| L(s) = 1 | + 2-s + 2.96·3-s + 4-s − 0.667·5-s + 2.96·6-s + 2.16·7-s + 8-s + 5.76·9-s − 0.667·10-s − 5.26·11-s + 2.96·12-s + 5.61·13-s + 2.16·14-s − 1.97·15-s + 16-s + 5.27·17-s + 5.76·18-s + 5.20·19-s − 0.667·20-s + 6.41·21-s − 5.26·22-s − 23-s + 2.96·24-s − 4.55·25-s + 5.61·26-s + 8.17·27-s + 2.16·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.70·3-s + 0.5·4-s − 0.298·5-s + 1.20·6-s + 0.819·7-s + 0.353·8-s + 1.92·9-s − 0.211·10-s − 1.58·11-s + 0.854·12-s + 1.55·13-s + 0.579·14-s − 0.510·15-s + 0.250·16-s + 1.28·17-s + 1.35·18-s + 1.19·19-s − 0.149·20-s + 1.40·21-s − 1.12·22-s − 0.208·23-s + 0.604·24-s − 0.910·25-s + 1.10·26-s + 1.57·27-s + 0.409·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.756525460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.756525460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 0.667T + 5T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 + 5.26T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 29 | \( 1 + 0.706T + 29T^{2} \) |
| 31 | \( 1 + 9.38T + 31T^{2} \) |
| 37 | \( 1 + 0.748T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 9.50T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 - 6.52T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 - 0.515T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.995426092451152968487865855092, −7.69654701092946347312350397010, −6.91939811591620299666264173605, −5.51712428902259019196671446036, −5.36031474956798453710603707165, −4.06596698813297359215007499185, −3.61666934170390679885316759236, −2.96767464217030495040578583103, −2.09957900290618540667193726134, −1.27428966082037564335250600080,
1.27428966082037564335250600080, 2.09957900290618540667193726134, 2.96767464217030495040578583103, 3.61666934170390679885316759236, 4.06596698813297359215007499185, 5.36031474956798453710603707165, 5.51712428902259019196671446036, 6.91939811591620299666264173605, 7.69654701092946347312350397010, 7.995426092451152968487865855092
