| L(s) = 1 | − 2-s − 0.680·3-s + 4-s + 3.16·5-s + 0.680·6-s − 4.20·7-s − 8-s − 2.53·9-s − 3.16·10-s + 0.745·11-s − 0.680·12-s + 4.20·14-s − 2.15·15-s + 16-s − 0.781·17-s + 2.53·18-s − 19-s + 3.16·20-s + 2.86·21-s − 0.745·22-s + 6.32·23-s + 0.680·24-s + 5.03·25-s + 3.76·27-s − 4.20·28-s − 4.50·29-s + 2.15·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.392·3-s + 0.5·4-s + 1.41·5-s + 0.277·6-s − 1.59·7-s − 0.353·8-s − 0.845·9-s − 1.00·10-s + 0.224·11-s − 0.196·12-s + 1.12·14-s − 0.556·15-s + 0.250·16-s − 0.189·17-s + 0.598·18-s − 0.229·19-s + 0.708·20-s + 0.624·21-s − 0.158·22-s + 1.31·23-s + 0.138·24-s + 1.00·25-s + 0.724·27-s − 0.795·28-s − 0.837·29-s + 0.393·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.680T + 3T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 - 0.745T + 11T^{2} \) |
| 17 | \( 1 + 0.781T + 17T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 4.69T + 41T^{2} \) |
| 43 | \( 1 - 0.647T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 + 5.51T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 8.29T + 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 - 0.169T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.57446249511503535709433559020, −6.73252408388414001320247170305, −6.35145989284788167759304559457, −5.74753979680291695081835610657, −5.18699582755061982921294661993, −3.81301823799150696278694553753, −2.87164145073531469980833947646, −2.36375271985643795180632507690, −1.14210063905168267718326881115, 0,
1.14210063905168267718326881115, 2.36375271985643795180632507690, 2.87164145073531469980833947646, 3.81301823799150696278694553753, 5.18699582755061982921294661993, 5.74753979680291695081835610657, 6.35145989284788167759304559457, 6.73252408388414001320247170305, 7.57446249511503535709433559020
