| L(s) = 1 | − 2·5-s − 2·17-s + 2·25-s + 2·29-s + 4·37-s + 4·41-s − 4·61-s − 2·73-s + 81-s + 4·85-s + 2·97-s − 4·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·5-s − 2·17-s + 2·25-s + 2·29-s + 4·37-s + 4·41-s − 4·61-s − 2·73-s + 81-s + 4·85-s + 2·97-s − 4·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5249393616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5249393616\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.38148339963609185222930062352, −5.98598972745594129458514995769, −5.90032178215830543248602271594, −5.78372108708983851532078614276, −5.62668248278256712182863518850, −5.03585923563431173113643545123, −4.83082661343267252436901643703, −4.82429089575324629435273148887, −4.46863654289251538628080483949, −4.37846922207251840726286887221, −4.28743561787703092616917186461, −4.12262601156797750176801894205, −4.02687499181533259968519147528, −3.54429478534341812860920301305, −3.36685376128482842510757183104, −3.12722208512587809263406871312, −2.79915261356301970685975486834, −2.61985193593352546955837338700, −2.50930555492044522717238162855, −2.33060199183292897044637170764, −2.01200081749381880998069491998, −1.20437912669463358398065554136, −1.16639595903471030800521402821, −1.09270127979804506080579312899, −0.31685156408038657137136491756,
0.31685156408038657137136491756, 1.09270127979804506080579312899, 1.16639595903471030800521402821, 1.20437912669463358398065554136, 2.01200081749381880998069491998, 2.33060199183292897044637170764, 2.50930555492044522717238162855, 2.61985193593352546955837338700, 2.79915261356301970685975486834, 3.12722208512587809263406871312, 3.36685376128482842510757183104, 3.54429478534341812860920301305, 4.02687499181533259968519147528, 4.12262601156797750176801894205, 4.28743561787703092616917186461, 4.37846922207251840726286887221, 4.46863654289251538628080483949, 4.82429089575324629435273148887, 4.83082661343267252436901643703, 5.03585923563431173113643545123, 5.62668248278256712182863518850, 5.78372108708983851532078614276, 5.90032178215830543248602271594, 5.98598972745594129458514995769, 6.38148339963609185222930062352
