| L(s) = 1 | + 3·9-s − 3·37-s − 3·41-s − 3·47-s + 6·81-s − 3·107-s − 125-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 | + 3·9-s − 3·37-s − 3·41-s − 3·47-s + 6·81-s − 3·107-s − 125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 139^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 139^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8439494429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8439494429\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 139 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 71 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 83 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.788661601315888432384405961416, −9.625159032886537329294117355695, −9.518039797260456856166629787138, −8.922045382921785223474803956639, −8.558821683987648837087736214198, −8.261856121642881804259311215134, −8.209769430083702859476409816669, −7.53937905612630565620310013975, −7.39097478646087166708314193680, −7.04659144658388240465858287608, −6.80369532624684592090137605871, −6.50947122576484676971056830156, −6.45845934079262220818056326496, −5.75495505566950166917321723389, −5.11944368240211163346727816931, −5.03527022274373633855495350583, −4.90069761193032046705637342732, −4.35884754923170880510400858619, −3.94130641228020241389082586604, −3.50328878102433499777978324623, −3.49610123320827543077676186837, −2.77675758033068059221685451914, −1.92607170671426838563203550583, −1.59748910079787490516288002376, −1.48526173356460361114413472379,
1.48526173356460361114413472379, 1.59748910079787490516288002376, 1.92607170671426838563203550583, 2.77675758033068059221685451914, 3.49610123320827543077676186837, 3.50328878102433499777978324623, 3.94130641228020241389082586604, 4.35884754923170880510400858619, 4.90069761193032046705637342732, 5.03527022274373633855495350583, 5.11944368240211163346727816931, 5.75495505566950166917321723389, 6.45845934079262220818056326496, 6.50947122576484676971056830156, 6.80369532624684592090137605871, 7.04659144658388240465858287608, 7.39097478646087166708314193680, 7.53937905612630565620310013975, 8.209769430083702859476409816669, 8.261856121642881804259311215134, 8.558821683987648837087736214198, 8.922045382921785223474803956639, 9.518039797260456856166629787138, 9.625159032886537329294117355695, 9.788661601315888432384405961416
