| L(s) = 1 | + 2·3-s + 9-s + 4·25-s − 2·27-s − 2·43-s − 49-s − 2·61-s + 8·75-s + 4·79-s − 4·81-s + 4·103-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 4·183-s + ⋯ |
| L(s) = 1 | + 2·3-s + 9-s + 4·25-s − 2·27-s − 2·43-s − 49-s − 2·61-s + 8·75-s + 4·79-s − 4·81-s + 4·103-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 4·183-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.574591062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.574591062\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85462729702733099696646125987, −6.31050184442944251618123836383, −6.24318326861462832447797062995, −6.21112043807360551317695804149, −6.16319285793911262094613056588, −5.54175481087427930192594200113, −5.41712929706599604611400727063, −5.00700659218045369388411140485, −4.92857228958139738958830065654, −4.86340897215149149484346394028, −4.55962472012792335869934460211, −4.50437504460506390536692081644, −3.75585220173567784397022089098, −3.69550870296359372515825236219, −3.67658124194440964129395766014, −3.30345605606611983111480920873, −3.08334106500975603790824671712, −2.95704701128708717350004676892, −2.72678631219657749376863488759, −2.39804883398504794502047851769, −2.11482243059191104292384473397, −1.98849666271633357621488127018, −1.40825964537388028448826641200, −1.30144481294405488552726294369, −0.69874770320213459157993883171,
0.69874770320213459157993883171, 1.30144481294405488552726294369, 1.40825964537388028448826641200, 1.98849666271633357621488127018, 2.11482243059191104292384473397, 2.39804883398504794502047851769, 2.72678631219657749376863488759, 2.95704701128708717350004676892, 3.08334106500975603790824671712, 3.30345605606611983111480920873, 3.67658124194440964129395766014, 3.69550870296359372515825236219, 3.75585220173567784397022089098, 4.50437504460506390536692081644, 4.55962472012792335869934460211, 4.86340897215149149484346394028, 4.92857228958139738958830065654, 5.00700659218045369388411140485, 5.41712929706599604611400727063, 5.54175481087427930192594200113, 6.16319285793911262094613056588, 6.21112043807360551317695804149, 6.24318326861462832447797062995, 6.31050184442944251618123836383, 6.85462729702733099696646125987
