| L(s) = 1 | + 4-s + 2·7-s − 12·13-s + 7·25-s + 2·28-s + 18·31-s + 28·37-s + 4·43-s − 11·49-s − 12·52-s + 24·61-s − 64-s − 4·67-s + 20·79-s − 24·91-s + 48·97-s + 7·100-s + 66·103-s + 52·109-s − 13·121-s + 18·124-s + 127-s + 131-s + 137-s + 139-s + 28·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 3.32·13-s + 7/5·25-s + 0.377·28-s + 3.23·31-s + 4.60·37-s + 0.609·43-s − 1.57·49-s − 1.66·52-s + 3.07·61-s − 1/8·64-s − 0.488·67-s + 2.25·79-s − 2.51·91-s + 4.87·97-s + 7/10·100-s + 6.50·103-s + 4.98·109-s − 1.18·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.242822903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.242822903\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 22 T^{2} - 357 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 65 T^{2} + 2544 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.03418399690251759340898120916, −6.76128950534948400907262808511, −6.62344609712104583226900944734, −6.30501807027488483119562506812, −6.21345724775961075345903546128, −6.03684456371214306074980975359, −5.76545083040654029480937705635, −5.33621199539565006588941389433, −4.94478165139916874673703700370, −4.87894960539796418511523985708, −4.87502639878310935530244578874, −4.53310312398938953251370954156, −4.48973225285547744852435069868, −4.20216296576331615354727486095, −3.67990058248785797011976299119, −3.35726101544635639341801549924, −3.13512592337517155366068589247, −2.71127631175545733891639855371, −2.65749417175467379978049526952, −2.24320054594658711130895688567, −2.11202299295919567409364366201, −2.05262079403207153106863830081, −1.02842906320136551675501132810, −0.802412669384906573776552830287, −0.68766237499812458945480432282,
0.68766237499812458945480432282, 0.802412669384906573776552830287, 1.02842906320136551675501132810, 2.05262079403207153106863830081, 2.11202299295919567409364366201, 2.24320054594658711130895688567, 2.65749417175467379978049526952, 2.71127631175545733891639855371, 3.13512592337517155366068589247, 3.35726101544635639341801549924, 3.67990058248785797011976299119, 4.20216296576331615354727486095, 4.48973225285547744852435069868, 4.53310312398938953251370954156, 4.87502639878310935530244578874, 4.87894960539796418511523985708, 4.94478165139916874673703700370, 5.33621199539565006588941389433, 5.76545083040654029480937705635, 6.03684456371214306074980975359, 6.21345724775961075345903546128, 6.30501807027488483119562506812, 6.62344609712104583226900944734, 6.76128950534948400907262808511, 7.03418399690251759340898120916
