| L(s) = 1 | + 2-s + 2·3-s + 5-s + 2·6-s − 8-s + 3·9-s + 10-s − 11-s + 13-s + 2·15-s − 16-s + 3·18-s + 2·19-s − 22-s − 2·23-s − 2·24-s + 26-s + 4·27-s + 29-s + 2·30-s − 31-s − 2·33-s + 2·37-s + 2·38-s + 2·39-s − 40-s − 2·43-s + ⋯ |
| L(s) = 1 | + 2-s + 2·3-s + 5-s + 2·6-s − 8-s + 3·9-s + 10-s − 11-s + 13-s + 2·15-s − 16-s + 3·18-s + 2·19-s − 22-s − 2·23-s − 2·24-s + 26-s + 4·27-s + 29-s + 2·30-s − 31-s − 2·33-s + 2·37-s + 2·38-s + 2·39-s − 40-s − 2·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.858483458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.858483458\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.458481674900585930749843365516, −8.906854326497644224817705193743, −8.481587622319228772714733540134, −8.437690235190450867866851667167, −7.76685114785454450950152220891, −7.62602943026192656124815087871, −7.15896241442155522048461738819, −6.63675418306843945865532155327, −6.04351643759529808395124727619, −5.83315053160085216435165708112, −5.53333573097022265468624607344, −4.73929953144796966459801110098, −4.54222238814258592925857890151, −4.13328043430445947807520093430, −3.51379219268461185730276573339, −3.11479149649103305521602675458, −2.94304726095212344658134304158, −2.40000491970116472393150043302, −1.68622570987426808772740945186, −1.39846795327624710321149104069,
1.39846795327624710321149104069, 1.68622570987426808772740945186, 2.40000491970116472393150043302, 2.94304726095212344658134304158, 3.11479149649103305521602675458, 3.51379219268461185730276573339, 4.13328043430445947807520093430, 4.54222238814258592925857890151, 4.73929953144796966459801110098, 5.53333573097022265468624607344, 5.83315053160085216435165708112, 6.04351643759529808395124727619, 6.63675418306843945865532155327, 7.15896241442155522048461738819, 7.62602943026192656124815087871, 7.76685114785454450950152220891, 8.437690235190450867866851667167, 8.481587622319228772714733540134, 8.906854326497644224817705193743, 9.458481674900585930749843365516
