| L(s) = 1 | + 7-s + 11-s − 4·13-s − 2·17-s + 6·23-s − 2·31-s − 8·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s + 2·61-s + 8·67-s − 8·71-s − 4·73-s + 77-s − 10·79-s + 6·83-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 1.25·23-s − 0.359·31-s − 1.31·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.256·61-s + 0.977·67-s − 0.949·71-s − 0.468·73-s + 0.113·77-s − 1.12·79-s + 0.658·83-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331735190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331735190\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.73266202353193, −12.23477768766052, −11.92823267908831, −11.33704040238121, −10.91523695922607, −10.60697352373132, −9.829096060980966, −9.639108707439959, −9.038582899233495, −8.574141416730948, −8.205055073047689, −7.472730240963697, −7.150943350550578, −6.767182742310522, −6.173941501645909, −5.523056450660502, −5.038035794814298, −4.709449604550752, −4.139673652246400, −3.460162467097602, −2.965709899353925, −2.363088387804856, −1.757882111045172, −1.212035751205121, −0.3113057908313381,
0.3113057908313381, 1.212035751205121, 1.757882111045172, 2.363088387804856, 2.965709899353925, 3.460162467097602, 4.139673652246400, 4.709449604550752, 5.038035794814298, 5.523056450660502, 6.173941501645909, 6.767182742310522, 7.150943350550578, 7.472730240963697, 8.205055073047689, 8.574141416730948, 9.038582899233495, 9.639108707439959, 9.829096060980966, 10.60697352373132, 10.91523695922607, 11.33704040238121, 11.92823267908831, 12.23477768766052, 12.73266202353193
