| L(s) = 1 | + 3-s + 7-s + 2·13-s + 19-s + 21-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s + 57-s − 2·61-s − 67-s − 73-s + 79-s − 81-s + 2·91-s + 93-s − 4·97-s − 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + 131-s + 133-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s + 2·13-s + 19-s + 21-s − 27-s + 31-s − 37-s + 2·39-s + 2·43-s + 57-s − 2·61-s − 67-s − 73-s + 79-s − 81-s + 2·91-s + 93-s − 4·97-s − 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + 131-s + 133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.270372983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270372983\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−9.258382909569547025805667674720, −9.063148766285286058111763923686, −8.694874687235925772125525205307, −8.314970630063814253153812377420, −7.924998305668453547481101029377, −7.87092920713023970547036921702, −7.19377377164200225508075526760, −6.94415730565846520036423164473, −6.18489209929863460314776102203, −5.99399776503391756253281080173, −5.56460077915549840454955658839, −5.12115623995207422647500419242, −4.47053160587192487142143812776, −4.20535025278161870876941770682, −3.64453286963641089160591927539, −3.23767976057072387388573039143, −2.80493092091741748280397684123, −2.25636826028939486999697662223, −1.37010616076441982381736623887, −1.31695176450497151177794280628,
1.31695176450497151177794280628, 1.37010616076441982381736623887, 2.25636826028939486999697662223, 2.80493092091741748280397684123, 3.23767976057072387388573039143, 3.64453286963641089160591927539, 4.20535025278161870876941770682, 4.47053160587192487142143812776, 5.12115623995207422647500419242, 5.56460077915549840454955658839, 5.99399776503391756253281080173, 6.18489209929863460314776102203, 6.94415730565846520036423164473, 7.19377377164200225508075526760, 7.87092920713023970547036921702, 7.924998305668453547481101029377, 8.314970630063814253153812377420, 8.694874687235925772125525205307, 9.063148766285286058111763923686, 9.258382909569547025805667674720
