| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 4·17-s − 25-s − 4·31-s + 6·32-s + 8·34-s − 2·49-s − 2·50-s − 8·62-s + 7·64-s + 12·68-s − 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s + 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 4·17-s − 25-s − 4·31-s + 6·32-s + 8·34-s − 2·49-s − 2·50-s − 8·62-s + 7·64-s + 12·68-s − 4·98-s − 3·100-s − 121-s − 12·124-s + 127-s + 8·128-s + 131-s + 16·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.096114238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.096114238\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−8.610129517064521063871200790209, −8.231853752708922034883465134851, −7.73457761912609279684093296340, −7.63124012438696137060726654141, −7.21266372005966326085810159255, −7.16929713735343097466295311344, −6.32173419051543895789902153128, −6.02575989141623421100365172667, −5.79965596381305496694007162054, −5.38116083081143528284756870931, −5.09486353341852729646412979775, −4.96518584966744755803389524303, −4.05991564912919533687928125093, −3.72461624341634139208034025797, −3.55230901344726836392846336629, −3.23362658755031601378844349088, −2.74015688377842915809085811906, −2.04403492347991883044002320281, −1.55272786336730007740934220132, −1.25772074744669751086525290423,
1.25772074744669751086525290423, 1.55272786336730007740934220132, 2.04403492347991883044002320281, 2.74015688377842915809085811906, 3.23362658755031601378844349088, 3.55230901344726836392846336629, 3.72461624341634139208034025797, 4.05991564912919533687928125093, 4.96518584966744755803389524303, 5.09486353341852729646412979775, 5.38116083081143528284756870931, 5.79965596381305496694007162054, 6.02575989141623421100365172667, 6.32173419051543895789902153128, 7.16929713735343097466295311344, 7.21266372005966326085810159255, 7.63124012438696137060726654141, 7.73457761912609279684093296340, 8.231853752708922034883465134851, 8.610129517064521063871200790209
