| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 9-s − 4·15-s + 10·17-s − 2·21-s − 6·25-s − 4·27-s + 2·35-s + 2·41-s + 12·43-s − 2·45-s − 8·47-s + 49-s + 20·51-s + 4·59-s − 63-s − 4·67-s − 12·75-s + 12·79-s − 11·81-s + 12·83-s − 20·85-s + 2·89-s − 2·101-s + 4·105-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.03·15-s + 2.42·17-s − 0.436·21-s − 6/5·25-s − 0.769·27-s + 0.338·35-s + 0.312·41-s + 1.82·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 2.80·51-s + 0.520·59-s − 0.125·63-s − 0.488·67-s − 1.38·75-s + 1.35·79-s − 1.22·81-s + 1.31·83-s − 2.16·85-s + 0.211·89-s − 0.199·101-s + 0.390·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256804806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256804806\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.094726207576002052327951237265, −7.83744678228188559402402949393, −7.58888161146460243251794695699, −7.22464157828388570293458543105, −6.48821678941594783657604009559, −5.95148335488680428828429687062, −5.59020824059883051468159177909, −5.06733987101923613791486571768, −4.27502890842435147332130377219, −3.85274177996473928583955728443, −3.44534495725308014455846595864, −3.08066518796494493239778549797, −2.42676430509681020355367385124, −1.68090237444709890758305579493, −0.68725605487355263280002196340,
0.68725605487355263280002196340, 1.68090237444709890758305579493, 2.42676430509681020355367385124, 3.08066518796494493239778549797, 3.44534495725308014455846595864, 3.85274177996473928583955728443, 4.27502890842435147332130377219, 5.06733987101923613791486571768, 5.59020824059883051468159177909, 5.95148335488680428828429687062, 6.48821678941594783657604009559, 7.22464157828388570293458543105, 7.58888161146460243251794695699, 7.83744678228188559402402949393, 8.094726207576002052327951237265
