| L(s) = 1 | + 3·2-s + 3-s + 6·4-s − 3·5-s + 3·6-s − 7-s + 10·8-s − 9·10-s + 6·12-s − 3·14-s − 3·15-s + 15·16-s − 18·20-s − 21-s + 23-s + 10·24-s + 6·25-s − 6·28-s − 29-s − 9·30-s + 21·32-s + 3·35-s − 30·40-s + 41-s − 3·42-s + 43-s + 3·46-s + ⋯ |
| L(s) = 1 | + 3·2-s + 3-s + 6·4-s − 3·5-s + 3·6-s − 7-s + 10·8-s − 9·10-s + 6·12-s − 3·14-s − 3·15-s + 15·16-s − 18·20-s − 21-s + 23-s + 10·24-s + 6·25-s − 6·28-s − 29-s − 9·30-s + 21·32-s + 3·35-s − 30·40-s + 41-s − 3·42-s + 43-s + 3·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(9.708256515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.708256515\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 67 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 97 | $C_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
−7.64149235544180435167828342490, −7.38312411870960796806161158205, −7.34807161531054468791818919319, −7.00141096322497464730891437412, −6.79801736372797350650081163072, −6.60791758856071618791712857529, −6.30614102924908065496690192338, −5.95895283790533200557267718296, −5.71710182643990300911823478823, −5.45041546502141874178775591622, −4.95268491260131279839602553412, −4.92716880141915874104983068418, −4.55280991833534420148658328640, −4.27216963543877248176487370095, −4.15911226533930997953477957098, −3.87140491298225688505437750109, −3.43952144961427468075317933194, −3.35365795525210714259229879084, −3.31249211780687260007979103096, −2.74083308366932046002062458915, −2.61536889314151579583028995548, −2.60271744438909728511529301567, −1.71395280661527343135618566265, −1.40115500077646666898954733612, −0.78260200312655329426528170447,
0.78260200312655329426528170447, 1.40115500077646666898954733612, 1.71395280661527343135618566265, 2.60271744438909728511529301567, 2.61536889314151579583028995548, 2.74083308366932046002062458915, 3.31249211780687260007979103096, 3.35365795525210714259229879084, 3.43952144961427468075317933194, 3.87140491298225688505437750109, 4.15911226533930997953477957098, 4.27216963543877248176487370095, 4.55280991833534420148658328640, 4.92716880141915874104983068418, 4.95268491260131279839602553412, 5.45041546502141874178775591622, 5.71710182643990300911823478823, 5.95895283790533200557267718296, 6.30614102924908065496690192338, 6.60791758856071618791712857529, 6.79801736372797350650081163072, 7.00141096322497464730891437412, 7.34807161531054468791818919319, 7.38312411870960796806161158205, 7.64149235544180435167828342490
