| L(s) = 1 | − 2.84·2-s − 4.64·3-s + 0.120·4-s − 12.8·5-s + 13.2·6-s + 22.4·8-s − 5.41·9-s + 36.6·10-s − 62.8·11-s − 0.561·12-s − 22.3·13-s + 59.8·15-s − 64.9·16-s + 57.9·17-s + 15.4·18-s − 71.3·19-s − 1.55·20-s + 179.·22-s − 49.5·23-s − 104.·24-s + 40.7·25-s + 63.8·26-s + 150.·27-s − 29·29-s − 170.·30-s − 62.9·31-s + 5.47·32-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.894·3-s + 0.0151·4-s − 1.15·5-s + 0.900·6-s + 0.992·8-s − 0.200·9-s + 1.16·10-s − 1.72·11-s − 0.0135·12-s − 0.477·13-s + 1.02·15-s − 1.01·16-s + 0.827·17-s + 0.202·18-s − 0.861·19-s − 0.0174·20-s + 1.73·22-s − 0.449·23-s − 0.887·24-s + 0.325·25-s + 0.481·26-s + 1.07·27-s − 0.185·29-s − 1.03·30-s − 0.364·31-s + 0.0302·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 2.84T + 8T^{2} \) |
| 3 | \( 1 + 4.64T + 27T^{2} \) |
| 5 | \( 1 + 12.8T + 125T^{2} \) |
| 11 | \( 1 + 62.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 62.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 919.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 868.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 83.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 187.T + 9.12e5T^{2} \) |
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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
−8.557197222944692627726485209323, −7.937481390974268943915226151604, −7.58984418257473294663331882960, −6.44673566670668903571643624723, −5.29469098033087521474705774759, −4.76438968089673336202124356172, −3.61670294164068066228932334707, −2.29078491845076573109712208150, −0.63829289487689418652528604047, 0,
0.63829289487689418652528604047, 2.29078491845076573109712208150, 3.61670294164068066228932334707, 4.76438968089673336202124356172, 5.29469098033087521474705774759, 6.44673566670668903571643624723, 7.58984418257473294663331882960, 7.937481390974268943915226151604, 8.557197222944692627726485209323
