| L(s) = 1 | − 0.868·2-s + 2.12·3-s − 1.24·4-s + 5-s − 1.84·6-s + 3.66·7-s + 2.81·8-s + 1.51·9-s − 0.868·10-s − 2.35·11-s − 2.64·12-s + 5.42·13-s − 3.17·14-s + 2.12·15-s + 0.0446·16-s + 4.31·17-s − 1.31·18-s + 2.73·19-s − 1.24·20-s + 7.77·21-s + 2.04·22-s − 3.59·23-s + 5.99·24-s + 25-s − 4.70·26-s − 3.15·27-s − 4.56·28-s + ⋯ |
| L(s) = 1 | − 0.613·2-s + 1.22·3-s − 0.623·4-s + 0.447·5-s − 0.753·6-s + 1.38·7-s + 0.996·8-s + 0.505·9-s − 0.274·10-s − 0.710·11-s − 0.764·12-s + 1.50·13-s − 0.849·14-s + 0.548·15-s + 0.0111·16-s + 1.04·17-s − 0.310·18-s + 0.627·19-s − 0.278·20-s + 1.69·21-s + 0.436·22-s − 0.750·23-s + 1.22·24-s + 0.200·25-s − 0.923·26-s − 0.606·27-s − 0.861·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.270334644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270334644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.868T + 2T^{2} \) |
| 3 | \( 1 - 2.12T + 3T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 - 4.30T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 - 0.533T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 - 8.20T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 + 8.77T + 97T^{2} \) |
| 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
−8.904319303367758071693748627894, −8.423648252659968739147784967884, −7.962185465294038304444788549621, −7.33870707040490806454098758344, −5.79003742130414056932821408709, −5.14897372313777264018160314784, −4.09018454279076517181782539690, −3.26014473812421174164700054285, −1.99028674555771286599643941326, −1.18292367762315562093619270993,
1.18292367762315562093619270993, 1.99028674555771286599643941326, 3.26014473812421174164700054285, 4.09018454279076517181782539690, 5.14897372313777264018160314784, 5.79003742130414056932821408709, 7.33870707040490806454098758344, 7.962185465294038304444788549621, 8.423648252659968739147784967884, 8.904319303367758071693748627894
