 18145448 T^{2} + 321230002760956 T^{4} - \)\(33\!\cdots\!24\)\( T^{6} + \)\(32\!\cdots\!86\)\( T^{8} - \)\(33\!\cdots\!24\)\( p^{8} T^{10} + 321230002760956 p^{16} T^{12} - 18145448 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 44142112 T^{2} + 1088926691812772 T^{4} - \)\(18\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!18\)\( T^{8} - \)\(18\!\cdots\!68\)\( p^{8} T^{10} + 1088926691812772 p^{16} T^{12} - 44142112 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 + 1280 T + 13865420 T^{2} + 17676181248 T^{3} + 395149923191270 T^{4} + 17676181248 p^{4} T^{5} + 13865420 p^{8} T^{6} + 1280 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 - 11892 T + 79780940 T^{2} - 378965460804 T^{3} + 1752487431659994 T^{4} - 378965460804 p^{4} T^{5} + 79780940 p^{8} T^{6} - 11892 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 + 13087616 T^{2} + 992211923650454 T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(31\!\cdots\!75\)\( T^{8} - \)\(11\!\cdots\!04\)\( p^{8} T^{10} + 992211923650454 p^{16} T^{12} + 13087616 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 + 6588 T + 111876188 T^{2} + 533549603340 T^{3} + 4720448405156442 T^{4} + 533549603340 p^{4} T^{5} + 111876188 p^{8} T^{6} + 6588 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4796 T + 124292444 T^{2} + 471046796940 T^{3} + 6993960834109274 T^{4} + 471046796940 p^{4} T^{5} + 124292444 p^{8} T^{6} + 4796 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 329875648 T^{2} + 49399645099250468 T^{4} - \)\(43\!\cdots\!88\)\( T^{6} + \)\(25\!\cdots\!78\)\( T^{8} - \)\(43\!\cdots\!88\)\( p^{8} T^{10} + 49399645099250468 p^{16} T^{12} - 329875648 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 49420160 T^{2} + 5451257749574998 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(31\!\cdots\!31\)\( T^{8} - \)\(47\!\cdots\!04\)\( p^{8} T^{10} + 5451257749574998 p^{16} T^{12} - 49420160 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 7008 T + 168353996 T^{2} + 1429649031072 T^{3} + 18573365118818982 T^{4} + 1429649031072 p^{4} T^{5} + 168353996 p^{8} T^{6} + 7008 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.32545283920799552824860322883, −4.19690710491412862262798159872, −4.13228401241459460325864897792, −4.03591844614972998988958645703, −3.58698784018896417907064872133, −3.46993666721800086059087491954, −3.39744879613712107459600073403, −3.38532514986569486799784649051, −3.11699167478030340110464905887, −2.97427671985590254136546223384, −2.81506357174174296847056613641, −2.78203060534989175820819682109, −2.59085786647827723682539287844, −2.19566692881102609097455282544, −2.07371700635607461067188461107, −1.86768549578199734363258936981, −1.85387923887005231103903729991, −1.27617103303613631443816099036, −1.13569369437216700635883305390, −1.05628378193242443144103720114, −1.00285219598287761496904386527, −0.74713715857206539878913860499, −0.62457226908631819786229960391, −0.17812309010728612900732288954, −0.15735331539341047386914777671,
0.15735331539341047386914777671, 0.17812309010728612900732288954, 0.62457226908631819786229960391, 0.74713715857206539878913860499, 1.00285219598287761496904386527, 1.05628378193242443144103720114, 1.13569369437216700635883305390, 1.27617103303613631443816099036, 1.85387923887005231103903729991, 1.86768549578199734363258936981, 2.07371700635607461067188461107, 2.19566692881102609097455282544, 2.59085786647827723682539287844, 2.78203060534989175820819682109, 2.81506357174174296847056613641, 2.97427671985590254136546223384, 3.11699167478030340110464905887, 3.38532514986569486799784649051, 3.39744879613712107459600073403, 3.46993666721800086059087491954, 3.58698784018896417907064872133, 4.03591844614972998988958645703, 4.13228401241459460325864897792, 4.19690710491412862262798159872, 4.32545283920799552824860322883
Plot not available for L-functions of degree greater than 10.
