| L(s) = 1 | + 7-s − 4·13-s + 3·19-s + 25-s + 3·31-s − 2·37-s + 61-s − 73-s − 4·91-s + 2·97-s − 109-s − 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 7-s − 4·13-s + 3·19-s + 25-s + 3·31-s − 2·37-s + 61-s − 73-s − 4·91-s + 2·97-s − 109-s − 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.370679161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370679161\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.146693244305479073729174041854, −8.777566555501710606406243005225, −8.098636429704326038154948349676, −7.899457029760147055175703262582, −7.71639180747147600790581683550, −7.09409056540165933756285349416, −6.96206432296125985913692154291, −6.83733614216917915903911867863, −5.91158456864191027091344266199, −5.46284629901417818555903742545, −5.04013905747245170667003445527, −4.92993896973130290987753981774, −4.70943401038473946306786990320, −4.20143540156681904659524904468, −3.24928138171835903276612853841, −3.06206064419376458676994437108, −2.56752116197349661251024892676, −2.19324552249228171479764119394, −1.45063366023305627670041675194, −0.75961566223952875040576801058,
0.75961566223952875040576801058, 1.45063366023305627670041675194, 2.19324552249228171479764119394, 2.56752116197349661251024892676, 3.06206064419376458676994437108, 3.24928138171835903276612853841, 4.20143540156681904659524904468, 4.70943401038473946306786990320, 4.92993896973130290987753981774, 5.04013905747245170667003445527, 5.46284629901417818555903742545, 5.91158456864191027091344266199, 6.83733614216917915903911867863, 6.96206432296125985913692154291, 7.09409056540165933756285349416, 7.71639180747147600790581683550, 7.899457029760147055175703262582, 8.098636429704326038154948349676, 8.777566555501710606406243005225, 9.146693244305479073729174041854
