| L(s) = 1 | − 1.82·5-s + 7-s − 4.82·11-s + 2.82·13-s − 0.171·17-s + 6.82·19-s + 4·23-s − 1.65·25-s − 2.82·29-s − 6.82·31-s − 1.82·35-s + 2.65·37-s − 3.82·41-s + 7.82·43-s − 8.65·47-s + 49-s + 2·53-s + 8.82·55-s − 0.656·59-s + 3.17·61-s − 5.17·65-s + 4·67-s − 1.65·71-s − 5.65·73-s − 4.82·77-s − 1.82·79-s + 5.34·83-s + ⋯ |
| L(s) = 1 | − 0.817·5-s + 0.377·7-s − 1.45·11-s + 0.784·13-s − 0.0416·17-s + 1.56·19-s + 0.834·23-s − 0.331·25-s − 0.525·29-s − 1.22·31-s − 0.309·35-s + 0.436·37-s − 0.597·41-s + 1.19·43-s − 1.26·47-s + 0.142·49-s + 0.274·53-s + 1.19·55-s − 0.0855·59-s + 0.406·61-s − 0.641·65-s + 0.488·67-s − 0.196·71-s − 0.662·73-s − 0.550·77-s − 0.205·79-s + 0.586·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444070676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444070676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 0.656T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
| 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
−7.83882530100806169770275033180, −7.62211547863444517469840690433, −6.85331781363989850795164637422, −5.68888507839198620424980888293, −5.30523797492048640477835757204, −4.45032106787935556341166527360, −3.55070289652267514163752378625, −2.96105683130417391798780148925, −1.81785854585588165341679511453, −0.62945926180880706436658611019,
0.62945926180880706436658611019, 1.81785854585588165341679511453, 2.96105683130417391798780148925, 3.55070289652267514163752378625, 4.45032106787935556341166527360, 5.30523797492048640477835757204, 5.68888507839198620424980888293, 6.85331781363989850795164637422, 7.62211547863444517469840690433, 7.83882530100806169770275033180
