| L(s) = 1 | − 4-s + 16-s − 25-s − 2·49-s − 4·61-s − 64-s + 100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 25-s − 2·49-s − 4·61-s − 64-s + 100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3843802773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3843802773\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−13.61416047276907475538878389672, −12.68773874131481316135968972332, −12.24786515957334543450342310498, −11.77201509376005350156308605147, −11.12243734817267578579130886343, −10.70856199700068145277697392096, −9.951066944493501011919944634999, −9.768152071362591591297198359402, −9.098082573711693085294217039423, −8.731949602901659574138655814886, −8.002705591916857497167492234651, −7.70984593560360609597004373017, −7.01138108804813254061843343097, −6.00544160009276436343889004620, −5.94207158086714460524284264874, −4.78133270384470936549343138359, −4.65105749693461699670374546515, −3.66834307526012773911986557907, −3.07697035633363634318314595901, −1.72822679618767786561111164747,
1.72822679618767786561111164747, 3.07697035633363634318314595901, 3.66834307526012773911986557907, 4.65105749693461699670374546515, 4.78133270384470936549343138359, 5.94207158086714460524284264874, 6.00544160009276436343889004620, 7.01138108804813254061843343097, 7.70984593560360609597004373017, 8.002705591916857497167492234651, 8.731949602901659574138655814886, 9.098082573711693085294217039423, 9.768152071362591591297198359402, 9.951066944493501011919944634999, 10.70856199700068145277697392096, 11.12243734817267578579130886343, 11.77201509376005350156308605147, 12.24786515957334543450342310498, 12.68773874131481316135968972332, 13.61416047276907475538878389672
