| L(s) = 1 | + 2-s + 0.673·3-s + 4-s + 2.41·5-s + 0.673·6-s − 3.55·7-s + 8-s − 2.54·9-s + 2.41·10-s − 11-s + 0.673·12-s + 2.22·13-s − 3.55·14-s + 1.62·15-s + 16-s − 1.80·17-s − 2.54·18-s − 3.15·19-s + 2.41·20-s − 2.39·21-s − 22-s − 6.20·23-s + 0.673·24-s + 0.815·25-s + 2.22·26-s − 3.73·27-s − 3.55·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.389·3-s + 0.5·4-s + 1.07·5-s + 0.275·6-s − 1.34·7-s + 0.353·8-s − 0.848·9-s + 0.762·10-s − 0.301·11-s + 0.194·12-s + 0.617·13-s − 0.949·14-s + 0.419·15-s + 0.250·16-s − 0.436·17-s − 0.600·18-s − 0.723·19-s + 0.539·20-s − 0.522·21-s − 0.213·22-s − 1.29·23-s + 0.137·24-s + 0.163·25-s + 0.436·26-s − 0.719·27-s − 0.671·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.673T + 3T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 6.20T + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 - 0.0895T + 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 + 0.0468T + 41T^{2} \) |
| 43 | \( 1 + 8.03T + 43T^{2} \) |
| 47 | \( 1 + 6.59T + 47T^{2} \) |
| 53 | \( 1 - 9.99T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 6.85T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 + 5.02T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 0.440T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−8.128496201230187186825288577716, −6.96704738722017235716683016772, −6.36760670069291188832180728291, −5.82840443170320291121030548364, −5.28416832141604771561188866084, −3.96985636170731257599706567506, −3.42144736475226726494275246255, −2.48699091273925700176011799267, −1.89445113941989551590856758233, 0,
1.89445113941989551590856758233, 2.48699091273925700176011799267, 3.42144736475226726494275246255, 3.96985636170731257599706567506, 5.28416832141604771561188866084, 5.82840443170320291121030548364, 6.36760670069291188832180728291, 6.96704738722017235716683016772, 8.128496201230187186825288577716
