| L(s) = 1 | + 7-s + 5·11-s − 13-s − 7·17-s + 6·19-s + 3·23-s + 2·29-s + 2·31-s + 7·37-s − 9·41-s − 8·43-s + 10·47-s − 6·49-s − 5·53-s − 5·61-s − 4·67-s − 9·71-s + 6·73-s + 5·77-s − 3·79-s + 4·83-s − 11·89-s − 91-s + 11·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.50·11-s − 0.277·13-s − 1.69·17-s + 1.37·19-s + 0.625·23-s + 0.371·29-s + 0.359·31-s + 1.15·37-s − 1.40·41-s − 1.21·43-s + 1.45·47-s − 6/7·49-s − 0.686·53-s − 0.640·61-s − 0.488·67-s − 1.06·71-s + 0.702·73-s + 0.569·77-s − 0.337·79-s + 0.439·83-s − 1.16·89-s − 0.104·91-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.818099166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.818099166\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
| 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
−13.10337931555715, −12.72919149588153, −11.97785088390898, −11.65500922291124, −11.43256604117413, −10.88954273894011, −10.23310448600365, −9.801664903515970, −9.207753160722865, −8.929840288988234, −8.493435853432010, −7.790404956790369, −7.350327134029103, −6.707350637464007, −6.514473155463059, −5.894477785790306, −5.175196502517511, −4.647627258396077, −4.373216693027880, −3.612174743670363, −3.117266383562823, −2.470543218930641, −1.705884536277555, −1.278365389575276, −0.4949779527004590,
0.4949779527004590, 1.278365389575276, 1.705884536277555, 2.470543218930641, 3.117266383562823, 3.612174743670363, 4.373216693027880, 4.647627258396077, 5.175196502517511, 5.894477785790306, 6.514473155463059, 6.707350637464007, 7.350327134029103, 7.790404956790369, 8.493435853432010, 8.929840288988234, 9.207753160722865, 9.801664903515970, 10.23310448600365, 10.88954273894011, 11.43256604117413, 11.65500922291124, 11.97785088390898, 12.72919149588153, 13.10337931555715
