| L(s) = 1 | + 4·4-s − 2·7-s + 12·16-s + 14·19-s + 10·25-s − 8·28-s + 4·31-s − 11·49-s + 32·64-s − 10·67-s + 56·76-s − 38·97-s + 40·100-s + 26·103-s + 4·109-s − 24·112-s − 22·121-s + 16·124-s + 127-s + 131-s − 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.755·7-s + 3·16-s + 3.21·19-s + 2·25-s − 1.51·28-s + 0.718·31-s − 1.57·49-s + 4·64-s − 1.22·67-s + 6.42·76-s − 3.85·97-s + 4·100-s + 2.56·103-s + 0.383·109-s − 2.26·112-s − 2·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s − 2.42·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.883607009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.883607009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−10.32690231549105871194129638164, −10.13021293583949227853670056708, −9.571688339889755240365150040673, −9.442055924525642148383127748431, −8.644916087102384032355507276855, −8.204400482718955113842690674705, −7.63112010312810817981098812721, −7.37435186975013637376682514525, −6.94393278264258738690409182587, −6.68973930469252513392504444109, −6.03101496197021426929207776160, −5.86633754488478670797407847829, −4.99945579488982066570362484438, −4.97714237611108990981561212136, −3.67939537835153271974096857617, −3.35084389068272271896108279206, −2.76048971805755213199400036943, −2.68598811595265917193386350893, −1.42665255847969558158114517367, −1.12171727430466845388982885073,
1.12171727430466845388982885073, 1.42665255847969558158114517367, 2.68598811595265917193386350893, 2.76048971805755213199400036943, 3.35084389068272271896108279206, 3.67939537835153271974096857617, 4.97714237611108990981561212136, 4.99945579488982066570362484438, 5.86633754488478670797407847829, 6.03101496197021426929207776160, 6.68973930469252513392504444109, 6.94393278264258738690409182587, 7.37435186975013637376682514525, 7.63112010312810817981098812721, 8.204400482718955113842690674705, 8.644916087102384032355507276855, 9.442055924525642148383127748431, 9.571688339889755240365150040673, 10.13021293583949227853670056708, 10.32690231549105871194129638164
