| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 4·8-s + 8·10-s − 4·13-s + 8·16-s + 18·17-s + 8·20-s + 5·25-s − 8·26-s − 30·29-s + 8·32-s + 36·34-s + 8·37-s + 16·40-s − 8·41-s + 10·50-s − 8·52-s − 18·53-s − 60·58-s + 12·61-s + 8·64-s − 16·65-s + 36·68-s + 22·73-s + 16·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 1.41·8-s + 2.52·10-s − 1.10·13-s + 2·16-s + 4.36·17-s + 1.78·20-s + 25-s − 1.56·26-s − 5.57·29-s + 1.41·32-s + 6.17·34-s + 1.31·37-s + 2.52·40-s − 1.24·41-s + 1.41·50-s − 1.10·52-s − 2.47·53-s − 7.87·58-s + 1.53·61-s + 64-s − 1.98·65-s + 4.36·68-s + 2.57·73-s + 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.504299306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.504299306\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−9.074273705898291619234774719251, −8.104413872002194615228126487794, −8.043179881847796517470189879025, −7.891906948525138400754440484932, −7.76060342643501637808852111903, −7.43676414340324666503852406342, −7.16673655526534998506436151277, −6.75224499775027604822645345594, −6.73315773317909223756985344797, −5.86599321735576373734177041226, −5.77281979180818852304890823105, −5.75929501830852505795665290386, −5.67112232024223033877849778261, −5.16176124874150957673076551612, −5.01944362843254632731667796285, −4.86841036686274959895721625800, −4.14945183036791228076153411124, −3.86363283640761030422397635587, −3.54931253791625302788617244709, −3.27534518875654376006040080532, −3.13842831450876510636122087383, −2.18121682226130946216705889744, −2.11373433996424690343575959023, −1.64892373512171025844326425368, −1.18890830579121130575397829367,
1.18890830579121130575397829367, 1.64892373512171025844326425368, 2.11373433996424690343575959023, 2.18121682226130946216705889744, 3.13842831450876510636122087383, 3.27534518875654376006040080532, 3.54931253791625302788617244709, 3.86363283640761030422397635587, 4.14945183036791228076153411124, 4.86841036686274959895721625800, 5.01944362843254632731667796285, 5.16176124874150957673076551612, 5.67112232024223033877849778261, 5.75929501830852505795665290386, 5.77281979180818852304890823105, 5.86599321735576373734177041226, 6.73315773317909223756985344797, 6.75224499775027604822645345594, 7.16673655526534998506436151277, 7.43676414340324666503852406342, 7.76060342643501637808852111903, 7.891906948525138400754440484932, 8.043179881847796517470189879025, 8.104413872002194615228126487794, 9.074273705898291619234774719251
