| L(s) = 1 | + 2-s + 2·4-s − 6·5-s + 5·8-s − 6·10-s + 5·16-s − 24·17-s − 12·20-s + 11·25-s − 20·29-s + 10·32-s − 24·34-s − 30·40-s + 7·49-s + 11·50-s + 14·53-s − 20·58-s − 36·61-s + 17·64-s − 48·68-s − 24·73-s − 30·80-s + 144·85-s + 36·89-s + 7·98-s + 22·100-s + 24·101-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 2.68·5-s + 1.76·8-s − 1.89·10-s + 5/4·16-s − 5.82·17-s − 2.68·20-s + 11/5·25-s − 3.71·29-s + 1.76·32-s − 4.11·34-s − 4.74·40-s + 49-s + 1.55·50-s + 1.92·53-s − 2.62·58-s − 4.60·61-s + 17/8·64-s − 5.82·68-s − 2.80·73-s − 3.35·80-s + 15.6·85-s + 3.81·89-s + 0.707·98-s + 11/5·100-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3191636122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3191636122\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 15 T^{2} + 104 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 41 T^{2} + 720 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 10 T^{2} - 2109 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 7 T - 4 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 71 T^{2} + 1560 T^{4} + 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 18 T + 169 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 95 T^{2} + 2784 T^{4} + 95 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.767822640030567965475869603707, −8.766920788941284831168553052464, −7.996339895083816699335545132849, −7.81473865434245165259396368970, −7.68383187160324922890331287662, −7.61005937844386687749653414619, −7.01382259083865975437757443296, −6.99821487854827112175172527102, −6.95774656473383070921967341300, −6.53963723115059543675804966223, −5.92912862406213373160656705118, −5.91083327771285487746022135395, −5.72277884919420742665908163744, −4.73585417859642499420512598767, −4.66344419076261624492788545550, −4.58612774939956256033318742322, −4.30251358564896830558275449404, −4.10474769712630771323683858379, −3.60773835850154606200124063080, −3.54969451825365226956347064858, −2.97472740750053989110719882266, −2.14874943459000897578375773083, −2.07212400141016427372685816478, −1.94630676757404214750977267972, −0.24107456398891390177635877369,
0.24107456398891390177635877369, 1.94630676757404214750977267972, 2.07212400141016427372685816478, 2.14874943459000897578375773083, 2.97472740750053989110719882266, 3.54969451825365226956347064858, 3.60773835850154606200124063080, 4.10474769712630771323683858379, 4.30251358564896830558275449404, 4.58612774939956256033318742322, 4.66344419076261624492788545550, 4.73585417859642499420512598767, 5.72277884919420742665908163744, 5.91083327771285487746022135395, 5.92912862406213373160656705118, 6.53963723115059543675804966223, 6.95774656473383070921967341300, 6.99821487854827112175172527102, 7.01382259083865975437757443296, 7.61005937844386687749653414619, 7.68383187160324922890331287662, 7.81473865434245165259396368970, 7.996339895083816699335545132849, 8.766920788941284831168553052464, 8.767822640030567965475869603707
