| L(s) = 1 | − 3-s + 9-s + 4·11-s − 8·17-s − 4·19-s − 2·25-s − 27-s − 4·33-s + 8·41-s + 4·43-s − 2·49-s + 8·51-s + 4·57-s + 4·59-s − 20·67-s − 4·73-s + 2·75-s + 81-s + 12·83-s − 12·97-s + 4·99-s + 4·107-s − 16·113-s − 6·121-s − 8·123-s + 127-s − 4·129-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.94·17-s − 0.917·19-s − 2/5·25-s − 0.192·27-s − 0.696·33-s + 1.24·41-s + 0.609·43-s − 2/7·49-s + 1.12·51-s + 0.529·57-s + 0.520·59-s − 2.44·67-s − 0.468·73-s + 0.230·75-s + 1/9·81-s + 1.31·83-s − 1.21·97-s + 0.402·99-s + 0.386·107-s − 1.50·113-s − 0.545·121-s − 0.721·123-s + 0.0887·127-s − 0.352·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.480753135490049228586140128298, −7.86913981829753340661674128814, −7.37229557056512333671712332413, −6.75454235229346471718400556909, −6.63589516987261747260701508822, −5.98405012427494401768273778119, −5.79272268440187514578879922310, −4.89746315468719336411913196192, −4.39089240504499683168279669968, −4.18916783457034067401852062032, −3.54735237847928272174122226580, −2.60693178492940340527133913246, −2.05394882685218966723140209550, −1.23284588025260695922078998101, 0,
1.23284588025260695922078998101, 2.05394882685218966723140209550, 2.60693178492940340527133913246, 3.54735237847928272174122226580, 4.18916783457034067401852062032, 4.39089240504499683168279669968, 4.89746315468719336411913196192, 5.79272268440187514578879922310, 5.98405012427494401768273778119, 6.63589516987261747260701508822, 6.75454235229346471718400556909, 7.37229557056512333671712332413, 7.86913981829753340661674128814, 8.480753135490049228586140128298
