L(s) = 1 | − 8·7-s + 16-s + 4·25-s + 8·31-s + 28·49-s + 4·73-s − 4·79-s − 8·97-s − 8·103-s − 8·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 8·7-s + 16-s + 4·25-s + 8·31-s + 28·49-s + 4·73-s − 4·79-s − 8·97-s − 8·103-s − 8·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03197544857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03197544857\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 \) |
| 19 | \( ( 1 + T^{2} )^{4} \) |
good | 5 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 7 | \( ( 1 + T + T^{2} )^{8} \) |
| 11 | \( ( 1 + T^{4} )^{4} \) |
| 13 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 29 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} )^{8} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 41 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 43 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 73 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 79 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 + T + T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.27142567997613092875469612002, −4.02882418011929879375592175176, −3.99360414659820788064266153440, −3.97806517890741944639866573825, −3.78394146839429030441905537050, −3.44161558601796967538204356814, −3.43896730823926328716090196817, −3.42204358598977679980193540041, −3.09047952805879842062926080407, −3.08273986455522512506891148363, −3.06827917098239451321384913778, −2.83745098967168823708320291643, −2.79484770197549617263948078139, −2.75534878326116380895287480850, −2.72317405135435340491006686293, −2.57061950849077855512158796412, −2.53257321682193062792767812108, −2.24081155348599780146611428612, −1.81321724579706628788823073695, −1.45157312927322497613592631426, −1.13076255205986667565091005338, −1.11498877643200776361233230571, −1.09987514374400221716913900359, −0.898426640689715484740847367470, −0.12219064685438725820572970355,
0.12219064685438725820572970355, 0.898426640689715484740847367470, 1.09987514374400221716913900359, 1.11498877643200776361233230571, 1.13076255205986667565091005338, 1.45157312927322497613592631426, 1.81321724579706628788823073695, 2.24081155348599780146611428612, 2.53257321682193062792767812108, 2.57061950849077855512158796412, 2.72317405135435340491006686293, 2.75534878326116380895287480850, 2.79484770197549617263948078139, 2.83745098967168823708320291643, 3.06827917098239451321384913778, 3.08273986455522512506891148363, 3.09047952805879842062926080407, 3.42204358598977679980193540041, 3.43896730823926328716090196817, 3.44161558601796967538204356814, 3.78394146839429030441905537050, 3.97806517890741944639866573825, 3.99360414659820788064266153440, 4.02882418011929879375592175176, 4.27142567997613092875469612002
Plot not available for L-functions of degree greater than 10.