| L(s) = 1 | + 2·5-s + 2·13-s + 3·25-s − 2·29-s + 2·41-s − 49-s + 2·61-s + 4·65-s − 2·97-s + 2·101-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·13-s + 3·25-s − 2·29-s + 2·41-s − 49-s + 2·61-s + 4·65-s − 2·97-s + 2·101-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.331593264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331593264\) |
| \(L(1)\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−6.87934197690069529202837490502, −6.52232047004454338084035725251, −6.45671470629239178836423714981, −5.94774051497234796275808604794, −5.93230029871973798035470081817, −5.90081304946985317127610828765, −5.68389307123091056654112544851, −5.49776867271052208112105938889, −5.24248666134070464560400147146, −4.90457158059993175440515498001, −4.66259989987023571304349153822, −4.46405782510757234638656158648, −4.38707309781617242343030375863, −3.73866777472437973757336750750, −3.59078343140711757516828772445, −3.53293102406919621341779264794, −3.48679898519357348431707504018, −2.74950149137168911424797330324, −2.62627162503586019791877367352, −2.38665520872311839944329547963, −2.20187990991086008338375275662, −1.70005822714374503798060240629, −1.56310958133187847681320844723, −1.12131364256116508425898221565, −0.987392758379767511786723518841,
0.987392758379767511786723518841, 1.12131364256116508425898221565, 1.56310958133187847681320844723, 1.70005822714374503798060240629, 2.20187990991086008338375275662, 2.38665520872311839944329547963, 2.62627162503586019791877367352, 2.74950149137168911424797330324, 3.48679898519357348431707504018, 3.53293102406919621341779264794, 3.59078343140711757516828772445, 3.73866777472437973757336750750, 4.38707309781617242343030375863, 4.46405782510757234638656158648, 4.66259989987023571304349153822, 4.90457158059993175440515498001, 5.24248666134070464560400147146, 5.49776867271052208112105938889, 5.68389307123091056654112544851, 5.90081304946985317127610828765, 5.93230029871973798035470081817, 5.94774051497234796275808604794, 6.45671470629239178836423714981, 6.52232047004454338084035725251, 6.87934197690069529202837490502
