| L(s) = 1 | + 4·7-s − 4·11-s + 13-s + 2·17-s + 4·19-s − 2·23-s + 2·29-s − 6·31-s − 2·37-s + 2·41-s + 4·43-s − 12·47-s + 9·49-s − 8·59-s + 6·61-s − 10·67-s + 8·71-s − 4·73-s − 16·77-s − 4·79-s − 4·83-s − 2·89-s + 4·91-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.417·23-s + 0.371·29-s − 1.07·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 1.04·59-s + 0.768·61-s − 1.22·67-s + 0.949·71-s − 0.468·73-s − 1.82·77-s − 0.450·79-s − 0.439·83-s − 0.211·89-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.09858027578225, −13.70083492112103, −13.05061436328610, −12.64906010835509, −12.05660430806980, −11.47259339099418, −11.22291350964708, −10.66082293284064, −10.16225131746698, −9.726557606332486, −8.936732474008677, −8.543014801393998, −7.945426040914166, −7.577080614077000, −7.301178489159753, −6.332280397477666, −5.791242764540980, −5.173734223094340, −4.958091982828192, −4.295886232972490, −3.537217488877257, −2.979226578523440, −2.216865243778051, −1.644203462280645, −1.007800332481298, 0,
1.007800332481298, 1.644203462280645, 2.216865243778051, 2.979226578523440, 3.537217488877257, 4.295886232972490, 4.958091982828192, 5.173734223094340, 5.791242764540980, 6.332280397477666, 7.301178489159753, 7.577080614077000, 7.945426040914166, 8.543014801393998, 8.936732474008677, 9.726557606332486, 10.16225131746698, 10.66082293284064, 11.22291350964708, 11.47259339099418, 12.05660430806980, 12.64906010835509, 13.05061436328610, 13.70083492112103, 14.09858027578225
