| L(s) = 1 | + 1.61·2-s + 0.292·3-s + 0.594·4-s − 3.98·5-s + 0.470·6-s + 4.67·7-s − 2.26·8-s − 2.91·9-s − 6.41·10-s − 2.68·11-s + 0.173·12-s + 1.93·13-s + 7.53·14-s − 1.16·15-s − 4.83·16-s − 0.305·17-s − 4.69·18-s + 19-s − 2.36·20-s + 1.36·21-s − 4.31·22-s − 0.285·23-s − 0.661·24-s + 10.8·25-s + 3.11·26-s − 1.72·27-s + 2.78·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.168·3-s + 0.297·4-s − 1.78·5-s + 0.192·6-s + 1.76·7-s − 0.800·8-s − 0.971·9-s − 2.02·10-s − 0.808·11-s + 0.0501·12-s + 0.535·13-s + 2.01·14-s − 0.300·15-s − 1.20·16-s − 0.0740·17-s − 1.10·18-s + 0.229·19-s − 0.529·20-s + 0.298·21-s − 0.920·22-s − 0.0595·23-s − 0.135·24-s + 2.17·25-s + 0.610·26-s − 0.332·27-s + 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.137402705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137402705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.292T + 3T^{2} \) |
| 5 | \( 1 + 3.98T + 5T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 + 0.305T + 17T^{2} \) |
| 23 | \( 1 + 0.285T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 6.05T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 6.89T + 97T^{2} \) |
| 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
−7.925894296420340802473734533389, −7.63801775034729575468247845462, −6.59139445573077759510783723436, −5.55746776296991655553000691924, −5.01234323835409424897372545438, −4.51122751846731872038290746667, −3.72657072075343011989738393474, −3.16143385227646648039114736639, −2.17650018848400154013965414974, −0.62464181469316159325384019408,
0.62464181469316159325384019408, 2.17650018848400154013965414974, 3.16143385227646648039114736639, 3.72657072075343011989738393474, 4.51122751846731872038290746667, 5.01234323835409424897372545438, 5.55746776296991655553000691924, 6.59139445573077759510783723436, 7.63801775034729575468247845462, 7.925894296420340802473734533389
