| L(s) = 1 | − 8·11-s − 16·13-s − 8·37-s + 48·47-s − 32·59-s − 40·61-s + 24·83-s + 64·97-s + 16·107-s − 32·109-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 128·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 4.43·13-s − 1.31·37-s + 7.00·47-s − 4.16·59-s − 5.12·61-s + 2.63·83-s + 6.49·97-s + 1.54·107-s − 3.06·109-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.7·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7421823846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7421823846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 36 T^{4} + 1126 T^{8} - 36 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 142 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 56 T^{3} + 62 T^{4} + 56 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + 274 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 612 T^{4} - 384602 T^{8} - 612 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 80 T^{2} + 3074 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 120 T^{2} + 6514 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 53 | \( 1 - 4388 T^{4} + 10107366 T^{8} - 4388 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 16 T + 128 T^{2} + 560 T^{3} + 1874 T^{4} + 560 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 20 T + 200 T^{2} + 1100 T^{3} + 6014 T^{4} + 1100 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 4516 T^{4} + 43788006 T^{8} + 4516 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 100 T^{2} + 5414 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 60 T^{2} + 9766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 48 T^{2} + 9026 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 12 T + 72 T^{2} + 300 T^{3} - 10414 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 144 T^{2} + 9826 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 230 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.77567037135600090835473708118, −3.54184330438436017861238564448, −3.32301647953333059968779938193, −3.31608915378822602859906485225, −3.20443046223421931147318411012, −3.11671918437421293294682606581, −3.08751342036721097181182341929, −3.01172859064016368825186161672, −2.68175810448610393181701786111, −2.52609498634542161018983091134, −2.45517984129134354210366263970, −2.45178195391633857090456911867, −2.35402857228229840295596833234, −2.08363783911096590762084080864, −1.99017875440674645688320945468, −1.93702213027266212880665447466, −1.89380228257844443205358829284, −1.69385847957087126049316656400, −1.30067307837659604835628955409, −1.06623092714846685377071271499, −1.01706857206708345040071740210, −0.63754524525460717533649850051, −0.59208385718503861023620850191, −0.22629666858793277242949230111, −0.17926745778782669681928445959,
0.17926745778782669681928445959, 0.22629666858793277242949230111, 0.59208385718503861023620850191, 0.63754524525460717533649850051, 1.01706857206708345040071740210, 1.06623092714846685377071271499, 1.30067307837659604835628955409, 1.69385847957087126049316656400, 1.89380228257844443205358829284, 1.93702213027266212880665447466, 1.99017875440674645688320945468, 2.08363783911096590762084080864, 2.35402857228229840295596833234, 2.45178195391633857090456911867, 2.45517984129134354210366263970, 2.52609498634542161018983091134, 2.68175810448610393181701786111, 3.01172859064016368825186161672, 3.08751342036721097181182341929, 3.11671918437421293294682606581, 3.20443046223421931147318411012, 3.31608915378822602859906485225, 3.32301647953333059968779938193, 3.54184330438436017861238564448, 3.77567037135600090835473708118
Plot not available for L-functions of degree greater than 10.
