| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3916897324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3916897324\) |
| \(L(1)\) |
|
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$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−7.898304037184150965892262877441, −7.78568402852589979850302836332, −7.58483552922305159556451678421, −7.43982266164931105211698921052, −7.06299408771861551595904807137, −6.92166442159257411947785103629, −6.68578315958118128747274737475, −6.25909602560367724146302844124, −6.00700754698867846014773369746, −5.72934784548585606706766407868, −5.67830604541890026810108894041, −5.44725657725167988838580996213, −4.99712873200408410429398761002, −4.80202232634207885304732900559, −4.43708997874958087758069894207, −4.35847235339394158100138240385, −3.92581744283433777952756655802, −3.66809801714371974600384617121, −3.52757895222731274896147097187, −2.51610797159669187321861614201, −2.41317590431588517460914701473, −1.73070337497445164535221643354, −1.51489068005775694053626102262, −1.16696546040772382780751521796, −0.49730740771534572588371024792,
0.49730740771534572588371024792, 1.16696546040772382780751521796, 1.51489068005775694053626102262, 1.73070337497445164535221643354, 2.41317590431588517460914701473, 2.51610797159669187321861614201, 3.52757895222731274896147097187, 3.66809801714371974600384617121, 3.92581744283433777952756655802, 4.35847235339394158100138240385, 4.43708997874958087758069894207, 4.80202232634207885304732900559, 4.99712873200408410429398761002, 5.44725657725167988838580996213, 5.67830604541890026810108894041, 5.72934784548585606706766407868, 6.00700754698867846014773369746, 6.25909602560367724146302844124, 6.68578315958118128747274737475, 6.92166442159257411947785103629, 7.06299408771861551595904807137, 7.43982266164931105211698921052, 7.58483552922305159556451678421, 7.78568402852589979850302836332, 7.898304037184150965892262877441
