L(s) = 1 | − 2-s + 2.41·3-s + 4-s − 5-s − 2.41·6-s − 3.18·7-s − 8-s + 2.82·9-s + 10-s − 0.378·11-s + 2.41·12-s − 0.964·13-s + 3.18·14-s − 2.41·15-s + 16-s − 3.19·17-s − 2.82·18-s + 7.39·19-s − 20-s − 7.68·21-s + 0.378·22-s − 2.41·24-s + 25-s + 0.964·26-s − 0.414·27-s − 3.18·28-s + 0.545·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.39·3-s + 0.5·4-s − 0.447·5-s − 0.985·6-s − 1.20·7-s − 0.353·8-s + 0.942·9-s + 0.316·10-s − 0.114·11-s + 0.696·12-s − 0.267·13-s + 0.850·14-s − 0.623·15-s + 0.250·16-s − 0.776·17-s − 0.666·18-s + 1.69·19-s − 0.223·20-s − 1.67·21-s + 0.0807·22-s − 0.492·24-s + 0.200·25-s + 0.189·26-s − 0.0797·27-s − 0.601·28-s + 0.101·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 13 | \( 1 + 0.964T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 7.39T + 19T^{2} \) |
| 29 | \( 1 - 0.545T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 6.26T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 9.58T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 6.78T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.033816696054131144758704363023, −7.25274993848276705956348083930, −6.78370753270762315038717886752, −5.87981632319047065050514629881, −4.74454081340175108310320659817, −3.74295348318019564844856639038, −3.01162889579235843533623774104, −2.65585447215124780112975082037, −1.38251252386063533402117113728, 0,
1.38251252386063533402117113728, 2.65585447215124780112975082037, 3.01162889579235843533623774104, 3.74295348318019564844856639038, 4.74454081340175108310320659817, 5.87981632319047065050514629881, 6.78370753270762315038717886752, 7.25274993848276705956348083930, 8.033816696054131144758704363023