| L(s) = 1 | + 4·3-s + 8·7-s + 4·9-s + 10·16-s + 8·19-s + 32·21-s − 4·27-s + 8·31-s − 4·37-s + 40·48-s + 32·49-s + 32·57-s − 56·61-s + 32·63-s + 8·67-s − 28·73-s + 16·79-s − 10·81-s + 32·93-s + 32·97-s − 64·109-s − 16·111-s + 80·112-s + 127-s + 131-s + 64·133-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 4/3·9-s + 5/2·16-s + 1.83·19-s + 6.98·21-s − 0.769·27-s + 1.43·31-s − 0.657·37-s + 5.77·48-s + 32/7·49-s + 4.23·57-s − 7.17·61-s + 4.03·63-s + 0.977·67-s − 3.27·73-s + 1.80·79-s − 1.11·81-s + 3.31·93-s + 3.24·97-s − 6.13·109-s − 1.51·111-s + 7.55·112-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.84532899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.84532899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - 5 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 22 T^{4} + 939 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( 1 - 100 T^{4} + 4134 T^{8} - 100 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 38 T^{2} + 831 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T + 8 T^{2} - 60 T^{3} + 434 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 34 T^{2} + 1863 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 8 T^{2} - 36 T^{3} - 322 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} - p^{2} T^{4} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 2474 T^{4} + 2947659 T^{8} + 2474 p^{4} T^{12} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 190 T^{2} + 14535 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 6980 T^{4} + 34417254 T^{8} + 6980 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 7 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 4 T + 8 T^{2} - 60 T^{3} - 2254 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 11684 T^{4} + 81904134 T^{8} + 11684 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 17476 T^{4} + 200482374 T^{8} - 17476 p^{4} T^{12} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 16 T + 128 T^{2} - 1200 T^{3} + 10766 T^{4} - 1200 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.84587586974155783040704100822, −4.71609598732893158752847633732, −4.53594734464482209061803838853, −4.47870776692308441017448149064, −4.11078235732421747882092760109, −4.02389652927999046409228721409, −3.99998641039788071664614668989, −3.67779179707514730927934544636, −3.62202520363089056225242024169, −3.36559672767332452165088586022, −3.18162726785623306526391600049, −3.11504070707579075629307607938, −2.95619030771835067367110588427, −2.94254894935334599259644899458, −2.81146271871794290744135972907, −2.61635569794629682528658338350, −2.27764584973697186684329294871, −2.10265355031866332586722121472, −1.89297381903310559239221867201, −1.81752162278138020617486767639, −1.42527144426796813949704219879, −1.35076656734382330975805788655, −1.17871630498751899758297164018, −1.11615308883659204162559493260, −0.44118078367902573697528814189,
0.44118078367902573697528814189, 1.11615308883659204162559493260, 1.17871630498751899758297164018, 1.35076656734382330975805788655, 1.42527144426796813949704219879, 1.81752162278138020617486767639, 1.89297381903310559239221867201, 2.10265355031866332586722121472, 2.27764584973697186684329294871, 2.61635569794629682528658338350, 2.81146271871794290744135972907, 2.94254894935334599259644899458, 2.95619030771835067367110588427, 3.11504070707579075629307607938, 3.18162726785623306526391600049, 3.36559672767332452165088586022, 3.62202520363089056225242024169, 3.67779179707514730927934544636, 3.99998641039788071664614668989, 4.02389652927999046409228721409, 4.11078235732421747882092760109, 4.47870776692308441017448149064, 4.53594734464482209061803838853, 4.71609598732893158752847633732, 4.84587586974155783040704100822
Plot not available for L-functions of degree greater than 10.
