| L(s) = 1 | − 2·9-s − 2·16-s − 4·49-s + 3·81-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2·16-s − 4·49-s + 3·81-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2034407944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2034407944\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−9.440866714430305903006383467153, −9.257280885009195301406715077245, −8.768657388659605761397972046429, −8.545141310823371774682264401599, −8.469663218549072951032124486086, −8.232234247667342082018338164558, −7.80245856384292925587611057570, −7.70808450273608444746121703780, −7.11846067440603834816578203640, −6.94789283718109461319336657615, −6.67272123123419501087238197541, −6.43590165842527185289645134336, −6.00811072252381319557068017303, −5.76768944560953259787734307338, −5.70857816027252926790076630429, −4.88969764023033145209072299567, −4.84633011308720520843676092457, −4.78366010462336074980744642991, −4.21797388937070812530291557551, −3.66403376311213264109839900037, −3.29967813781756604352129154086, −3.09860104945933376631564321676, −2.52383809435635533548851648622, −2.23288252098447639708228577616, −1.66422429043957898322457300816,
1.66422429043957898322457300816, 2.23288252098447639708228577616, 2.52383809435635533548851648622, 3.09860104945933376631564321676, 3.29967813781756604352129154086, 3.66403376311213264109839900037, 4.21797388937070812530291557551, 4.78366010462336074980744642991, 4.84633011308720520843676092457, 4.88969764023033145209072299567, 5.70857816027252926790076630429, 5.76768944560953259787734307338, 6.00811072252381319557068017303, 6.43590165842527185289645134336, 6.67272123123419501087238197541, 6.94789283718109461319336657615, 7.11846067440603834816578203640, 7.70808450273608444746121703780, 7.80245856384292925587611057570, 8.232234247667342082018338164558, 8.469663218549072951032124486086, 8.545141310823371774682264401599, 8.768657388659605761397972046429, 9.257280885009195301406715077245, 9.440866714430305903006383467153
