| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 9-s − 2·11-s + 35·16-s + 4·18-s − 8·22-s − 2·25-s + 56·32-s + 10·36-s − 2·43-s − 20·44-s − 8·50-s + 84·64-s − 4·67-s + 20·72-s − 8·86-s − 40·88-s − 2·99-s − 20·100-s + 2·107-s + 4·113-s + 3·121-s + 127-s + 120·128-s + 131-s + ⋯ |
| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 9-s − 2·11-s + 35·16-s + 4·18-s − 8·22-s − 2·25-s + 56·32-s + 10·36-s − 2·43-s − 20·44-s − 8·50-s + 84·64-s − 4·67-s + 20·72-s − 8·86-s − 40·88-s − 2·99-s − 20·100-s + 2·107-s + 4·113-s + 3·121-s + 127-s + 120·128-s + 131-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(34.06981238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(34.06981238\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.18256351694547769078107074552, −5.95591483206267339170616089488, −5.78752159459201593395873178730, −5.57866585753006904270252893606, −5.43779420884832283705897081469, −5.21975086219487835120749084357, −4.85409548575324260322186074397, −4.83694006765172745712026721795, −4.76048138334343864681289687447, −4.37610181681956975406612854120, −4.32420235388087467493381312829, −4.21718260032973309155577624806, −3.76881990575735506633775637596, −3.61088685752560499873008458928, −3.35646294671467833176834480307, −3.22467670364229765008594810217, −3.17711767920968668380279876335, −2.72372848457672153679145536314, −2.55336194721316292126627412318, −2.28205846763283607213959957695, −2.14525653474081361164154021050, −1.75848866271229996519326139013, −1.58926033695801821960870656194, −1.50192276516152470065178035986, −0.859911790455046369412508870855,
0.859911790455046369412508870855, 1.50192276516152470065178035986, 1.58926033695801821960870656194, 1.75848866271229996519326139013, 2.14525653474081361164154021050, 2.28205846763283607213959957695, 2.55336194721316292126627412318, 2.72372848457672153679145536314, 3.17711767920968668380279876335, 3.22467670364229765008594810217, 3.35646294671467833176834480307, 3.61088685752560499873008458928, 3.76881990575735506633775637596, 4.21718260032973309155577624806, 4.32420235388087467493381312829, 4.37610181681956975406612854120, 4.76048138334343864681289687447, 4.83694006765172745712026721795, 4.85409548575324260322186074397, 5.21975086219487835120749084357, 5.43779420884832283705897081469, 5.57866585753006904270252893606, 5.78752159459201593395873178730, 5.95591483206267339170616089488, 6.18256351694547769078107074552
