| L(s) = 1 | − 3-s + 3·4-s − 5-s + 3·11-s − 3·12-s + 15-s + 6·16-s − 3·20-s − 23-s − 31-s − 3·33-s − 37-s + 9·44-s − 47-s − 6·48-s + 3·49-s − 53-s − 3·55-s − 59-s + 3·60-s + 10·64-s + 6·67-s + 69-s − 71-s − 6·80-s − 89-s − 3·92-s + ⋯ |
| L(s) = 1 | − 3-s + 3·4-s − 5-s + 3·11-s − 3·12-s + 15-s + 6·16-s − 3·20-s − 23-s − 31-s − 3·33-s − 37-s + 9·44-s − 47-s − 6·48-s + 3·49-s − 53-s − 3·55-s − 59-s + 3·60-s + 10·64-s + 6·67-s + 69-s − 71-s − 6·80-s − 89-s − 3·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.003966629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003966629\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$ | \( ( 1 - T )^{6} \) |
| 71 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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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
−8.415793773469297177210293674414, −7.961988389777494940333401401128, −7.84251593840551186637965948423, −7.37200228352772388731079307821, −7.24302629205076807798147706619, −6.94919723085321288147877295584, −6.79640938010960423199608962464, −6.52550399761612344959507505872, −6.40568753789253779618788994161, −6.09943539099223884820044861384, −5.73449726855556323737518920849, −5.56659645466655133035878205957, −5.43136315875746284702129437035, −4.88218089659646155974992026746, −4.41904343795872049697922612782, −3.95510519673818623311143131366, −3.67127816214404962352341152175, −3.63519578290581189004729671776, −3.50146483921704539811110885925, −2.75153906743668587167462864341, −2.45652022995288226292771721013, −2.08108240462824727812101146251, −1.71579201571265490817666154187, −1.30460705206415413718097051199, −1.00581983433485497460450871395,
1.00581983433485497460450871395, 1.30460705206415413718097051199, 1.71579201571265490817666154187, 2.08108240462824727812101146251, 2.45652022995288226292771721013, 2.75153906743668587167462864341, 3.50146483921704539811110885925, 3.63519578290581189004729671776, 3.67127816214404962352341152175, 3.95510519673818623311143131366, 4.41904343795872049697922612782, 4.88218089659646155974992026746, 5.43136315875746284702129437035, 5.56659645466655133035878205957, 5.73449726855556323737518920849, 6.09943539099223884820044861384, 6.40568753789253779618788994161, 6.52550399761612344959507505872, 6.79640938010960423199608962464, 6.94919723085321288147877295584, 7.24302629205076807798147706619, 7.37200228352772388731079307821, 7.84251593840551186637965948423, 7.961988389777494940333401401128, 8.415793773469297177210293674414
