L(s) = 1 | − 2·7-s − 2·11-s − 13-s − 2·17-s − 4·19-s + 23-s − 29-s + 2·31-s − 12·41-s + 5·43-s + 4·47-s − 3·49-s + 9·53-s + 8·59-s + 7·61-s − 14·67-s − 6·73-s + 4·77-s + 15·79-s − 4·83-s + 18·89-s + 2·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s − 0.185·29-s + 0.359·31-s − 1.87·41-s + 0.762·43-s + 0.583·47-s − 3/7·49-s + 1.23·53-s + 1.04·59-s + 0.896·61-s − 1.71·67-s − 0.702·73-s + 0.455·77-s + 1.68·79-s − 0.439·83-s + 1.90·89-s + 0.209·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.83766902635370, −14.48510897291378, −13.59103592733741, −13.31007009799266, −12.93870016451687, −12.28570750940126, −11.83526874109890, −11.25367963495660, −10.48457581434144, −10.31992528939703, −9.710451720031584, −8.993450389328662, −8.655948462821802, −8.023390139613725, −7.314384908749803, −6.897560026296160, −6.260089840712996, −5.793336122974317, −5.007722356473867, −4.558944615432775, −3.750112310036905, −3.232619943012752, −2.421511117258014, −1.984113222927122, −0.7962268170411363, 0,
0.7962268170411363, 1.984113222927122, 2.421511117258014, 3.232619943012752, 3.750112310036905, 4.558944615432775, 5.007722356473867, 5.793336122974317, 6.260089840712996, 6.897560026296160, 7.314384908749803, 8.023390139613725, 8.655948462821802, 8.993450389328662, 9.710451720031584, 10.31992528939703, 10.48457581434144, 11.25367963495660, 11.83526874109890, 12.28570750940126, 12.93870016451687, 13.31007009799266, 13.59103592733741, 14.48510897291378, 14.83766902635370