L(s) = 1 | − 4·5-s + 10·25-s − 81-s − 4·121-s − 20·125-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 + ⋯ |
L(s) = 1 | − 4·5-s + 10·25-s − 81-s − 4·121-s − 20·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0002968355851\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0002968355851\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.22461103655919537040496537999, −6.06118736793739334712054466473, −5.78808989137020646873215763814, −5.42832202003097371252154260544, −5.40954698290835439253901618813, −5.01200135443536793589265335769, −4.88684051379152758422520559656, −4.67052556955220412421916382772, −4.61360348343072813955555176988, −4.45057210495473343033195865516, −3.96092236669099698664225630391, −3.90952483144489179310417362983, −3.81356137558899594991905410199, −3.63240466849811611118150125263, −3.48764252977294784069339095837, −3.15671669102598803735436745692, −2.96095573817742705856300078850, −2.57070507008022208528793041421, −2.47093531157134890589620554362, −2.46337935622586281392649732504, −1.68649776864335353221787410115, −1.31464935370800166507967300460, −1.12418347067374407920736451455, −0.907516935546230755221536131392, −0.008150018827572329029354412215,
0.008150018827572329029354412215, 0.907516935546230755221536131392, 1.12418347067374407920736451455, 1.31464935370800166507967300460, 1.68649776864335353221787410115, 2.46337935622586281392649732504, 2.47093531157134890589620554362, 2.57070507008022208528793041421, 2.96095573817742705856300078850, 3.15671669102598803735436745692, 3.48764252977294784069339095837, 3.63240466849811611118150125263, 3.81356137558899594991905410199, 3.90952483144489179310417362983, 3.96092236669099698664225630391, 4.45057210495473343033195865516, 4.61360348343072813955555176988, 4.67052556955220412421916382772, 4.88684051379152758422520559656, 5.01200135443536793589265335769, 5.40954698290835439253901618813, 5.42832202003097371252154260544, 5.78808989137020646873215763814, 6.06118736793739334712054466473, 6.22461103655919537040496537999