| L(s) = 1 | − 3·7-s + 11-s − 13-s − 5·17-s + 4·19-s + 2·23-s + 3·29-s − 5·31-s + 6·37-s + 8·41-s − 6·43-s + 47-s + 2·49-s + 11·53-s + 3·59-s − 5·61-s + 9·67-s + 8·71-s + 2·73-s − 3·77-s + 4·79-s + 3·83-s − 6·89-s + 3·91-s + 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.301·11-s − 0.277·13-s − 1.21·17-s + 0.917·19-s + 0.417·23-s + 0.557·29-s − 0.898·31-s + 0.986·37-s + 1.24·41-s − 0.914·43-s + 0.145·47-s + 2/7·49-s + 1.51·53-s + 0.390·59-s − 0.640·61-s + 1.09·67-s + 0.949·71-s + 0.234·73-s − 0.341·77-s + 0.450·79-s + 0.329·83-s − 0.635·89-s + 0.314·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-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
| \(L(1)\) |
\(\approx\) |
\(1.511658647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511658647\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.63347069850657, −13.98442381956777, −13.57252695333705, −12.95413925349683, −12.71195934905742, −12.06395775508238, −11.43114538767837, −11.08898131930743, −10.34931316803237, −9.875751521321037, −9.263033249916397, −9.071978773186502, −8.306573155284095, −7.610462817708167, −7.071538233258405, −6.551832727947177, −6.142780631428461, −5.354201777376250, −4.852563189702134, −3.983852235655148, −3.629547843825064, −2.727901617809251, −2.377385787186084, −1.280346982544511, −0.4590057863552441,
0.4590057863552441, 1.280346982544511, 2.377385787186084, 2.727901617809251, 3.629547843825064, 3.983852235655148, 4.852563189702134, 5.354201777376250, 6.142780631428461, 6.551832727947177, 7.071538233258405, 7.610462817708167, 8.306573155284095, 9.071978773186502, 9.263033249916397, 9.875751521321037, 10.34931316803237, 11.08898131930743, 11.43114538767837, 12.06395775508238, 12.71195934905742, 12.95413925349683, 13.57252695333705, 13.98442381956777, 14.63347069850657
