L(s) = 1 | + 2·7-s + 4·11-s + 13-s + 2·19-s − 4·23-s − 2·31-s + 10·37-s − 2·41-s + 8·43-s − 3·49-s + 12·53-s + 12·59-s − 6·61-s − 6·67-s − 8·71-s + 2·73-s + 8·77-s − 12·79-s + 4·83-s − 14·89-s + 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.20·11-s + 0.277·13-s + 0.458·19-s − 0.834·23-s − 0.359·31-s + 1.64·37-s − 0.312·41-s + 1.21·43-s − 3/7·49-s + 1.64·53-s + 1.56·59-s − 0.768·61-s − 0.733·67-s − 0.949·71-s + 0.234·73-s + 0.911·77-s − 1.35·79-s + 0.439·83-s − 1.48·89-s + 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.276204051\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.276204051\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.51439474991693, −14.28114487130816, −13.57355183162048, −13.16050349495178, −12.45585860824332, −11.91809677163612, −11.48632881205694, −11.17688591299051, −10.42274167687350, −9.886566186930492, −9.365323576257384, −8.735228428096263, −8.404771275663295, −7.571223634786527, −7.317501967601303, −6.500858954704474, −5.948764080894207, −5.514756971852355, −4.626122514257772, −4.193542202115348, −3.660153136547982, −2.820254357962660, −2.045851881761975, −1.374721657449212, −0.6912731398318442,
0.6912731398318442, 1.374721657449212, 2.045851881761975, 2.820254357962660, 3.660153136547982, 4.193542202115348, 4.626122514257772, 5.514756971852355, 5.948764080894207, 6.500858954704474, 7.317501967601303, 7.571223634786527, 8.404771275663295, 8.735228428096263, 9.365323576257384, 9.886566186930492, 10.42274167687350, 11.17688591299051, 11.48632881205694, 11.91809677163612, 12.45585860824332, 13.16050349495178, 13.57355183162048, 14.28114487130816, 14.51439474991693