| L(s) = 1 | + 2·3-s − 4·4-s + 3·9-s − 8·12-s + 12·16-s − 4·19-s + 10·27-s − 12·36-s − 40·43-s + 24·48-s + 28·49-s − 8·57-s − 32·64-s − 28·67-s − 4·73-s + 16·76-s + 20·81-s − 40·97-s − 40·108-s − 14·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 36·144-s + 56·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 9-s − 2.30·12-s + 3·16-s − 0.917·19-s + 1.92·27-s − 2·36-s − 6.09·43-s + 3.46·48-s + 4·49-s − 1.05·57-s − 4·64-s − 3.42·67-s − 0.468·73-s + 1.83·76-s + 20/9·81-s − 4.06·97-s − 3.84·108-s − 1.27·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3·144-s + 4.61·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.063192221157579709872570939354, −7.41489738662164645969653340826, −7.20080468393670606822166107233, −7.15353935003074134333905240113, −6.71279429984671108498173026002, −6.66403648872221264584960976180, −6.21103141315897839907165688203, −6.03762194948727775029233376261, −5.52128835741893852585331153609, −5.45352945312898964327907056488, −5.22691123116832256825036586079, −4.96874481543554595847445963402, −4.54914426125641948665039221922, −4.31900200431173120586697709808, −4.18685295762032038829384992694, −4.13790241315388008446135736539, −3.44406971042512228721925907572, −3.28008792738676122344106674428, −3.23052830823535974226787529587, −2.76222378724683294052043134413, −2.42652197709125494655113207438, −1.70497655933536255691644910678, −1.65242994560059223596761261869, −1.10660997444743774770840227069, −0.32205561421005123883743418971,
0.32205561421005123883743418971, 1.10660997444743774770840227069, 1.65242994560059223596761261869, 1.70497655933536255691644910678, 2.42652197709125494655113207438, 2.76222378724683294052043134413, 3.23052830823535974226787529587, 3.28008792738676122344106674428, 3.44406971042512228721925907572, 4.13790241315388008446135736539, 4.18685295762032038829384992694, 4.31900200431173120586697709808, 4.54914426125641948665039221922, 4.96874481543554595847445963402, 5.22691123116832256825036586079, 5.45352945312898964327907056488, 5.52128835741893852585331153609, 6.03762194948727775029233376261, 6.21103141315897839907165688203, 6.66403648872221264584960976180, 6.71279429984671108498173026002, 7.15353935003074134333905240113, 7.20080468393670606822166107233, 7.41489738662164645969653340826, 8.063192221157579709872570939354
