L(s) = 1 | + 2·9-s − 2·25-s + 4·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4·225-s + 227-s + ⋯ |
L(s) = 1 | + 2·9-s − 2·25-s + 4·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4·225-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.013339085\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013339085\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.15302193392129514545646118067, −6.09975052674398827965950187913, −5.81441181110851486691691011909, −5.51702390320995492468018590750, −5.29037926303853323696174108194, −5.28886286155025898388959890409, −4.84211575406691721862244066989, −4.68206754014833178583060037693, −4.66629496976202453060037807522, −4.44956608790212105882203027814, −3.98375307145101670305929932306, −3.97851471962543880594130770281, −3.94389515701335131694057178352, −3.64394913070110952399276718580, −3.33550101485453386116903008289, −2.97064759651776962625400815123, −2.84011768945163756429693933205, −2.48917894556172260893005336881, −2.39958354495418204134704667338, −2.24979900847689935477682389354, −1.59646771079412109620937701749, −1.49788453221545786838645193554, −1.29618632104580819877361630414, −1.12405357878212052987835012073, −0.50673974312971847292206995767,
0.50673974312971847292206995767, 1.12405357878212052987835012073, 1.29618632104580819877361630414, 1.49788453221545786838645193554, 1.59646771079412109620937701749, 2.24979900847689935477682389354, 2.39958354495418204134704667338, 2.48917894556172260893005336881, 2.84011768945163756429693933205, 2.97064759651776962625400815123, 3.33550101485453386116903008289, 3.64394913070110952399276718580, 3.94389515701335131694057178352, 3.97851471962543880594130770281, 3.98375307145101670305929932306, 4.44956608790212105882203027814, 4.66629496976202453060037807522, 4.68206754014833178583060037693, 4.84211575406691721862244066989, 5.28886286155025898388959890409, 5.29037926303853323696174108194, 5.51702390320995492468018590750, 5.81441181110851486691691011909, 6.09975052674398827965950187913, 6.15302193392129514545646118067