| L(s) = 1 | + 2·7-s − 3·9-s + 4·11-s − 2·17-s − 4·19-s + 8·23-s + 2·25-s + 4·29-s + 10·31-s − 4·37-s − 2·41-s − 4·43-s + 8·47-s + 6·49-s − 4·53-s + 8·59-s − 4·61-s − 6·63-s − 8·67-s − 4·73-s + 8·77-s + 2·79-s + 9·81-s − 4·83-s + 2·89-s − 16·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 1.20·11-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 2/5·25-s + 0.742·29-s + 1.79·31-s − 0.657·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 6/7·49-s − 0.549·53-s + 1.04·59-s − 0.512·61-s − 0.755·63-s − 0.977·67-s − 0.468·73-s + 0.911·77-s + 0.225·79-s + 81-s − 0.439·83-s + 0.211·89-s − 1.62·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392975217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392975217\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−15.0365893991, −14.7060718723, −14.0921723835, −13.7933171416, −13.3950182328, −12.6838267100, −12.1006960686, −11.8949116769, −11.2326589465, −11.0038304427, −10.4195316398, −9.87646060117, −8.98699011091, −8.83496538531, −8.44057671922, −7.81532325553, −6.94312186480, −6.65130383171, −6.04281082919, −5.27429885526, −4.67814945649, −4.14222357498, −3.15655809885, −2.44910189234, −1.21802991224,
1.21802991224, 2.44910189234, 3.15655809885, 4.14222357498, 4.67814945649, 5.27429885526, 6.04281082919, 6.65130383171, 6.94312186480, 7.81532325553, 8.44057671922, 8.83496538531, 8.98699011091, 9.87646060117, 10.4195316398, 11.0038304427, 11.2326589465, 11.8949116769, 12.1006960686, 12.6838267100, 13.3950182328, 13.7933171416, 14.0921723835, 14.7060718723, 15.0365893991
