| L(s) = 1 | − 4·7-s − 2·16-s + 4·43-s + 8·49-s + 8·61-s − 4·73-s + 8·112-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·7-s − 2·16-s + 4·43-s + 8·49-s + 8·61-s − 4·73-s + 8·112-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3471844890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3471844890\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$ | \( ( 1 - T )^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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
−7.31780188478307991858998662991, −7.19891578715409309517316877698, −7.02905413148285075207399633218, −6.83883075009207009662782640316, −6.69683490653387019619337918651, −6.31225992798739807931633303070, −6.29691858149005023084512091144, −5.95762773566764123329529615527, −5.77638106324670461903577395945, −5.61311521998488495491058004213, −5.19409813399229753798455324267, −5.14362758230769134078721484760, −4.51340070619943166188766706478, −4.29713547017525582512552863543, −4.00382118451888696270977422416, −3.98682867765082996015417922914, −3.64900819978712373072822762962, −3.37133175909054293983531186810, −2.98062933772815863863763820858, −2.75184923333917728736360534650, −2.56780158157675065733925476292, −2.24614946098989810571290502622, −2.07441109748410964014599822809, −1.04879198636909619473240441296, −0.59014907399946156899795443648,
0.59014907399946156899795443648, 1.04879198636909619473240441296, 2.07441109748410964014599822809, 2.24614946098989810571290502622, 2.56780158157675065733925476292, 2.75184923333917728736360534650, 2.98062933772815863863763820858, 3.37133175909054293983531186810, 3.64900819978712373072822762962, 3.98682867765082996015417922914, 4.00382118451888696270977422416, 4.29713547017525582512552863543, 4.51340070619943166188766706478, 5.14362758230769134078721484760, 5.19409813399229753798455324267, 5.61311521998488495491058004213, 5.77638106324670461903577395945, 5.95762773566764123329529615527, 6.29691858149005023084512091144, 6.31225992798739807931633303070, 6.69683490653387019619337918651, 6.83883075009207009662782640316, 7.02905413148285075207399633218, 7.19891578715409309517316877698, 7.31780188478307991858998662991
