| L(s) = 1 | + 2·4-s + 4·11-s + 16-s − 4·29-s + 8·44-s + 2·49-s − 2·64-s − 4·89-s − 4·101-s − 4·109-s − 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·4-s + 4·11-s + 16-s − 4·29-s + 8·44-s + 2·49-s − 2·64-s − 4·89-s − 4·101-s − 4·109-s − 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.254548272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.254548272\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 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
−6.97521659562400289068037110914, −6.93685517473482952822409025129, −6.79702715156859495192709331649, −6.44196141677873941593080443324, −6.41102835994116180479947010341, −5.99960822400291924195681016217, −5.87035153941037343147387467410, −5.71871128636259064715408258652, −5.38250658370185532485174349822, −5.37591952760100189981128218137, −4.89157052261506185710334771109, −4.39497443868652757035883322767, −4.08686641423656999667181487190, −4.05344393099094602045311876221, −4.04157646832285740614025453381, −3.82123535619916242822078092884, −3.35522843904284125642139184055, −3.00698395384819701814263092080, −2.87274236070004107949456765827, −2.52174986913497857654678087032, −2.12252140003167944483305309268, −1.69598919815065488925671967601, −1.67038125642667276524652896145, −1.53299647145158683911279144529, −1.01035329716552294712814623760,
1.01035329716552294712814623760, 1.53299647145158683911279144529, 1.67038125642667276524652896145, 1.69598919815065488925671967601, 2.12252140003167944483305309268, 2.52174986913497857654678087032, 2.87274236070004107949456765827, 3.00698395384819701814263092080, 3.35522843904284125642139184055, 3.82123535619916242822078092884, 4.04157646832285740614025453381, 4.05344393099094602045311876221, 4.08686641423656999667181487190, 4.39497443868652757035883322767, 4.89157052261506185710334771109, 5.37591952760100189981128218137, 5.38250658370185532485174349822, 5.71871128636259064715408258652, 5.87035153941037343147387467410, 5.99960822400291924195681016217, 6.41102835994116180479947010341, 6.44196141677873941593080443324, 6.79702715156859495192709331649, 6.93685517473482952822409025129, 6.97521659562400289068037110914
