| L(s) = 1 | − 1.43·2-s + 3-s + 0.0711·4-s − 2.31·5-s − 1.43·6-s − 4.44·7-s + 2.77·8-s + 9-s + 3.33·10-s − 4.52·11-s + 0.0711·12-s + 1.17·13-s + 6.39·14-s − 2.31·15-s − 4.13·16-s − 1.43·18-s + 4.86·19-s − 0.164·20-s − 4.44·21-s + 6.51·22-s − 0.625·23-s + 2.77·24-s + 0.354·25-s − 1.69·26-s + 27-s − 0.316·28-s − 1.51·29-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.577·3-s + 0.0355·4-s − 1.03·5-s − 0.587·6-s − 1.67·7-s + 0.981·8-s + 0.333·9-s + 1.05·10-s − 1.36·11-s + 0.0205·12-s + 0.325·13-s + 1.70·14-s − 0.597·15-s − 1.03·16-s − 0.339·18-s + 1.11·19-s − 0.0368·20-s − 0.969·21-s + 1.38·22-s − 0.130·23-s + 0.566·24-s + 0.0708·25-s − 0.331·26-s + 0.192·27-s − 0.0597·28-s − 0.280·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5013585459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5013585459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 + 0.625T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 8.79T + 37T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 + 1.51T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−9.968091043311665206408667424576, −9.353783158220116051307579510123, −8.476984509794353684409752900246, −7.72993320572643067079592516777, −7.24823260186862730728224320015, −6.00306742164692419145957588225, −4.60517030844873146783504372343, −3.56344929371181999464990851841, −2.68203688119091048961921254681, −0.62300062748284261966763834209,
0.62300062748284261966763834209, 2.68203688119091048961921254681, 3.56344929371181999464990851841, 4.60517030844873146783504372343, 6.00306742164692419145957588225, 7.24823260186862730728224320015, 7.72993320572643067079592516777, 8.476984509794353684409752900246, 9.353783158220116051307579510123, 9.968091043311665206408667424576
