| L(s) = 1 | − 9-s − 2·25-s − 2·49-s + 4·73-s + 81-s − 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
| L(s) = 1 | − 9-s − 2·25-s − 2·49-s + 4·73-s + 81-s − 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4503693738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4503693738\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−12.97722363016898008735771969506, −12.50641606759299101579290172988, −11.96652054339395058620004956477, −11.59840981927542118533216491975, −11.01370503130495906728019173891, −10.82222367714644356398020955651, −9.849516072503568628474637697775, −9.666114570195211975042793979673, −9.126967973020062447789672133728, −8.369648201467567148737270093994, −8.018139637964617364667919671293, −7.63017294754426849302037950971, −6.58586934399064214388814283248, −6.43322565428916493073505123627, −5.47366894557275997654851625383, −5.28239110236326378801837962126, −4.24790763788515601708158344657, −3.62954125332872921564855330771, −2.81568032663380696799101060580, −1.90019379025942779197305225894,
1.90019379025942779197305225894, 2.81568032663380696799101060580, 3.62954125332872921564855330771, 4.24790763788515601708158344657, 5.28239110236326378801837962126, 5.47366894557275997654851625383, 6.43322565428916493073505123627, 6.58586934399064214388814283248, 7.63017294754426849302037950971, 8.018139637964617364667919671293, 8.369648201467567148737270093994, 9.126967973020062447789672133728, 9.666114570195211975042793979673, 9.849516072503568628474637697775, 10.82222367714644356398020955651, 11.01370503130495906728019173891, 11.59840981927542118533216491975, 11.96652054339395058620004956477, 12.50641606759299101579290172988, 12.97722363016898008735771969506
