| L(s) = 1 | + 14·7-s − 4·13-s + 46·19-s − 32·25-s + 94·31-s + 110·37-s − 92·43-s + 49·49-s − 208·61-s − 194·67-s − 130·73-s + 226·79-s − 56·91-s + 416·97-s + 238·103-s + 98·109-s − 80·121-s + 127-s + 131-s + 644·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·7-s − 0.307·13-s + 2.42·19-s − 1.27·25-s + 3.03·31-s + 2.97·37-s − 2.13·43-s + 49-s − 3.40·61-s − 2.89·67-s − 1.78·73-s + 2.86·79-s − 0.615·91-s + 4.28·97-s + 2.31·103-s + 0.899·109-s − 0.661·121-s + 0.00787·127-s + 0.00763·131-s + 4.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.645122506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.645122506\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 80 T^{2} - 8241 T^{4} + 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 290 T^{2} + 579 T^{4} + 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 23 T + 168 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 47 T + 1248 T^{2} - 47 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 55 T + 1656 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4400 T^{2} + 14480319 T^{4} + 4400 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3026 T^{2} + 1266195 T^{4} + 3026 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 82 T + 3243 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} )( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + 104 T + 7095 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 97 T + 4920 T^{2} + 97 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 65 T - 1104 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 113 T + 6528 T^{2} - 113 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12896 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
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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
−7.01105009072913780921646206953, −6.67002207849982045744605004101, −6.33969121326458103733786129829, −6.12552813725567231487018178080, −6.10575069936303329912572467751, −5.93992675712439646958305689787, −5.55362827281221382404826825682, −5.24212375375952715274864843038, −4.88756085715340661261181725415, −4.76014893376309859007756981923, −4.71628310605330985062119529121, −4.60281246362804469973087956494, −4.22555762241875516712526811612, −3.94812296232787813094696104652, −3.39058594129012499683810750698, −3.21600918766239151894838956819, −3.18911329434959911988336000570, −2.74850016733247159906303725724, −2.32808092776805742781258798862, −2.23168019773635448125308335648, −1.56385783069821463230697734231, −1.53778401678891906041078390295, −1.17607177420588161840035516127, −0.818293108166654686252154648320, −0.37066812012022831688340842107,
0.37066812012022831688340842107, 0.818293108166654686252154648320, 1.17607177420588161840035516127, 1.53778401678891906041078390295, 1.56385783069821463230697734231, 2.23168019773635448125308335648, 2.32808092776805742781258798862, 2.74850016733247159906303725724, 3.18911329434959911988336000570, 3.21600918766239151894838956819, 3.39058594129012499683810750698, 3.94812296232787813094696104652, 4.22555762241875516712526811612, 4.60281246362804469973087956494, 4.71628310605330985062119529121, 4.76014893376309859007756981923, 4.88756085715340661261181725415, 5.24212375375952715274864843038, 5.55362827281221382404826825682, 5.93992675712439646958305689787, 6.10575069936303329912572467751, 6.12552813725567231487018178080, 6.33969121326458103733786129829, 6.67002207849982045744605004101, 7.01105009072913780921646206953
