L(s) = 1 | − 2·3-s − 2·7-s + 2·9-s + 4·21-s + 2·23-s − 2·27-s + 2·43-s + 2·47-s + 2·49-s − 4·63-s + 2·67-s − 4·69-s + 3·81-s + 2·83-s − 4·101-s − 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s − 4·147-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 2·3-s − 2·7-s + 2·9-s + 4·21-s + 2·23-s − 2·27-s + 2·43-s + 2·47-s + 2·49-s − 4·63-s + 2·67-s − 4·69-s + 3·81-s + 2·83-s − 4·101-s − 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s − 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3497200583\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3497200583\) |
\(L(1)\) |
|
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 | | \( 1 \) |
good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−9.729460302158698731675019460483, −9.466791997473487130280737475219, −9.060901116853319297913509992627, −9.043928265447382677534423569293, −7.917833732876467541241229579751, −7.85505237748298831364543957271, −6.95941736209400434148230776282, −6.92340797262602558140441530626, −6.55986998524535030237384893101, −6.22041960582588081252379114895, −5.60929089008795643751488387041, −5.43784671243576231369934894367, −5.18714801062533176031294069748, −4.36706039008445730316710681308, −3.94599839881235936888079873317, −3.58261539079594797861871942478, −2.70410768916376011757813084711, −2.56395168842232300890867732186, −1.24958978499490677720461180141, −0.59147959482019037054521395577,
0.59147959482019037054521395577, 1.24958978499490677720461180141, 2.56395168842232300890867732186, 2.70410768916376011757813084711, 3.58261539079594797861871942478, 3.94599839881235936888079873317, 4.36706039008445730316710681308, 5.18714801062533176031294069748, 5.43784671243576231369934894367, 5.60929089008795643751488387041, 6.22041960582588081252379114895, 6.55986998524535030237384893101, 6.92340797262602558140441530626, 6.95941736209400434148230776282, 7.85505237748298831364543957271, 7.917833732876467541241229579751, 9.043928265447382677534423569293, 9.060901116853319297913509992627, 9.466791997473487130280737475219, 9.729460302158698731675019460483