L(s) = 1 | − 4-s − 3·16-s + 8·19-s − 40·49-s − 5·64-s − 48·71-s − 8·76-s + 4·81-s − 16·101-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 40·196-s + 197-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 3/4·16-s + 1.83·19-s − 5.71·49-s − 5/8·64-s − 5.69·71-s − 0.917·76-s + 4/9·81-s − 1.59·101-s + 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 20/7·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.424441743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424441743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T^{2} + p^{2} T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( 1 - 4 T^{4} + 16 T^{6} + 70 T^{8} + 16 p^{2} T^{10} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 + 20 T^{2} + 244 T^{4} + 2364 T^{6} + 18678 T^{8} + 2364 p^{2} T^{10} + 244 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 14424 p^{2} T^{10} + 892 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 4 p T^{2} + 1556 T^{4} + 31660 T^{6} + 477814 T^{8} + 31660 p^{2} T^{10} + 1556 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 84 T^{2} + 212 p T^{4} + 101164 T^{6} + 2018902 T^{8} + 101164 p^{2} T^{10} + 212 p^{5} T^{12} + 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 2 T + 40 T^{2} - 42 T^{3} + 766 T^{4} - 42 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 - 4 p T^{2} + 4420 T^{4} - 144852 T^{6} + 3702742 T^{8} - 144852 p^{2} T^{10} + 4420 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 52 T^{2} + 112 T^{3} + 1286 T^{4} + 112 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 170900 p^{2} T^{10} + 5412 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 68 T^{2} + 4788 T^{4} + 250460 T^{6} + 9311478 T^{8} + 250460 p^{2} T^{10} + 4788 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 2456248 p^{2} T^{10} + 32668 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 320 T^{2} + 45756 T^{4} - 3816624 T^{6} + 203071110 T^{8} - 3816624 p^{2} T^{10} + 45756 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 140 T^{2} + 284 p T^{4} - 854628 T^{6} + 45549910 T^{8} - 854628 p^{2} T^{10} + 284 p^{5} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 312 T^{2} + 45948 T^{4} - 4212040 T^{6} + 265479398 T^{8} - 4212040 p^{2} T^{10} + 45948 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 3304648 p^{2} T^{10} + 41500 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 3783016 p^{2} T^{10} + 39452 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 176 T^{2} + 15964 T^{4} - 993216 T^{6} + 63719302 T^{8} - 993216 p^{2} T^{10} + 15964 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 12 T + 220 T^{2} + 1564 T^{3} + 18854 T^{4} + 1564 p T^{5} + 220 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 344 T^{2} + 60316 T^{4} - 7095528 T^{6} + 604376710 T^{8} - 7095528 p^{2} T^{10} + 60316 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 18135668 p^{2} T^{10} + 125604 p^{4} T^{12} - 524 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 512 T^{2} + 125148 T^{4} + 18790160 T^{6} + 1886502918 T^{8} + 18790160 p^{2} T^{10} + 125148 p^{4} T^{12} + 512 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 8336760 p^{2} T^{10} + 61788 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 408 T^{2} + 72412 T^{4} - 7810088 T^{6} + 720805318 T^{8} - 7810088 p^{2} T^{10} + 72412 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.93664428919771614263526792890, −2.76645854160017676116640780899, −2.69490745605049380088313652632, −2.59529279406596239747145831951, −2.46245770550679564616647207401, −2.37895822866885816830078079434, −2.30650396967709347763536781537, −2.30268909714447250067437783516, −2.29550417764970507851758942790, −2.21838706209154803012952388394, −2.07571312863065881747715948903, −1.78255500560737514818526682853, −1.67589276211129804961176080442, −1.64355104061000545083358813022, −1.62117513450111748204195663702, −1.37745482804058425512077373682, −1.31685093124457451312465225366, −1.24399258855644690271547033737, −1.19845135681385212840665154662, −1.18677140194481391223941513813, −1.09116702674477044598071291462, −0.58847974629454440193177187989, −0.36871250568059751031547253042, −0.28470287858164367942346339029, −0.19040740785265302950565128852,
0.19040740785265302950565128852, 0.28470287858164367942346339029, 0.36871250568059751031547253042, 0.58847974629454440193177187989, 1.09116702674477044598071291462, 1.18677140194481391223941513813, 1.19845135681385212840665154662, 1.24399258855644690271547033737, 1.31685093124457451312465225366, 1.37745482804058425512077373682, 1.62117513450111748204195663702, 1.64355104061000545083358813022, 1.67589276211129804961176080442, 1.78255500560737514818526682853, 2.07571312863065881747715948903, 2.21838706209154803012952388394, 2.29550417764970507851758942790, 2.30268909714447250067437783516, 2.30650396967709347763536781537, 2.37895822866885816830078079434, 2.46245770550679564616647207401, 2.59529279406596239747145831951, 2.69490745605049380088313652632, 2.76645854160017676116640780899, 2.93664428919771614263526792890
Plot not available for L-functions of degree greater than 10.