| L(s) = 1 | + 4·7-s + 8·49-s + 4·73-s − 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯ |
| L(s) = 1 | + 4·7-s + 8·49-s + 4·73-s − 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.063683439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063683439\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 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
−7.23877932339360326864512391072, −6.62240641491018134764554524249, −6.57920763497318230169131748881, −6.42423265312709727222128965060, −6.31141017574478206698481934846, −5.72530047700319781805796430884, −5.49478325187509058579704908580, −5.36110192252952478220764486783, −5.28060923241097287067246004484, −5.03784283417454863002743097183, −4.82763904591644992128280019524, −4.68335828631591790826189762781, −4.42053086846019867613949769744, −4.05074312259481213836843855944, −3.89937790894559906117786684628, −3.69127556985221512978170843970, −3.61721760789822778265946992051, −2.69607859618904268413154831238, −2.65384282431495531795497896803, −2.57850643967214927653842104658, −2.18961229420917321246268900238, −1.67726755341189007465008378223, −1.51646690258066704947517234067, −1.36293816973987177981554515946, −1.01192412048972559087686914864,
1.01192412048972559087686914864, 1.36293816973987177981554515946, 1.51646690258066704947517234067, 1.67726755341189007465008378223, 2.18961229420917321246268900238, 2.57850643967214927653842104658, 2.65384282431495531795497896803, 2.69607859618904268413154831238, 3.61721760789822778265946992051, 3.69127556985221512978170843970, 3.89937790894559906117786684628, 4.05074312259481213836843855944, 4.42053086846019867613949769744, 4.68335828631591790826189762781, 4.82763904591644992128280019524, 5.03784283417454863002743097183, 5.28060923241097287067246004484, 5.36110192252952478220764486783, 5.49478325187509058579704908580, 5.72530047700319781805796430884, 6.31141017574478206698481934846, 6.42423265312709727222128965060, 6.57920763497318230169131748881, 6.62240641491018134764554524249, 7.23877932339360326864512391072
