| L(s) = 1 | + 8·17-s − 4·25-s − 8·53-s − 8·113-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 + ⋯ |
| L(s) = 1 | + 8·17-s − 4·25-s − 8·53-s − 8·113-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4967958048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4967958048\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{8} \) |
| 3 | \( 1 \) |
| 17 | \( ( 1 - T )^{8} \) |
| good | 5 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 7 | \( 1 + T^{16} \) |
| 11 | \( 1 + T^{16} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 + T^{8} )^{2} \) |
| 23 | \( 1 + T^{16} \) |
| 29 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 31 | \( 1 + T^{16} \) |
| 37 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 41 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 43 | \( ( 1 + T^{8} )^{2} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T )^{8}( 1 + T^{4} )^{2} \) |
| 59 | \( ( 1 + T^{8} )^{2} \) |
| 61 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
| 67 | \( ( 1 + T^{2} )^{8} \) |
| 71 | \( 1 + T^{16} \) |
| 73 | \( ( 1 + T^{2} )^{4}( 1 + T^{8} ) \) |
| 79 | \( 1 + T^{16} \) |
| 83 | \( ( 1 + T^{8} )^{2} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{4} )^{2}( 1 + T^{8} ) \) |
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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
−5.01156730462063699634492137950, −5.00136283245327848669886978761, −4.52039325131189009735890188433, −4.51235540323843296994654618596, −4.48009954556624008772954521187, −4.05591414517420848349194432337, −3.95052163281459538357068317143, −3.87703335443540583419675649859, −3.79931342554656817506439103851, −3.50961532652946266177739285328, −3.43269358256016012346316172922, −3.40485582274871193485538222775, −3.27373230463397299030741684572, −3.20517197291101005208747580897, −2.87917217485117057657813732622, −2.79396775493337057527157589933, −2.60254402975994719784269745056, −2.44464991905724102492053880512, −2.27143703112488575003161141293, −1.70587401190127518820653082402, −1.60226829869039267854178917823, −1.49011892331071608037846245329, −1.38951556636505413954817367003, −1.37296305230285723526260808229, −0.990123950790725098627366273874,
0.990123950790725098627366273874, 1.37296305230285723526260808229, 1.38951556636505413954817367003, 1.49011892331071608037846245329, 1.60226829869039267854178917823, 1.70587401190127518820653082402, 2.27143703112488575003161141293, 2.44464991905724102492053880512, 2.60254402975994719784269745056, 2.79396775493337057527157589933, 2.87917217485117057657813732622, 3.20517197291101005208747580897, 3.27373230463397299030741684572, 3.40485582274871193485538222775, 3.43269358256016012346316172922, 3.50961532652946266177739285328, 3.79931342554656817506439103851, 3.87703335443540583419675649859, 3.95052163281459538357068317143, 4.05591414517420848349194432337, 4.48009954556624008772954521187, 4.51235540323843296994654618596, 4.52039325131189009735890188433, 5.00136283245327848669886978761, 5.01156730462063699634492137950
Plot not available for L-functions of degree greater than 10.
