| L(s) = 1 | + 4-s + 4·5-s − 2·9-s + 4·20-s + 10·25-s − 2·36-s + 4·41-s − 8·45-s + 49-s − 6·61-s − 64-s + 3·81-s − 2·89-s + 10·100-s − 4·101-s − 4·109-s − 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + ⋯ |
| L(s) = 1 | + 4-s + 4·5-s − 2·9-s + 4·20-s + 10·25-s − 2·36-s + 4·41-s − 8·45-s + 49-s − 6·61-s − 64-s + 3·81-s − 2·89-s + 10·100-s − 4·101-s − 4·109-s − 4·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.706879934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.706879934\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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
−6.81107651978977083227175357615, −6.68223063811269235170266728578, −6.62136047080126541631488667744, −6.48349043539424484280093575736, −6.28634862325814414317861584422, −5.85071292438049271412489111453, −5.75206331008691630025875432234, −5.72377583140326229317873768971, −5.63688792482180903277787507903, −5.24208560702033184641257073411, −5.23860739767220038446927803754, −4.63177102056873403481277271958, −4.58871052608067503367866511407, −4.23524928762237560044066381937, −4.06571804842304884912483226843, −3.38330758534954897823108681622, −3.03076502324193322716916496028, −2.92012683967461966255106766222, −2.72231303935733839428552139249, −2.52909892528752814293594857122, −2.31246311293130482964913751944, −2.17321762946528768849451510680, −1.47622606909111370400746695295, −1.33749376944039342095287133960, −1.27775670246214999211020276102,
1.27775670246214999211020276102, 1.33749376944039342095287133960, 1.47622606909111370400746695295, 2.17321762946528768849451510680, 2.31246311293130482964913751944, 2.52909892528752814293594857122, 2.72231303935733839428552139249, 2.92012683967461966255106766222, 3.03076502324193322716916496028, 3.38330758534954897823108681622, 4.06571804842304884912483226843, 4.23524928762237560044066381937, 4.58871052608067503367866511407, 4.63177102056873403481277271958, 5.23860739767220038446927803754, 5.24208560702033184641257073411, 5.63688792482180903277787507903, 5.72377583140326229317873768971, 5.75206331008691630025875432234, 5.85071292438049271412489111453, 6.28634862325814414317861584422, 6.48349043539424484280093575736, 6.62136047080126541631488667744, 6.68223063811269235170266728578, 6.81107651978977083227175357615
