| L(s) = 1 | − 2-s − 2.03·3-s + 4-s − 3.10·5-s + 2.03·6-s − 0.000158·7-s − 8-s + 1.14·9-s + 3.10·10-s − 1.15·11-s − 2.03·12-s − 1.92·13-s + 0.000158·14-s + 6.31·15-s + 16-s − 2.77·17-s − 1.14·18-s + 3.95·19-s − 3.10·20-s + 0.000322·21-s + 1.15·22-s − 1.25·23-s + 2.03·24-s + 4.62·25-s + 1.92·26-s + 3.78·27-s − 0.000158·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.17·3-s + 0.5·4-s − 1.38·5-s + 0.830·6-s − 5.99e − 5·7-s − 0.353·8-s + 0.380·9-s + 0.980·10-s − 0.347·11-s − 0.587·12-s − 0.533·13-s + 4.23e−5·14-s + 1.62·15-s + 0.250·16-s − 0.673·17-s − 0.269·18-s + 0.906·19-s − 0.693·20-s + 7.04e−5·21-s + 0.245·22-s − 0.261·23-s + 0.415·24-s + 0.924·25-s + 0.377·26-s + 0.727·27-s − 2.99e − 5·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06550138263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06550138263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.03T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 + 0.000158T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 7.67T + 29T^{2} \) |
| 31 | \( 1 + 5.07T + 31T^{2} \) |
| 37 | \( 1 - 1.88T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.252438743440423945290526260851, −7.73415226704721552566714101110, −7.06136677015392437057462632140, −6.42513148702535435185061066798, −5.41379265531550957877874333645, −4.87750100684941659080791663872, −3.84579020050555861090044411879, −2.99533960868298803395805968259, −1.62080227433207496634208151126, −0.17107383307098032105018341814,
0.17107383307098032105018341814, 1.62080227433207496634208151126, 2.99533960868298803395805968259, 3.84579020050555861090044411879, 4.87750100684941659080791663872, 5.41379265531550957877874333645, 6.42513148702535435185061066798, 7.06136677015392437057462632140, 7.73415226704721552566714101110, 8.252438743440423945290526260851
