| L(s) = 1 | + 2·5-s − 2·9-s − 2·13-s + 2·25-s + 6·29-s + 4·37-s + 4·41-s − 4·45-s − 4·65-s + 2·73-s + 81-s − 2·89-s − 2·97-s + 4·109-s + 4·117-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·9-s − 2·13-s + 2·25-s + 6·29-s + 4·37-s + 4·41-s − 4·45-s − 4·65-s + 2·73-s + 81-s − 2·89-s − 2·97-s + 4·109-s + 4·117-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.721838695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721838695\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.16973630695720549709955745130, −6.14298998455370009298131736416, −5.95590815885459182680499672331, −5.77454395282607514960973054792, −5.55500193687132870116964110503, −5.35701155000773536628563734028, −4.99080894977485011405071949089, −4.95276314727428422449442762680, −4.86635912528479448975027052556, −4.32070130265477081095739799137, −4.29823811408053858463768280750, −4.28303909530734965116018440426, −4.13542678883333485709505908580, −3.25578719646000899301107582398, −3.17581378630306381035729390321, −3.07440001321462754717056425364, −2.65883080894049094210515550327, −2.55769374648017486053446027619, −2.52429185860616203246947830923, −2.38815573006241071183332501094, −2.18185987473414201355319546283, −1.62234241986152846072797458785, −1.02724387459941760741150787824, −0.892198906343324535952034737952, −0.867005665124100306582583730909,
0.867005665124100306582583730909, 0.892198906343324535952034737952, 1.02724387459941760741150787824, 1.62234241986152846072797458785, 2.18185987473414201355319546283, 2.38815573006241071183332501094, 2.52429185860616203246947830923, 2.55769374648017486053446027619, 2.65883080894049094210515550327, 3.07440001321462754717056425364, 3.17581378630306381035729390321, 3.25578719646000899301107582398, 4.13542678883333485709505908580, 4.28303909530734965116018440426, 4.29823811408053858463768280750, 4.32070130265477081095739799137, 4.86635912528479448975027052556, 4.95276314727428422449442762680, 4.99080894977485011405071949089, 5.35701155000773536628563734028, 5.55500193687132870116964110503, 5.77454395282607514960973054792, 5.95590815885459182680499672331, 6.14298998455370009298131736416, 6.16973630695720549709955745130
