| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5229048140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5229048140\) |
| \(L(1)\) |
|
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
−8.292539683389288985112163405555, −7.71621506664115351524724123027, −7.68448898919329146098893357109, −7.28625155324289672698272817718, −6.99658498534019354268492847128, −6.91906838436844395268577660582, −6.57361497476501462714389800745, −6.33749050401458408994139787064, −6.07897706859304921688609895872, −6.06636784162407590066444391644, −5.37791348032863760307696527283, −5.26844723531087514107282166394, −5.05295758961349030166920206413, −4.83731914147215123808324809836, −4.53080275361845858840565894443, −4.35136885190179736969885903481, −4.02100546134468548233210777415, −3.69950209953707938992029285562, −3.13302973908668216888296534943, −2.98179984216700250557221153793, −2.06303999472383132311104479016, −1.93003054497361431391125367017, −1.47579939597712814270817544533, −1.05380231752988230136180855778, −0.62600961268415776705364245821,
0.62600961268415776705364245821, 1.05380231752988230136180855778, 1.47579939597712814270817544533, 1.93003054497361431391125367017, 2.06303999472383132311104479016, 2.98179984216700250557221153793, 3.13302973908668216888296534943, 3.69950209953707938992029285562, 4.02100546134468548233210777415, 4.35136885190179736969885903481, 4.53080275361845858840565894443, 4.83731914147215123808324809836, 5.05295758961349030166920206413, 5.26844723531087514107282166394, 5.37791348032863760307696527283, 6.06636784162407590066444391644, 6.07897706859304921688609895872, 6.33749050401458408994139787064, 6.57361497476501462714389800745, 6.91906838436844395268577660582, 6.99658498534019354268492847128, 7.28625155324289672698272817718, 7.68448898919329146098893357109, 7.71621506664115351524724123027, 8.292539683389288985112163405555
