| L(s) = 1 | − 8·41-s + 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 8·41-s + 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05287574144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05287574144\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 + T^{2} )^{8} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T + T^{2} )^{8} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{2} )^{8} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 67 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 71 | \( ( 1 + T^{2} )^{8} \) |
| 73 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 83 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 97 | \( ( 1 + T^{4} )^{4} \) |
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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
−3.71032120465903672724342247658, −3.67494598080229290651240763701, −3.63111463087516152656123284271, −3.55606075210030279674597227654, −3.33916659617099566238518528290, −3.17249076093474153133820463284, −3.15933704244650959081054926518, −2.97537827091353287376745197482, −2.96938910282853536737491927868, −2.66413080337361737448110774033, −2.65340595301336041708800028876, −2.56337491938000895277093103153, −2.40006706866104849175705735381, −2.16174029248050337525541524261, −2.15776361094464177006956170578, −1.76151560694083765650570597611, −1.73728097273076743701399895757, −1.66107809232308796411211818102, −1.65481608587261929640668256165, −1.57461801071528784926108088365, −1.35239498874749541923251808711, −0.925422232863320276145882019951, −0.880895053263373324823330517697, −0.73649380079350580759001213942, −0.06028003099381638918542596440,
0.06028003099381638918542596440, 0.73649380079350580759001213942, 0.880895053263373324823330517697, 0.925422232863320276145882019951, 1.35239498874749541923251808711, 1.57461801071528784926108088365, 1.65481608587261929640668256165, 1.66107809232308796411211818102, 1.73728097273076743701399895757, 1.76151560694083765650570597611, 2.15776361094464177006956170578, 2.16174029248050337525541524261, 2.40006706866104849175705735381, 2.56337491938000895277093103153, 2.65340595301336041708800028876, 2.66413080337361737448110774033, 2.96938910282853536737491927868, 2.97537827091353287376745197482, 3.15933704244650959081054926518, 3.17249076093474153133820463284, 3.33916659617099566238518528290, 3.55606075210030279674597227654, 3.63111463087516152656123284271, 3.67494598080229290651240763701, 3.71032120465903672724342247658
Plot not available for L-functions of degree greater than 10.
