| L(s) = 1 | − 3·5-s + 2·7-s − 9-s + 3·17-s + 19-s − 3·23-s + 5·25-s − 6·35-s + 3·45-s + 2·47-s + 3·49-s − 2·63-s − 2·83-s − 9·85-s − 3·95-s + 3·101-s + 9·115-s + 6·119-s − 121-s − 6·125-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3·5-s + 2·7-s − 9-s + 3·17-s + 19-s − 3·23-s + 5·25-s − 6·35-s + 3·45-s + 2·47-s + 3·49-s − 2·63-s − 2·83-s − 9·85-s − 3·95-s + 3·101-s + 9·115-s + 6·119-s − 121-s − 6·125-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4528384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8123927287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8123927287\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.447518534286920168368871786696, −8.659330615228117947383402275993, −8.470507690608842951038007020025, −8.246857951311914064867923837188, −7.911992987174439869646672830492, −7.57033008933390230494106598631, −7.40471070474306594963870286449, −7.30165757746488259783406120048, −6.22611235115667258567388439865, −5.62415852236648704299354914853, −5.61091398054100387808665124673, −5.05820642195303694774687866612, −4.37513906216833030932583372844, −4.27568027461664598340555195316, −3.69695136520859988420538150719, −3.46851343277349167427868613284, −2.95540018165320772393406118273, −2.21075619522109966132077998261, −1.35089012953080285445316301409, −0.72206824961062071798645446657,
0.72206824961062071798645446657, 1.35089012953080285445316301409, 2.21075619522109966132077998261, 2.95540018165320772393406118273, 3.46851343277349167427868613284, 3.69695136520859988420538150719, 4.27568027461664598340555195316, 4.37513906216833030932583372844, 5.05820642195303694774687866612, 5.61091398054100387808665124673, 5.62415852236648704299354914853, 6.22611235115667258567388439865, 7.30165757746488259783406120048, 7.40471070474306594963870286449, 7.57033008933390230494106598631, 7.911992987174439869646672830492, 8.246857951311914064867923837188, 8.470507690608842951038007020025, 8.659330615228117947383402275993, 9.447518534286920168368871786696
