L(s) = 1 | − 2·3-s + 9-s − 4·11-s + 4·13-s + 8·23-s − 2·25-s + 4·27-s + 8·33-s + 4·37-s − 8·39-s − 2·49-s + 4·59-s + 20·61-s − 16·69-s + 8·71-s + 4·73-s + 4·75-s − 11·81-s + 20·83-s − 4·97-s − 4·99-s + 4·107-s − 12·109-s − 8·111-s + 4·117-s + 6·121-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 1.66·23-s − 2/5·25-s + 0.769·27-s + 1.39·33-s + 0.657·37-s − 1.28·39-s − 2/7·49-s + 0.520·59-s + 2.56·61-s − 1.92·69-s + 0.949·71-s + 0.468·73-s + 0.461·75-s − 1.22·81-s + 2.19·83-s − 0.406·97-s − 0.402·99-s + 0.386·107-s − 1.14·109-s − 0.759·111-s + 0.369·117-s + 6/11·121-s + 0.0887·127-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
\(L(1)\) |
\(\approx\) |
\(0.8004367739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8004367739\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
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$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.57117027388674033150373501160, −9.980234498858972240800426785293, −9.359432931956235852299822576830, −8.761253126142270829593658544362, −8.200959075936605180783682994475, −7.75377023612537773032444559189, −6.91644360728498526615914333306, −6.55784348471676331264571227520, −5.89833014178485861074934344891, −5.24969047550422206798735288876, −5.06643453152369135477258592643, −4.09379989521195289492080539242, −3.27626560022899986319936689496, −2.38303978348464108346924679457, −0.896458712162833399049925227729,
0.896458712162833399049925227729, 2.38303978348464108346924679457, 3.27626560022899986319936689496, 4.09379989521195289492080539242, 5.06643453152369135477258592643, 5.24969047550422206798735288876, 5.89833014178485861074934344891, 6.55784348471676331264571227520, 6.91644360728498526615914333306, 7.75377023612537773032444559189, 8.200959075936605180783682994475, 8.761253126142270829593658544362, 9.359432931956235852299822576830, 9.980234498858972240800426785293, 10.57117027388674033150373501160