| L(s) = 1 | + 1.41·2-s + 1.12·3-s + 2.00·4-s + 1.58·6-s + 6.17i·7-s + 2.82·8-s − 7.74·9-s + 2.77·11-s + 2.24·12-s − 10.6·13-s + 8.72i·14-s + 4.00·16-s + 24.4i·17-s − 10.9·18-s + (−12.9 + 13.9i)19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.373·3-s + 0.500·4-s + 0.264·6-s + 0.881i·7-s + 0.353·8-s − 0.860·9-s + 0.252·11-s + 0.186·12-s − 0.822·13-s + 0.623i·14-s + 0.250·16-s + 1.44i·17-s − 0.608·18-s + (−0.681 + 0.732i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.238289260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238289260\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (12.9 - 13.9i)T \) |
| good | 3 | \( 1 - 1.12T + 9T^{2} \) |
| 7 | \( 1 - 6.17iT - 49T^{2} \) |
| 11 | \( 1 - 2.77T + 121T^{2} \) |
| 13 | \( 1 + 10.6T + 169T^{2} \) |
| 17 | \( 1 - 24.4iT - 289T^{2} \) |
| 23 | \( 1 - 9.51iT - 529T^{2} \) |
| 29 | \( 1 + 9.64iT - 841T^{2} \) |
| 31 | \( 1 - 10.3iT - 961T^{2} \) |
| 37 | \( 1 - 38.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 0.171iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.26iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 33.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 63.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 35.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 99.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 114. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 24.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 93.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169.T + 9.40e3T^{2} \) |
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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
−10.11425858311522595274278863190, −9.202249590752803797851107687008, −8.343253384971984193621444773776, −7.71271671542609069293003354605, −6.31598938221122996807983970119, −5.88717687891955282881258286339, −4.83261809754106604372979995988, −3.75370807710220126313219639032, −2.76057337220584715960740841056, −1.83700246477878612707207434234,
0.48805004080323580125628467224, 2.30864258617448816293958755481, 3.12574350198857591294558517322, 4.27974694830938500953238260609, 5.01752715312850829156970824301, 6.11546274721554589628997269455, 7.09653858810821855382935736120, 7.65541955601801149142857010847, 8.807333267943128666253044782448, 9.552608747626977412868640838189
