| L(s) = 1 | − 10·7-s + 76·19-s + 31·25-s + 170·43-s + 123·49-s + 206·61-s + 50·73-s − 233·121-s + 127-s + 131-s − 760·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s − 310·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.42·7-s + 4·19-s + 1.23·25-s + 3.95·43-s + 2.51·49-s + 3.37·61-s + 0.684·73-s − 1.92·121-s + 0.00787·127-s + 0.00763·131-s − 5.71·133-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 + 4·169-s + 0.00578·173-s − 1.77·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.978832456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.978832456\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 5 | $C_2^3$ | \( 1 - 31 T^{2} + 336 T^{4} - 31 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 233 T^{2} + 39648 T^{4} + 233 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 353 T^{2} + 41088 T^{4} + 353 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T + 5376 T^{2} - 85 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1207 T^{2} - 3422832 T^{4} - 1207 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 5678 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−7.42662078739164856114452663119, −7.10219887956866018569697394286, −6.89199562181837237763244105796, −6.61333178367685072636144820323, −6.56073586274598806390546434394, −6.03208125620247785894557334116, −5.77197329402580987089122235561, −5.76845037720168001067726041610, −5.45965206714618937082383471334, −5.22575714098768762590399189514, −4.88616679476817671827706909345, −4.85901110000881065939756669072, −4.31217927205019777937871646645, −3.79433736886027074657477955305, −3.77808947308780840039653908236, −3.75499855966981562531210366991, −3.23499576765945060923441191991, −2.85198076745675211866879106011, −2.68838761327096086067058740339, −2.52970888796765841691896996975, −2.15802858015011876427449359354, −1.31846967635821519710982968441, −1.05209058676810083659733323207, −0.870296799611138544644627679045, −0.45778269775088052822274685662,
0.45778269775088052822274685662, 0.870296799611138544644627679045, 1.05209058676810083659733323207, 1.31846967635821519710982968441, 2.15802858015011876427449359354, 2.52970888796765841691896996975, 2.68838761327096086067058740339, 2.85198076745675211866879106011, 3.23499576765945060923441191991, 3.75499855966981562531210366991, 3.77808947308780840039653908236, 3.79433736886027074657477955305, 4.31217927205019777937871646645, 4.85901110000881065939756669072, 4.88616679476817671827706909345, 5.22575714098768762590399189514, 5.45965206714618937082383471334, 5.76845037720168001067726041610, 5.77197329402580987089122235561, 6.03208125620247785894557334116, 6.56073586274598806390546434394, 6.61333178367685072636144820323, 6.89199562181837237763244105796, 7.10219887956866018569697394286, 7.42662078739164856114452663119
