| L(s) = 1 | − 2·25-s + 2·49-s + 4·73-s − 4·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2·25-s + 2·49-s + 4·73-s − 4·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.037810246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037810246\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
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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
−6.96554693003475030939259925085, −6.69667009869626762415287951612, −6.56384369398420986721041059111, −6.01058269903611076093421220034, −5.85764038646417197966560838426, −5.85216650966682629769221314303, −5.71458768147761223305377234574, −5.36461389378710630803584534790, −5.10648996493238634169990687584, −5.01468287539011956956548146856, −4.46875602335939669220137224221, −4.46256558369872648350477891297, −4.35252477490520141037581438516, −3.73471513026515533627593127315, −3.72500822319344091911680340217, −3.69020245367746803452221434075, −3.34216952906879499504036469782, −2.87473423681427240939004859718, −2.65024315508821018248780196458, −2.38766893729123080263566463660, −2.15091601406651104804112614109, −1.90124744269204405741049955552, −1.48913112638116237446730664632, −1.12668990585797744293638940924, −0.60370191646727068006050159353,
0.60370191646727068006050159353, 1.12668990585797744293638940924, 1.48913112638116237446730664632, 1.90124744269204405741049955552, 2.15091601406651104804112614109, 2.38766893729123080263566463660, 2.65024315508821018248780196458, 2.87473423681427240939004859718, 3.34216952906879499504036469782, 3.69020245367746803452221434075, 3.72500822319344091911680340217, 3.73471513026515533627593127315, 4.35252477490520141037581438516, 4.46256558369872648350477891297, 4.46875602335939669220137224221, 5.01468287539011956956548146856, 5.10648996493238634169990687584, 5.36461389378710630803584534790, 5.71458768147761223305377234574, 5.85216650966682629769221314303, 5.85764038646417197966560838426, 6.01058269903611076093421220034, 6.56384369398420986721041059111, 6.69667009869626762415287951612, 6.96554693003475030939259925085
