| L(s) = 1 | + 2·3-s + 2·9-s + 2·23-s + 25-s + 2·37-s + 4·47-s − 4·53-s − 4·59-s − 2·67-s + 4·69-s + 2·75-s − 3·81-s − 2·97-s + 4·103-s + 4·111-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 8·141-s + 149-s + 151-s + 157-s − 8·159-s + 163-s + ⋯ |
| L(s) = 1 | + 2·3-s + 2·9-s + 2·23-s + 25-s + 2·37-s + 4·47-s − 4·53-s − 4·59-s − 2·67-s + 4·69-s + 2·75-s − 3·81-s − 2·97-s + 4·103-s + 4·111-s − 2·113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 8·141-s + 149-s + 151-s + 157-s − 8·159-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.792525845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792525845\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−7.54060717930936533961767526011, −7.50795037723133448432457129126, −7.19248134360475378628894512528, −6.88830834718349923319945872170, −6.77605229499364546240785989781, −6.22094143731718373599574831723, −6.05658600125537919109297168332, −5.99249959261409142109989320512, −5.91139212460404454348784898472, −5.37159322268994136238573860278, −4.88540701164703095623888673719, −4.86204698256454426250430980842, −4.63768991866909904581888063658, −4.51015432566838211828285673862, −4.03988941904665264019732286370, −3.75478048464613548416210600467, −3.56159649848261142436627024358, −3.32592014572024620001501582188, −2.83074672534406765355507858593, −2.79721978330171152997967239463, −2.50056518450525596810840107745, −2.47176325094174984925014367806, −1.65509299013494404753012030906, −1.30898036881670797950025826654, −1.29913537355058552031294107842,
1.29913537355058552031294107842, 1.30898036881670797950025826654, 1.65509299013494404753012030906, 2.47176325094174984925014367806, 2.50056518450525596810840107745, 2.79721978330171152997967239463, 2.83074672534406765355507858593, 3.32592014572024620001501582188, 3.56159649848261142436627024358, 3.75478048464613548416210600467, 4.03988941904665264019732286370, 4.51015432566838211828285673862, 4.63768991866909904581888063658, 4.86204698256454426250430980842, 4.88540701164703095623888673719, 5.37159322268994136238573860278, 5.91139212460404454348784898472, 5.99249959261409142109989320512, 6.05658600125537919109297168332, 6.22094143731718373599574831723, 6.77605229499364546240785989781, 6.88830834718349923319945872170, 7.19248134360475378628894512528, 7.50795037723133448432457129126, 7.54060717930936533961767526011
