| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s + 3·19-s − 27-s + 21·32-s − 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s + 18·76-s − 9·94-s − 6·108-s + 3·121-s + 127-s + 36·128-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯ |
| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s + 3·19-s − 27-s + 21·32-s − 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s + 18·76-s − 9·94-s − 6·108-s + 3·121-s + 127-s + 36·128-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(17.31642653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.31642653\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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
−7.60832019804577456202142826120, −7.17152920915400361154831934643, −7.02678264425828841444314941909, −6.96872483613053770117503391512, −6.77885521607385788888835106452, −6.20992304248875064512757145322, −6.06449188826151788193384752646, −5.88004712399777255206150725396, −5.66455582646833053538356700807, −5.30413696606658519391343074171, −4.99621449367105578287382345469, −4.95200572745320401436093592935, −4.87313184259353025648862862607, −4.47188360610106782843205787481, −3.96768468813104163989317715687, −3.66230199216777753740237343710, −3.60587576232893532899609522099, −3.27841165432932801604987387238, −3.16519910098316355134791707938, −2.83834727189313551016203009367, −2.41338293578327042636646666491, −1.95667818261153896396445420352, −1.83520021970847513060548334122, −1.24394717335128879486726389519, −1.22480060681873618762671818129,
1.22480060681873618762671818129, 1.24394717335128879486726389519, 1.83520021970847513060548334122, 1.95667818261153896396445420352, 2.41338293578327042636646666491, 2.83834727189313551016203009367, 3.16519910098316355134791707938, 3.27841165432932801604987387238, 3.60587576232893532899609522099, 3.66230199216777753740237343710, 3.96768468813104163989317715687, 4.47188360610106782843205787481, 4.87313184259353025648862862607, 4.95200572745320401436093592935, 4.99621449367105578287382345469, 5.30413696606658519391343074171, 5.66455582646833053538356700807, 5.88004712399777255206150725396, 6.06449188826151788193384752646, 6.20992304248875064512757145322, 6.77885521607385788888835106452, 6.96872483613053770117503391512, 7.02678264425828841444314941909, 7.17152920915400361154831934643, 7.60832019804577456202142826120
