| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s + 3·29-s − 3·41-s − 6·72-s + 15·81-s + 9·87-s − 3·103-s − 9·123-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 + ⋯ |
| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s + 3·29-s − 3·41-s − 6·72-s + 15·81-s + 9·87-s − 3·103-s − 9·123-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.515133046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.515133046\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.408644787817091106536694094947, −7.991267099831890594699457123940, −7.86389041499769093945870291767, −7.66275555624468185180681167655, −7.31490326498810007029635375582, −6.80849427207487440746421475832, −6.72796410866918711335748441382, −6.68791974876211866439026023359, −6.34545213277789513529148227185, −6.02107150574453049313613739909, −5.33594900268624110175744969100, −5.19004193341438494320000767310, −4.93788023362431131508179829057, −4.47194071957267156608671701903, −4.33743472566190014343759443314, −3.98505075658530088173752754451, −3.54680228661875330699676854351, −3.49223951512014967179331160779, −2.94212328958274825648708830591, −2.83614782990033014927984572165, −2.74687679771882961240533101605, −2.13497346391147626444047201176, −1.93764703926138004375136775211, −1.25939835582413232369720352277, −1.18432516009006482961781076353,
1.18432516009006482961781076353, 1.25939835582413232369720352277, 1.93764703926138004375136775211, 2.13497346391147626444047201176, 2.74687679771882961240533101605, 2.83614782990033014927984572165, 2.94212328958274825648708830591, 3.49223951512014967179331160779, 3.54680228661875330699676854351, 3.98505075658530088173752754451, 4.33743472566190014343759443314, 4.47194071957267156608671701903, 4.93788023362431131508179829057, 5.19004193341438494320000767310, 5.33594900268624110175744969100, 6.02107150574453049313613739909, 6.34545213277789513529148227185, 6.68791974876211866439026023359, 6.72796410866918711335748441382, 6.80849427207487440746421475832, 7.31490326498810007029635375582, 7.66275555624468185180681167655, 7.86389041499769093945870291767, 7.991267099831890594699457123940, 8.408644787817091106536694094947
