| L(s) = 1 | + 0.755·2-s + 0.0659·3-s − 1.42·4-s + 1.80·5-s + 0.0497·6-s − 1.15·7-s − 2.58·8-s − 2.99·9-s + 1.36·10-s + 0.520·11-s − 0.0942·12-s + 0.157·13-s − 0.874·14-s + 0.119·15-s + 0.903·16-s − 17-s − 2.26·18-s − 7.60·19-s − 2.58·20-s − 0.0763·21-s + 0.392·22-s + 3.13·23-s − 0.170·24-s − 1.73·25-s + 0.118·26-s − 0.395·27-s + 1.65·28-s + ⋯ |
| L(s) = 1 | + 0.533·2-s + 0.0380·3-s − 0.714·4-s + 0.808·5-s + 0.0203·6-s − 0.437·7-s − 0.915·8-s − 0.998·9-s + 0.431·10-s + 0.156·11-s − 0.0272·12-s + 0.0435·13-s − 0.233·14-s + 0.0307·15-s + 0.225·16-s − 0.242·17-s − 0.533·18-s − 1.74·19-s − 0.577·20-s − 0.0166·21-s + 0.0837·22-s + 0.653·23-s − 0.0348·24-s − 0.346·25-s + 0.0232·26-s − 0.0760·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 0.755T + 2T^{2} \) |
| 3 | \( 1 - 0.0659T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 0.520T + 11T^{2} \) |
| 13 | \( 1 - 0.157T + 13T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 - 3.13T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 9.13T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.794T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 9.60T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 - 3.05T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + 5.35T + 97T^{2} \) |
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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
−9.244661860039526687615574628865, −9.043881804573866754644433208551, −8.064107272506339056292833057853, −6.69929612420019800026769108837, −5.92230665000640989373400412465, −5.32978227192875163451256898611, −4.20410833137328163166098007272, −3.25923811348773000930791179493, −2.09916840458894438729466411107, 0,
2.09916840458894438729466411107, 3.25923811348773000930791179493, 4.20410833137328163166098007272, 5.32978227192875163451256898611, 5.92230665000640989373400412465, 6.69929612420019800026769108837, 8.064107272506339056292833057853, 9.043881804573866754644433208551, 9.244661860039526687615574628865
