| L(s) = 1 | − 4·49-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 251-s + ⋯ |
| L(s) = 1 | − 4·49-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 251-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8057698256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8057698256\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−7.27303512395662761156427371385, −6.82479710757392465753180640923, −6.73283601041645292928138877343, −6.62493620497504100776350074793, −6.36037881166079769658999251801, −6.24464345949396607014312935750, −5.77975721144684164692655753849, −5.77914247163796537194159575314, −5.39229759305115307216307119454, −5.12333072742999378008992649051, −5.01147990142026700891968148701, −4.65905070814679167101539162825, −4.65305696879996544277111566046, −4.19946092318241915619863649394, −3.94250452564572894717490706497, −3.69224925260578778810051882166, −3.49831560605953047502453314750, −3.21110176112132681611861235441, −2.80269654150583840512696627740, −2.62381756048251923175897022981, −2.48919903996984099744771376927, −1.69588644847303352507954150731, −1.66556699678536343319573575223, −1.47995591810136245514367711367, −0.64291267251029234845886976374,
0.64291267251029234845886976374, 1.47995591810136245514367711367, 1.66556699678536343319573575223, 1.69588644847303352507954150731, 2.48919903996984099744771376927, 2.62381756048251923175897022981, 2.80269654150583840512696627740, 3.21110176112132681611861235441, 3.49831560605953047502453314750, 3.69224925260578778810051882166, 3.94250452564572894717490706497, 4.19946092318241915619863649394, 4.65305696879996544277111566046, 4.65905070814679167101539162825, 5.01147990142026700891968148701, 5.12333072742999378008992649051, 5.39229759305115307216307119454, 5.77914247163796537194159575314, 5.77975721144684164692655753849, 6.24464345949396607014312935750, 6.36037881166079769658999251801, 6.62493620497504100776350074793, 6.73283601041645292928138877343, 6.82479710757392465753180640923, 7.27303512395662761156427371385
