| L(s) = 1 | − 2·49-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 2·49-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.269999160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269999160\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.81602805952662671527989733065, −6.41620939846779624137200615489, −6.39954639714155647403833438354, −5.91417226871951285500437089425, −5.87805167280411652657719349527, −5.75967228228394173153998636269, −5.61589766192628420569618602160, −5.04820317817149915537790032745, −5.01696770359342235398471554338, −4.72460924295937248678957118743, −4.57565632907693409972007276983, −4.57235103219738370142888263103, −4.10731287785860528203320251107, −3.69769869507050417664612806438, −3.62851429915875349752060224539, −3.55734639438890375042461572094, −3.01658380050073067093677843413, −2.95097373307377632743712870248, −2.75275375171096306086207964270, −2.32073328036948585527311670018, −1.85049143501637111046529227519, −1.83053504815339100853283845117, −1.66575440404263102548913708498, −0.911336349285537840221428926381, −0.68315849343281942353411925020,
0.68315849343281942353411925020, 0.911336349285537840221428926381, 1.66575440404263102548913708498, 1.83053504815339100853283845117, 1.85049143501637111046529227519, 2.32073328036948585527311670018, 2.75275375171096306086207964270, 2.95097373307377632743712870248, 3.01658380050073067093677843413, 3.55734639438890375042461572094, 3.62851429915875349752060224539, 3.69769869507050417664612806438, 4.10731287785860528203320251107, 4.57235103219738370142888263103, 4.57565632907693409972007276983, 4.72460924295937248678957118743, 5.01696770359342235398471554338, 5.04820317817149915537790032745, 5.61589766192628420569618602160, 5.75967228228394173153998636269, 5.87805167280411652657719349527, 5.91417226871951285500437089425, 6.39954639714155647403833438354, 6.41620939846779624137200615489, 6.81602805952662671527989733065
