L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5353003331\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5353003331\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−10.75221685985421843874095155583, −10.52182806991915867942435966588, −10.03268865522979731250990617072, −9.516285885690007470604571955858, −9.099879670111464566614379913831, −8.354335030273642507666476163441, −7.987549958995614787489586635325, −7.79309233068281090636853498539, −7.23989387664129141346164886590, −6.64426959537030984101543600395, −6.05031302786525693150766574450, −5.94671590199537883366208087967, −5.35118785830813710866323432898, −4.90110252422022193942073224709, −4.68241429286250276903957052486, −4.04655376704796702448425754963, −3.53643054532470293052896772089, −2.16891100401709272297954391108, −1.91087069452013848845759771743, −0.912779280286979310418994085698,
0.912779280286979310418994085698, 1.91087069452013848845759771743, 2.16891100401709272297954391108, 3.53643054532470293052896772089, 4.04655376704796702448425754963, 4.68241429286250276903957052486, 4.90110252422022193942073224709, 5.35118785830813710866323432898, 5.94671590199537883366208087967, 6.05031302786525693150766574450, 6.64426959537030984101543600395, 7.23989387664129141346164886590, 7.79309233068281090636853498539, 7.987549958995614787489586635325, 8.354335030273642507666476163441, 9.099879670111464566614379913831, 9.516285885690007470604571955858, 10.03268865522979731250990617072, 10.52182806991915867942435966588, 10.75221685985421843874095155583