| L(s) = 1 | + 9-s + 4·17-s − 2·25-s + 2·41-s − 2·49-s + 4·73-s − 8·89-s − 2·97-s − 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 9-s + 4·17-s − 2·25-s + 2·41-s − 2·49-s + 4·73-s − 8·89-s − 2·97-s − 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.237816568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.237816568\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{8} \) |
| 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
−7.40545179070304723265444748795, −6.94447880615143813301849895722, −6.82358281772922801825386637426, −6.66913407687611776171316649505, −6.34009142564284027374456469336, −5.94533847087419271686435575298, −5.86529393079910395565725270640, −5.63194683064345890290452089239, −5.47712973199914126508290944881, −5.32646169398997055802109854777, −5.05263250295119888734388951383, −4.71411806411841636565813913717, −4.47335688787094529705439088121, −4.06298705114712841407454826840, −3.84748085446877552486036163676, −3.77225455404695305355969498953, −3.71335588738781758439519472150, −3.06883693146696507906034106071, −2.93578396386770892692027149065, −2.61970212251826839234668221787, −2.49402314046716422586945968291, −1.63722745860050185923826561579, −1.60322794885275983020278715933, −1.36436868098597528508970491646, −0.906721785050873677663317096060,
0.906721785050873677663317096060, 1.36436868098597528508970491646, 1.60322794885275983020278715933, 1.63722745860050185923826561579, 2.49402314046716422586945968291, 2.61970212251826839234668221787, 2.93578396386770892692027149065, 3.06883693146696507906034106071, 3.71335588738781758439519472150, 3.77225455404695305355969498953, 3.84748085446877552486036163676, 4.06298705114712841407454826840, 4.47335688787094529705439088121, 4.71411806411841636565813913717, 5.05263250295119888734388951383, 5.32646169398997055802109854777, 5.47712973199914126508290944881, 5.63194683064345890290452089239, 5.86529393079910395565725270640, 5.94533847087419271686435575298, 6.34009142564284027374456469336, 6.66913407687611776171316649505, 6.82358281772922801825386637426, 6.94447880615143813301849895722, 7.40545179070304723265444748795
