| L(s) = 1 | + 4-s − 2·9-s + 6·31-s − 2·36-s + 49-s − 64-s + 4·71-s + 2·79-s + 81-s − 6·89-s − 2·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s − 2·9-s + 6·31-s − 2·36-s + 49-s − 64-s + 4·71-s + 2·79-s + 81-s − 6·89-s − 2·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.223311715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223311715\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.86947400123642982821511951848, −6.69196782425426827306123981798, −6.63532427232161056470101490968, −6.55040576083291127361691162647, −6.17247791406425204569962969618, −5.95474826851794784163257743483, −5.72505500947506354828497585868, −5.67685385553752730178630997665, −5.34142690993631580743450149420, −5.02172893201302514963705171709, −4.88010053989908111659696060361, −4.56140998207264830748760057485, −4.34113853677316512842851802228, −4.25678018355909960867551659412, −3.74530240559608923087225185491, −3.47160226851284291616910371761, −3.35286231215934185708487548345, −2.81208054691695784311617190260, −2.69880606878090054973636048477, −2.54372587052313966033813673154, −2.45628824756722494273533155425, −2.17577986626440423989841620957, −1.31612938331364370024605107047, −1.30472567027342386452105907802, −0.74138255448435302885048051543,
0.74138255448435302885048051543, 1.30472567027342386452105907802, 1.31612938331364370024605107047, 2.17577986626440423989841620957, 2.45628824756722494273533155425, 2.54372587052313966033813673154, 2.69880606878090054973636048477, 2.81208054691695784311617190260, 3.35286231215934185708487548345, 3.47160226851284291616910371761, 3.74530240559608923087225185491, 4.25678018355909960867551659412, 4.34113853677316512842851802228, 4.56140998207264830748760057485, 4.88010053989908111659696060361, 5.02172893201302514963705171709, 5.34142690993631580743450149420, 5.67685385553752730178630997665, 5.72505500947506354828497585868, 5.95474826851794784163257743483, 6.17247791406425204569962969618, 6.55040576083291127361691162647, 6.63532427232161056470101490968, 6.69196782425426827306123981798, 6.86947400123642982821511951848
