| L(s) = 1 | + 4·19-s + 2·49-s − 4·61-s − 4·79-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 + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 4·19-s + 2·49-s − 4·61-s − 4·79-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 + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.090136027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090136027\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−7.44975303052875230523096987207, −7.02236779013932368493349869111, −6.82041338045018042892834334766, −6.70516456762575239617454662887, −6.60789092096609919081771770500, −5.84197774650761826389011109558, −5.83546181186525117480088096184, −5.74051901653664801545713939703, −5.64874430044957381201714166989, −5.39488487704278120896722878890, −4.85267865907708623276420154481, −4.74684725050412620870627595115, −4.73583662097147993020629515256, −4.26603112404279290351173036892, −3.96030308687788534337122033046, −3.81595426820235872111729045947, −3.32993078421676406979888763454, −3.20423035322691142272678170533, −2.87581608948235562614761370286, −2.85907310983920148881395774872, −2.48326890417605766335319548930, −1.82798030261596932416146894108, −1.62372811312655153054184665337, −1.21180907453429959613752507040, −0.927855472286996525520908739439,
0.927855472286996525520908739439, 1.21180907453429959613752507040, 1.62372811312655153054184665337, 1.82798030261596932416146894108, 2.48326890417605766335319548930, 2.85907310983920148881395774872, 2.87581608948235562614761370286, 3.20423035322691142272678170533, 3.32993078421676406979888763454, 3.81595426820235872111729045947, 3.96030308687788534337122033046, 4.26603112404279290351173036892, 4.73583662097147993020629515256, 4.74684725050412620870627595115, 4.85267865907708623276420154481, 5.39488487704278120896722878890, 5.64874430044957381201714166989, 5.74051901653664801545713939703, 5.83546181186525117480088096184, 5.84197774650761826389011109558, 6.60789092096609919081771770500, 6.70516456762575239617454662887, 6.82041338045018042892834334766, 7.02236779013932368493349869111, 7.44975303052875230523096987207
