| L(s) = 1 | − 3-s + 3·13-s + 3·19-s − 25-s + 27-s − 2·31-s + 3·37-s − 3·39-s − 43-s − 49-s − 3·57-s − 3·73-s + 75-s + 2·79-s − 81-s + 2·93-s + 2·97-s + 3·103-s + 2·109-s − 3·111-s − 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 147-s + ⋯ |
| L(s) = 1 | − 3-s + 3·13-s + 3·19-s − 25-s + 27-s − 2·31-s + 3·37-s − 3·39-s − 43-s − 49-s − 3·57-s − 3·73-s + 75-s + 2·79-s − 81-s + 2·93-s + 2·97-s + 3·103-s + 2·109-s − 3·111-s − 121-s + 127-s + 129-s + 131-s + 137-s + 139-s + 147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9763530388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9763530388\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.867429201776597638038841545212, −9.440465664961935287953666871326, −9.140400379825146924920021190663, −8.763665438343648671039901265592, −8.262841052908605593267057004385, −7.83626791720227650767396892698, −7.46678206856262876879541337323, −7.15199839785478074993209405750, −6.30000420357904096288706192879, −6.18591892325746597072404099952, −5.72082785141743650162454159779, −5.69219407864362707219510463281, −4.86576586945234488161609585313, −4.65035761907430305662006884632, −3.68697446470634988705270047630, −3.52559967904300016003776207883, −3.20560429384883308821798574757, −2.25734666938253826775592471845, −1.27035627952216217071655140511, −1.09417080174029216707941007277,
1.09417080174029216707941007277, 1.27035627952216217071655140511, 2.25734666938253826775592471845, 3.20560429384883308821798574757, 3.52559967904300016003776207883, 3.68697446470634988705270047630, 4.65035761907430305662006884632, 4.86576586945234488161609585313, 5.69219407864362707219510463281, 5.72082785141743650162454159779, 6.18591892325746597072404099952, 6.30000420357904096288706192879, 7.15199839785478074993209405750, 7.46678206856262876879541337323, 7.83626791720227650767396892698, 8.262841052908605593267057004385, 8.763665438343648671039901265592, 9.140400379825146924920021190663, 9.440465664961935287953666871326, 9.867429201776597638038841545212
