| L(s) = 1 | + 9-s + 2·25-s − 6·41-s + 2·49-s − 4·73-s − 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
| L(s) = 1 | + 9-s + 2·25-s − 6·41-s + 2·49-s − 4·73-s − 2·97-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6732433869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6732433869\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.082684795086260868929177818776, −7.49225222265461180192907109696, −7.47440832375795933917676830325, −7.18165579167383673745690429316, −7.06455592311538659600152287601, −6.83401686200139843374107042774, −6.61004794729969118970921584743, −6.33286240630181358207240000623, −6.22700937088384161602121393312, −5.50040927306322818781316997100, −5.48491417883794814218081601630, −5.46310201546624131464893493877, −4.98754688349010874943566305594, −4.66303731426903007899028002723, −4.52399339859595714453183933364, −4.30307733460604796364957181372, −3.84968304718626400168524045296, −3.68277954911902791232040918981, −3.09682451146858744606590839211, −3.07759556465071361272217360998, −2.88552034487942242599068169871, −2.21398352661825444463316834287, −1.72760905627092993267903785277, −1.62958493013945976422657223905, −1.09416233549237762612237209965,
1.09416233549237762612237209965, 1.62958493013945976422657223905, 1.72760905627092993267903785277, 2.21398352661825444463316834287, 2.88552034487942242599068169871, 3.07759556465071361272217360998, 3.09682451146858744606590839211, 3.68277954911902791232040918981, 3.84968304718626400168524045296, 4.30307733460604796364957181372, 4.52399339859595714453183933364, 4.66303731426903007899028002723, 4.98754688349010874943566305594, 5.46310201546624131464893493877, 5.48491417883794814218081601630, 5.50040927306322818781316997100, 6.22700937088384161602121393312, 6.33286240630181358207240000623, 6.61004794729969118970921584743, 6.83401686200139843374107042774, 7.06455592311538659600152287601, 7.18165579167383673745690429316, 7.47440832375795933917676830325, 7.49225222265461180192907109696, 8.082684795086260868929177818776
