| L(s) = 1 | + 4·3-s + 10·9-s − 2·11-s − 25-s + 20·27-s − 8·33-s + 2·41-s + 2·43-s − 49-s − 2·59-s − 2·67-s − 4·75-s + 35·81-s − 2·83-s − 2·97-s − 20·99-s + 2·113-s + 3·121-s + 8·123-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4·3-s + 10·9-s − 2·11-s − 25-s + 20·27-s − 8·33-s + 2·41-s + 2·43-s − 49-s − 2·59-s − 2·67-s − 4·75-s + 35·81-s − 2·83-s − 2·97-s − 20·99-s + 2·113-s + 3·121-s + 8·123-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.302096386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.302096386\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.97129979093545917792429425937, −6.48062743167900917369860093380, −6.05745763999122792673598904235, −6.04824374810446674370881438100, −5.87998280559972185775847874820, −5.60708871737076146614406785605, −5.15696840438912223360724588341, −5.05705537383646254974986354384, −4.65025248970634396307949902105, −4.49675016541336054337730132332, −4.36022173444552703701511118269, −4.16636189675723222547085089842, −4.16578292739703942738486668735, −3.57222466385717333512698902475, −3.31585607274060785964024531932, −3.24365989317625350439131086028, −3.23800607359521436650949810093, −2.67794752637382999723099780366, −2.54994454798396842372443519602, −2.37910290514335720822202058616, −2.37034161566975595597315861529, −1.83130937006298015985247716267, −1.61601654158749595441237871512, −1.32825054439069175272083794113, −0.935015557818875603127685377238,
0.935015557818875603127685377238, 1.32825054439069175272083794113, 1.61601654158749595441237871512, 1.83130937006298015985247716267, 2.37034161566975595597315861529, 2.37910290514335720822202058616, 2.54994454798396842372443519602, 2.67794752637382999723099780366, 3.23800607359521436650949810093, 3.24365989317625350439131086028, 3.31585607274060785964024531932, 3.57222466385717333512698902475, 4.16578292739703942738486668735, 4.16636189675723222547085089842, 4.36022173444552703701511118269, 4.49675016541336054337730132332, 4.65025248970634396307949902105, 5.05705537383646254974986354384, 5.15696840438912223360724588341, 5.60708871737076146614406785605, 5.87998280559972185775847874820, 6.04824374810446674370881438100, 6.05745763999122792673598904235, 6.48062743167900917369860093380, 6.97129979093545917792429425937
