| L(s) = 1 | + 2·7-s + 13-s + 4·17-s − 2·19-s − 6·23-s − 4·31-s + 2·37-s + 6·41-s + 4·43-s − 4·47-s − 3·49-s − 10·53-s + 8·59-s + 6·61-s − 8·67-s − 16·73-s + 4·83-s − 6·89-s + 2·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.277·13-s + 0.970·17-s − 0.458·19-s − 1.25·23-s − 0.718·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 1.37·53-s + 1.04·59-s + 0.768·61-s − 0.977·67-s − 1.87·73-s + 0.439·83-s − 0.635·89-s + 0.209·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.33142747418434, −13.56581779193619, −13.06895869659359, −12.60735826593321, −12.09201363297583, −11.55188187845919, −11.21870589691017, −10.58415661304070, −10.19268596272745, −9.588451630286159, −9.137944827091594, −8.427985917823569, −8.070351534200584, −7.617099714937811, −7.092953379435715, −6.333423770519960, −5.842148103281000, −5.453946525617435, −4.651170227506952, −4.278013272108136, −3.601403536198052, −3.014877416944783, −2.188643121131086, −1.670724051601972, −0.9670523901958852, 0,
0.9670523901958852, 1.670724051601972, 2.188643121131086, 3.014877416944783, 3.601403536198052, 4.278013272108136, 4.651170227506952, 5.453946525617435, 5.842148103281000, 6.333423770519960, 7.092953379435715, 7.617099714937811, 8.070351534200584, 8.427985917823569, 9.137944827091594, 9.588451630286159, 10.19268596272745, 10.58415661304070, 11.21870589691017, 11.55188187845919, 12.09201363297583, 12.60735826593321, 13.06895869659359, 13.56581779193619, 14.33142747418434
