| L(s) = 1 | − 16-s − 4·61-s − 4·79-s + 4·97-s + 4·103-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 16-s − 4·61-s − 4·79-s + 4·97-s + 4·103-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6868520637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6868520637\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.57756054427567159612012094881, −6.38984767765554831714236969744, −6.11291575861926805959168187419, −6.10055728291232742287314563741, −6.02080508406673625062044473748, −5.67446966181710748852418363260, −5.28538032218939531118593246962, −5.12733775927277385797823846845, −4.91329552612959679352130677452, −4.73286837016023124939932928017, −4.67306372519341234469690156285, −4.29337718196958449001127094337, −4.06782762135547836546210517358, −3.75139535317117595261612570909, −3.75032821598482199289521544378, −3.35839383917449221237263436594, −3.03901239264438875328493908012, −2.85830229168009217376887100185, −2.57723271062309493000018427452, −2.34840303986454687848843743689, −2.17243039326751140882828853663, −1.49683688304972513288255829459, −1.41097822543807620976198449082, −1.35529478392443849270202305131, −0.38826245540949152131986365804,
0.38826245540949152131986365804, 1.35529478392443849270202305131, 1.41097822543807620976198449082, 1.49683688304972513288255829459, 2.17243039326751140882828853663, 2.34840303986454687848843743689, 2.57723271062309493000018427452, 2.85830229168009217376887100185, 3.03901239264438875328493908012, 3.35839383917449221237263436594, 3.75032821598482199289521544378, 3.75139535317117595261612570909, 4.06782762135547836546210517358, 4.29337718196958449001127094337, 4.67306372519341234469690156285, 4.73286837016023124939932928017, 4.91329552612959679352130677452, 5.12733775927277385797823846845, 5.28538032218939531118593246962, 5.67446966181710748852418363260, 6.02080508406673625062044473748, 6.10055728291232742287314563741, 6.11291575861926805959168187419, 6.38984767765554831714236969744, 6.57756054427567159612012094881
