| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·17-s + 2·19-s − 25-s + 6·27-s + 4·33-s − 2·49-s − 4·51-s + 4·57-s − 2·59-s + 2·67-s − 2·73-s − 2·75-s + 9·81-s − 2·89-s + 6·99-s − 2·107-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 2·17-s + 2·19-s − 25-s + 6·27-s + 4·33-s − 2·49-s − 4·51-s + 4·57-s − 2·59-s + 2·67-s − 2·73-s − 2·75-s + 9·81-s − 2·89-s + 6·99-s − 2·107-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.656499629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656499629\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.96636907079564851656186985618, −6.55593093233507886417818046976, −6.44520638731945974304691180401, −6.42617563183082211375668176522, −6.16661348590221150116335886339, −5.68051694441794104773846147042, −5.47087354091277510379641971871, −5.42963537732886919071993501048, −4.76659039149825758193909530565, −4.71873513134575285795567537393, −4.61071388818698084422170556965, −4.42664199914241487760671127943, −4.19493583380334173053446224714, −3.74316100458043508979310139353, −3.73890431729541957538470280435, −3.36899549633895598120594500473, −3.35407054574075086358081396378, −2.81367233101159689074272448410, −2.70998939651767025189754511942, −2.62707088292234577016462025256, −1.99167113684591483642074168646, −1.90988638246042662286316963311, −1.44052215794769407315134583763, −1.31782827417036651652519749898, −1.00844144867985892814276997952,
1.00844144867985892814276997952, 1.31782827417036651652519749898, 1.44052215794769407315134583763, 1.90988638246042662286316963311, 1.99167113684591483642074168646, 2.62707088292234577016462025256, 2.70998939651767025189754511942, 2.81367233101159689074272448410, 3.35407054574075086358081396378, 3.36899549633895598120594500473, 3.73890431729541957538470280435, 3.74316100458043508979310139353, 4.19493583380334173053446224714, 4.42664199914241487760671127943, 4.61071388818698084422170556965, 4.71873513134575285795567537393, 4.76659039149825758193909530565, 5.42963537732886919071993501048, 5.47087354091277510379641971871, 5.68051694441794104773846147042, 6.16661348590221150116335886339, 6.42617563183082211375668176522, 6.44520638731945974304691180401, 6.55593093233507886417818046976, 6.96636907079564851656186985618
