| L(s) = 1 | + 4·4-s + 10·16-s − 4·25-s + 20·64-s + 4·97-s − 16·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 | + 4·4-s + 10·16-s − 4·25-s + 20·64-s + 4·97-s − 16·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(6.058372085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.058372085\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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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
−6.47896242437598672541613328868, −6.10000461034407803781073997851, −5.96527130959385211439344471664, −5.93224957270222222737158279360, −5.71557868154761541621362927897, −5.38588173126502152311447486104, −5.29381553457790105101882656502, −5.05556988636564458177397510377, −4.78494384024374958637999385871, −4.41526534156070664886655197547, −4.10294859997713759860091138422, −3.83425013407645239092474736472, −3.82070752023145197271420709173, −3.48356403681026982115537413227, −3.29401099533779911087651522055, −3.05550009227584157588296114351, −2.94873373982252916440742829324, −2.58563405145722755519026087337, −2.22634970037347322430525235748, −2.08904261085142173650190533901, −1.99624627511606155583601100987, −1.80857182052585382954070619772, −1.62842713776413164374217909733, −0.987091595782885604660525575453, −0.895633476524245239422701637981,
0.895633476524245239422701637981, 0.987091595782885604660525575453, 1.62842713776413164374217909733, 1.80857182052585382954070619772, 1.99624627511606155583601100987, 2.08904261085142173650190533901, 2.22634970037347322430525235748, 2.58563405145722755519026087337, 2.94873373982252916440742829324, 3.05550009227584157588296114351, 3.29401099533779911087651522055, 3.48356403681026982115537413227, 3.82070752023145197271420709173, 3.83425013407645239092474736472, 4.10294859997713759860091138422, 4.41526534156070664886655197547, 4.78494384024374958637999385871, 5.05556988636564458177397510377, 5.29381553457790105101882656502, 5.38588173126502152311447486104, 5.71557868154761541621362927897, 5.93224957270222222737158279360, 5.96527130959385211439344471664, 6.10000461034407803781073997851, 6.47896242437598672541613328868
