| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 5-s + 3·6-s − 10·8-s − 3·10-s + 11-s − 6·12-s + 3·13-s − 15-s + 15·16-s − 17-s − 3·19-s + 6·20-s − 3·22-s − 23-s + 10·24-s − 9·26-s − 29-s + 3·30-s + 31-s − 21·32-s − 33-s + 3·34-s + 9·38-s − 3·39-s + ⋯ |
| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 5-s + 3·6-s − 10·8-s − 3·10-s + 11-s − 6·12-s + 3·13-s − 15-s + 15·16-s − 17-s − 3·19-s + 6·20-s − 3·22-s − 23-s + 10·24-s − 9·26-s − 29-s + 3·30-s + 31-s − 21·32-s − 33-s + 3·34-s + 9·38-s − 3·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2570148661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2570148661\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 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$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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
−8.597372410334076705968472546740, −8.171115061881692281575825803260, −8.095694940412692997249916302713, −7.65637246570846374027281377735, −7.30359458512052013816420203348, −6.98944973124307986850343645572, −6.71082982267472174709157653771, −6.33779067823258511727055989275, −6.32617909256098070915864189259, −6.24411089302666578877808506691, −5.85252096952130247247296470975, −5.70719120903126176951762142495, −5.63977265835937661082218268570, −4.86290430349759325085884054619, −4.36917275450275040788806192295, −3.86165040315568452594110585376, −3.85124640996981290859089131850, −3.54130671427308571189024735141, −2.87932520979675273410206662979, −2.41777833009515074908332579348, −2.12337886594025453399754603529, −1.98184637677978698621991281539, −1.53297820829920470050134204519, −1.11548300461909459115785985143, −0.58978713185538617416976795906,
0.58978713185538617416976795906, 1.11548300461909459115785985143, 1.53297820829920470050134204519, 1.98184637677978698621991281539, 2.12337886594025453399754603529, 2.41777833009515074908332579348, 2.87932520979675273410206662979, 3.54130671427308571189024735141, 3.85124640996981290859089131850, 3.86165040315568452594110585376, 4.36917275450275040788806192295, 4.86290430349759325085884054619, 5.63977265835937661082218268570, 5.70719120903126176951762142495, 5.85252096952130247247296470975, 6.24411089302666578877808506691, 6.32617909256098070915864189259, 6.33779067823258511727055989275, 6.71082982267472174709157653771, 6.98944973124307986850343645572, 7.30359458512052013816420203348, 7.65637246570846374027281377735, 8.095694940412692997249916302713, 8.171115061881692281575825803260, 8.597372410334076705968472546740
