| L(s) = 1 | − 2·5-s + 16-s + 3·25-s − 2·31-s − 4·37-s − 2·47-s − 4·53-s + 2·59-s + 2·67-s + 4·71-s − 2·80-s − 2·97-s + 2·103-s − 2·113-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·5-s + 16-s + 3·25-s − 2·31-s − 4·37-s − 2·47-s − 4·53-s + 2·59-s + 2·67-s + 4·71-s − 2·80-s − 2·97-s + 2·103-s − 2·113-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0004278895229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0004278895229\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−6.45949723561087218192974298514, −6.08943884385057165434253129885, −6.08508482613631503527550080706, −5.58021581971990007543261638050, −5.32525680514042746651559890159, −5.12448797732212111668423389650, −5.11174408760262401217468717017, −5.01935891409478439701681690436, −4.84096493803381442125342042960, −4.53430283961575243409377519814, −4.07163203872269346009609372529, −3.96882884523849774906428846481, −3.74123514913048708029816218920, −3.56058669993556346975678768139, −3.41200343143198147628981986891, −3.35739706929834313465353539178, −3.24720463332197365772412733717, −2.58477148578749871926585032246, −2.45713765439714152096810372335, −2.24786420494106723844835935167, −1.85007236605427484197941941146, −1.46751715252266370944596190580, −1.27036646009138617738167413137, −1.04542263829292066345155849764, −0.008645651384033091613644502078,
0.008645651384033091613644502078, 1.04542263829292066345155849764, 1.27036646009138617738167413137, 1.46751715252266370944596190580, 1.85007236605427484197941941146, 2.24786420494106723844835935167, 2.45713765439714152096810372335, 2.58477148578749871926585032246, 3.24720463332197365772412733717, 3.35739706929834313465353539178, 3.41200343143198147628981986891, 3.56058669993556346975678768139, 3.74123514913048708029816218920, 3.96882884523849774906428846481, 4.07163203872269346009609372529, 4.53430283961575243409377519814, 4.84096493803381442125342042960, 5.01935891409478439701681690436, 5.11174408760262401217468717017, 5.12448797732212111668423389650, 5.32525680514042746651559890159, 5.58021581971990007543261638050, 6.08508482613631503527550080706, 6.08943884385057165434253129885, 6.45949723561087218192974298514
