| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 8-s − 10-s − 15-s − 16-s − 3·17-s + 3·19-s + 4·23-s − 24-s − 27-s − 30-s − 2·31-s − 3·34-s + 3·38-s + 40-s + 4·46-s − 48-s + 49-s − 3·51-s − 54-s + 3·57-s − 2·62-s + 64-s + 4·69-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 8-s − 10-s − 15-s − 16-s − 3·17-s + 3·19-s + 4·23-s − 24-s − 27-s − 30-s − 2·31-s − 3·34-s + 3·38-s + 40-s + 4·46-s − 48-s + 49-s − 3·51-s − 54-s + 3·57-s − 2·62-s + 64-s + 4·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3459600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.782994570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782994570\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.516962193755469465597681071554, −9.214616359839847528499995282563, −9.001822292236751917486771600141, −8.401048761558076803549812053946, −8.257443535524298187723441504142, −7.42373585561881789914747067857, −7.31130566752892153944858990890, −6.89583390768559142421167575313, −6.70921969504651832706622305987, −5.77836626439740611534073563731, −5.49556878936933059162312860346, −4.89920555448619163445224245418, −4.88970235896995765524089633446, −4.15860680314738563446412743096, −3.78043950603885932356583474779, −3.28560333936627113487291873288, −3.01449912105070437135905145228, −2.62222802112097520363363005613, −1.88122675929907580322288410650, −0.813064567638551932593266090222,
0.813064567638551932593266090222, 1.88122675929907580322288410650, 2.62222802112097520363363005613, 3.01449912105070437135905145228, 3.28560333936627113487291873288, 3.78043950603885932356583474779, 4.15860680314738563446412743096, 4.88970235896995765524089633446, 4.89920555448619163445224245418, 5.49556878936933059162312860346, 5.77836626439740611534073563731, 6.70921969504651832706622305987, 6.89583390768559142421167575313, 7.31130566752892153944858990890, 7.42373585561881789914747067857, 8.257443535524298187723441504142, 8.401048761558076803549812053946, 9.001822292236751917486771600141, 9.214616359839847528499995282563, 9.516962193755469465597681071554
