| L(s) = 1 | − 4·13-s − 16-s − 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4·13-s − 16-s − 4·37-s + 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1905090604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1905090604\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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
−9.561850654276350098949230212311, −9.133873685202997301501508553454, −9.082220587181601654647671606172, −8.961480299107341667896798827893, −8.579235042579777653555315927582, −7.88776530148163649023129236316, −7.88211989399789342591853770749, −7.79150562522137738898533873692, −7.37277902640396022440918763588, −6.86688554640222632887659246540, −6.81023583686771266981051501347, −6.78668755735606095915049414151, −6.42809373412478744110268449053, −5.64388873873458188404281857425, −5.36330249790401760675189644478, −5.31695617795032628558620790028, −4.89767024677194076794894509282, −4.60272500648166805124435939652, −4.52548958960700959022514853505, −3.79394951927279425916884978760, −3.44976765883068887818600552393, −3.09402917711668797726896871776, −2.42384697399779368363070105446, −2.19393100119277205174608225177, −1.93238772367325555093755388175,
1.93238772367325555093755388175, 2.19393100119277205174608225177, 2.42384697399779368363070105446, 3.09402917711668797726896871776, 3.44976765883068887818600552393, 3.79394951927279425916884978760, 4.52548958960700959022514853505, 4.60272500648166805124435939652, 4.89767024677194076794894509282, 5.31695617795032628558620790028, 5.36330249790401760675189644478, 5.64388873873458188404281857425, 6.42809373412478744110268449053, 6.78668755735606095915049414151, 6.81023583686771266981051501347, 6.86688554640222632887659246540, 7.37277902640396022440918763588, 7.79150562522137738898533873692, 7.88211989399789342591853770749, 7.88776530148163649023129236316, 8.579235042579777653555315927582, 8.961480299107341667896798827893, 9.082220587181601654647671606172, 9.133873685202997301501508553454, 9.561850654276350098949230212311
