| L(s) = 1 | − 2.60·2-s + 4.80·4-s − 0.565·5-s − 7.32·8-s + 1.47·10-s − 2.31·11-s − 13-s + 9.50·16-s + 3.03·17-s + 8.50·19-s − 2.71·20-s + 6.05·22-s − 2.33·23-s − 4.68·25-s + 2.60·26-s − 4.06·29-s − 0.245·31-s − 10.1·32-s − 7.92·34-s + 0.180·37-s − 22.1·38-s + 4.14·40-s − 3.29·41-s − 7.37·43-s − 11.1·44-s + 6.08·46-s + 10.5·47-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.40·4-s − 0.252·5-s − 2.59·8-s + 0.466·10-s − 0.699·11-s − 0.277·13-s + 2.37·16-s + 0.736·17-s + 1.95·19-s − 0.607·20-s + 1.29·22-s − 0.486·23-s − 0.936·25-s + 0.511·26-s − 0.754·29-s − 0.0440·31-s − 1.79·32-s − 1.35·34-s + 0.0297·37-s − 3.59·38-s + 0.654·40-s − 0.514·41-s − 1.12·43-s − 1.68·44-s + 0.897·46-s + 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 + 0.565T + 5T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 8.50T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 + 0.245T + 31T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 7.37T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 6.88T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.81285345946841316002547915179, −7.45908873175419579816753486034, −6.73757495097757227757008095819, −5.74597459019383065448784773346, −5.18791252369345623377331052810, −3.73092274877339736525218921418, −2.93726604274589101611008781986, −2.02846908437930995169408157163, −1.07709918414562583161960821051, 0,
1.07709918414562583161960821051, 2.02846908437930995169408157163, 2.93726604274589101611008781986, 3.73092274877339736525218921418, 5.18791252369345623377331052810, 5.74597459019383065448784773346, 6.73757495097757227757008095819, 7.45908873175419579816753486034, 7.81285345946841316002547915179
