| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·13-s − 3·17-s − 3·24-s + 10·27-s − 9·39-s + 3·49-s − 9·51-s − 3·59-s − 6·72-s + 15·81-s + 3·104-s − 18·117-s − 125-s + 127-s + 131-s + 3·136-s + 137-s + 139-s + 9·147-s + 149-s + 151-s − 18·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·13-s − 3·17-s − 3·24-s + 10·27-s − 9·39-s + 3·49-s − 9·51-s − 3·59-s − 6·72-s + 15·81-s + 3·104-s − 18·117-s − 125-s + 127-s + 131-s + 3·136-s + 137-s + 139-s + 9·147-s + 149-s + 151-s − 18·153-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 173^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 173^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.473631837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473631837\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 173 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 5 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−9.844947749369434761669096721369, −9.533763183008743960442797685706, −9.075927557539539072959309281406, −9.039355526492961333323009119728, −8.937873785282916147802921786139, −8.508179438516865945990149840596, −8.268463212767749178777201900414, −7.70248027015074319328410759311, −7.56019518931137475330869109535, −7.27846421355471961024682416328, −7.13062739175661670006817702224, −6.70092312068472312067729836329, −6.38694107018888587171574678837, −6.06850324518883959558728895011, −5.22708317617157767195727244088, −4.90767244939729125471027408110, −4.62307470209508745188468536426, −4.26701167201518961192013830474, −4.04755401118983425228002859268, −3.52542595512853469910814851195, −2.97246200018243753142458650210, −2.55192010465386358318022388375, −2.54012360041250452071602629720, −2.18390903213174704432551120741, −1.62173881495108181743445002753,
1.62173881495108181743445002753, 2.18390903213174704432551120741, 2.54012360041250452071602629720, 2.55192010465386358318022388375, 2.97246200018243753142458650210, 3.52542595512853469910814851195, 4.04755401118983425228002859268, 4.26701167201518961192013830474, 4.62307470209508745188468536426, 4.90767244939729125471027408110, 5.22708317617157767195727244088, 6.06850324518883959558728895011, 6.38694107018888587171574678837, 6.70092312068472312067729836329, 7.13062739175661670006817702224, 7.27846421355471961024682416328, 7.56019518931137475330869109535, 7.70248027015074319328410759311, 8.268463212767749178777201900414, 8.508179438516865945990149840596, 8.937873785282916147802921786139, 9.039355526492961333323009119728, 9.075927557539539072959309281406, 9.533763183008743960442797685706, 9.844947749369434761669096721369
