| L(s) = 1 | + 2-s + 1.51·3-s + 4-s − 0.563·5-s + 1.51·6-s + 2.76·7-s + 8-s − 0.689·9-s − 0.563·10-s − 5.11·11-s + 1.51·12-s + 3.17·13-s + 2.76·14-s − 0.855·15-s + 16-s − 4.89·17-s − 0.689·18-s + 19-s − 0.563·20-s + 4.20·21-s − 5.11·22-s + 3.22·23-s + 1.51·24-s − 4.68·25-s + 3.17·26-s − 5.60·27-s + 2.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.877·3-s + 0.5·4-s − 0.251·5-s + 0.620·6-s + 1.04·7-s + 0.353·8-s − 0.229·9-s − 0.178·10-s − 1.54·11-s + 0.438·12-s + 0.880·13-s + 0.740·14-s − 0.220·15-s + 0.250·16-s − 1.18·17-s − 0.162·18-s + 0.229·19-s − 0.125·20-s + 0.918·21-s − 1.09·22-s + 0.672·23-s + 0.310·24-s − 0.936·25-s + 0.622·26-s − 1.07·27-s + 0.523·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 + 0.563T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.36630056813812330853148770116, −7.12592997589832848942943868249, −5.72362035461037794082346615426, −5.44038309700538399494814111315, −4.67105283828919698657490289295, −3.68859421928351085030768298664, −3.32959630972763468531158895095, −2.13014705338856190643900394615, −1.88330243898225286044459748733, 0,
1.88330243898225286044459748733, 2.13014705338856190643900394615, 3.32959630972763468531158895095, 3.68859421928351085030768298664, 4.67105283828919698657490289295, 5.44038309700538399494814111315, 5.72362035461037794082346615426, 7.12592997589832848942943868249, 7.36630056813812330853148770116
