| L(s) = 1 | − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 4·17-s + 4·19-s + 4·31-s + 4·47-s − 4·53-s − 4·64-s + 8·68-s − 8·76-s + 4·79-s − 4·109-s + 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 8·188-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7941612696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7941612696\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−6.55090295747870129447713192329, −6.37868536208954240287024296718, −6.33625715295151641563856967775, −6.07356121775251998160959923331, −5.76346771413092523889889977936, −5.55978910583392285385163396250, −5.21723281758893574486414839805, −4.99390085124004374958723872221, −4.99282014844883649316989246006, −4.74793045520512264184654670387, −4.59898512648147778312873068240, −4.40886999868329333792093800180, −4.05802977341906386499302823398, −3.85749179312171973910118910423, −3.83492817778156374844949669927, −3.28956062057192296939018032324, −3.20377485212611317776624811218, −2.78365700511872524726253274281, −2.75183510207255080940771329695, −2.37350128787113147952784948725, −2.16327285279202566813070696185, −1.57335299944060337354564257844, −1.15776006786732501871274067496, −1.01710712023317427930031504358, −0.58725968488981043727340413373,
0.58725968488981043727340413373, 1.01710712023317427930031504358, 1.15776006786732501871274067496, 1.57335299944060337354564257844, 2.16327285279202566813070696185, 2.37350128787113147952784948725, 2.75183510207255080940771329695, 2.78365700511872524726253274281, 3.20377485212611317776624811218, 3.28956062057192296939018032324, 3.83492817778156374844949669927, 3.85749179312171973910118910423, 4.05802977341906386499302823398, 4.40886999868329333792093800180, 4.59898512648147778312873068240, 4.74793045520512264184654670387, 4.99282014844883649316989246006, 4.99390085124004374958723872221, 5.21723281758893574486414839805, 5.55978910583392285385163396250, 5.76346771413092523889889977936, 6.07356121775251998160959923331, 6.33625715295151641563856967775, 6.37868536208954240287024296718, 6.55090295747870129447713192329
