| L(s) = 1 | + 4-s + 2·25-s − 64-s + 4·79-s + 2·100-s + 2·121-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 + ⋯ |
| L(s) = 1 | + 4-s + 2·25-s − 64-s + 4·79-s + 2·100-s + 2·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.654821734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654821734\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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
−6.22873198247952456757762136805, −6.00038325677020310668582244367, −5.95028610396301907024631845044, −5.81247766232003993913943929622, −5.49350484176038121036309887863, −5.14179899529117137870446576027, −5.13917033434594768298071814951, −4.71314553698536924946225385993, −4.64090549380733303652997222573, −4.56196640887765899475342695365, −4.43852598387056006457517288443, −3.82766418044223564874127557290, −3.73733318327453000239573023905, −3.57013175903297849591344441358, −3.32191514592374821502674757866, −3.06422339134968997396719792631, −2.79658206931692803497334716176, −2.79268207423488883998197423758, −2.32975314150966046371305273841, −2.02889760666804390379638859042, −1.88294125771003227164373178683, −1.86896117990721474737616970524, −1.20832932415528453541867286532, −0.884844544768888568296854984541, −0.71351141845824042700881767578,
0.71351141845824042700881767578, 0.884844544768888568296854984541, 1.20832932415528453541867286532, 1.86896117990721474737616970524, 1.88294125771003227164373178683, 2.02889760666804390379638859042, 2.32975314150966046371305273841, 2.79268207423488883998197423758, 2.79658206931692803497334716176, 3.06422339134968997396719792631, 3.32191514592374821502674757866, 3.57013175903297849591344441358, 3.73733318327453000239573023905, 3.82766418044223564874127557290, 4.43852598387056006457517288443, 4.56196640887765899475342695365, 4.64090549380733303652997222573, 4.71314553698536924946225385993, 5.13917033434594768298071814951, 5.14179899529117137870446576027, 5.49350484176038121036309887863, 5.81247766232003993913943929622, 5.95028610396301907024631845044, 6.00038325677020310668582244367, 6.22873198247952456757762136805
