L(s) = 1 | + i·2-s − 4-s + i·5-s − i·8-s − 10-s + 16-s − 2i·17-s − i·20-s − 25-s + i·32-s + 2·34-s + 40-s − 49-s − i·50-s + 2i·53-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + i·5-s − i·8-s − 10-s + 16-s − 2i·17-s − i·20-s − 25-s + i·32-s + 2·34-s + 40-s − 49-s − i·50-s + 2i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6199840944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6199840944\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.61416047276907475538878389672, −12.24786515957334543450342310498, −11.12243734817267578579130886343, −9.951066944493501011919944634999, −9.098082573711693085294217039423, −7.70984593560360609597004373017, −7.01138108804813254061843343097, −5.94207158086714460524284264874, −4.65105749693461699670374546515, −3.07697035633363634318314595901,
1.72822679618767786561111164747, 3.66834307526012773911986557907, 4.78133270384470936549343138359, 6.00544160009276436343889004620, 8.002705591916857497167492234651, 8.731949602901659574138655814886, 9.768152071362591591297198359402, 10.70856199700068145277697392096, 11.77201509376005350156308605147, 12.68773874131481316135968972332