| L(s) = 1 | + 8-s − 3·23-s + 27-s + 3·49-s − 3·59-s − 3·101-s + 3·121-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 − 3·184-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-s + 223-s + ⋯ |
| L(s) = 1 | + 8-s − 3·23-s + 27-s + 3·49-s − 3·59-s − 3·101-s + 3·121-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 − 3·184-s + 191-s + 193-s + 197-s + 199-s + 211-s + 216-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7407263897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7407263897\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 3 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−9.831735902917001771170123080028, −9.566974037216345423159262952153, −9.178620434056004068424555058636, −8.906980985779761061261237373403, −8.559896395279451024917339072941, −8.294491515645224017743565176338, −7.82280815387553346203126681246, −7.72066815556037169657916700904, −7.62290659815204742245646005111, −6.90099261831250610072036111005, −6.87711148929099842553837276550, −6.47397233637545652841735518004, −5.87818792632542254313325019652, −5.85312810210945507781523823885, −5.54940914813480352812913792073, −5.01717691935053124454113117108, −4.53258870454623338382412597617, −4.28715086012524521325495839169, −4.15527193395974415839074896834, −3.65652723499038769190938768506, −3.09760388663792715481797239514, −2.73887186105316952670678253550, −2.02466515105394899431729173924, −1.89718424583192943844562804149, −1.12148389387848592855986208856,
1.12148389387848592855986208856, 1.89718424583192943844562804149, 2.02466515105394899431729173924, 2.73887186105316952670678253550, 3.09760388663792715481797239514, 3.65652723499038769190938768506, 4.15527193395974415839074896834, 4.28715086012524521325495839169, 4.53258870454623338382412597617, 5.01717691935053124454113117108, 5.54940914813480352812913792073, 5.85312810210945507781523823885, 5.87818792632542254313325019652, 6.47397233637545652841735518004, 6.87711148929099842553837276550, 6.90099261831250610072036111005, 7.62290659815204742245646005111, 7.72066815556037169657916700904, 7.82280815387553346203126681246, 8.294491515645224017743565176338, 8.559896395279451024917339072941, 8.906980985779761061261237373403, 9.178620434056004068424555058636, 9.566974037216345423159262952153, 9.831735902917001771170123080028
