| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 8·13-s − 8·17-s − 8·19-s − 4·21-s − 10·23-s + 6·27-s − 16·39-s − 8·41-s + 14·43-s − 6·47-s + 2·49-s − 16·51-s − 8·53-s − 16·57-s − 8·59-s + 16·61-s − 4·63-s − 6·67-s − 20·69-s + 8·73-s − 16·79-s + 11·81-s − 10·83-s + 16·91-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 2/3·9-s − 2.21·13-s − 1.94·17-s − 1.83·19-s − 0.872·21-s − 2.08·23-s + 1.15·27-s − 2.56·39-s − 1.24·41-s + 2.13·43-s − 0.875·47-s + 2/7·49-s − 2.24·51-s − 1.09·53-s − 2.11·57-s − 1.04·59-s + 2.04·61-s − 0.503·63-s − 0.733·67-s − 2.40·69-s + 0.936·73-s − 1.80·79-s + 11/9·81-s − 1.09·83-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3670314245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3670314245\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.878476042868914690461753608833, −9.192900721160815985217300973106, −8.651729898430027713675274265469, −8.608609749428778522541462133173, −8.089710328410930518911263394814, −7.67151786192096691761061187536, −7.23639904030759459189969826805, −6.80379654127223642158764400275, −6.51233916545006477869457783845, −6.11105529258979221719144960840, −5.57524113552240322005604675061, −4.67765169022726407765509898463, −4.65910509730366936616086594755, −4.15087887146774025401614048191, −3.69070812349028980362855878864, −2.97357296964952183081668291802, −2.46311500184935181434349186261, −2.27037120910283167330980891325, −1.78525176044017764225093375829, −0.19487628435779945869852605905,
0.19487628435779945869852605905, 1.78525176044017764225093375829, 2.27037120910283167330980891325, 2.46311500184935181434349186261, 2.97357296964952183081668291802, 3.69070812349028980362855878864, 4.15087887146774025401614048191, 4.65910509730366936616086594755, 4.67765169022726407765509898463, 5.57524113552240322005604675061, 6.11105529258979221719144960840, 6.51233916545006477869457783845, 6.80379654127223642158764400275, 7.23639904030759459189969826805, 7.67151786192096691761061187536, 8.089710328410930518911263394814, 8.608609749428778522541462133173, 8.651729898430027713675274265469, 9.192900721160815985217300973106, 9.878476042868914690461753608833
