| L(s) = 1 | + 2.14·2-s − 2.43·3-s + 2.58·4-s − 0.818·5-s − 5.22·6-s + 1.24·8-s + 2.94·9-s − 1.75·10-s + 0.116·11-s − 6.30·12-s + 0.869·13-s + 1.99·15-s − 2.49·16-s + 0.498·17-s + 6.31·18-s − 4.92·19-s − 2.11·20-s + 0.249·22-s − 23-s − 3.04·24-s − 4.33·25-s + 1.86·26-s + 0.128·27-s − 7.25·29-s + 4.27·30-s − 8.02·31-s − 7.83·32-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 1.40·3-s + 1.29·4-s − 0.366·5-s − 2.13·6-s + 0.441·8-s + 0.982·9-s − 0.554·10-s + 0.0350·11-s − 1.81·12-s + 0.241·13-s + 0.515·15-s − 0.623·16-s + 0.121·17-s + 1.48·18-s − 1.13·19-s − 0.472·20-s + 0.0531·22-s − 0.208·23-s − 0.621·24-s − 0.866·25-s + 0.365·26-s + 0.0246·27-s − 1.34·29-s + 0.780·30-s − 1.44·31-s − 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 + 0.818T + 5T^{2} \) |
| 11 | \( 1 - 0.116T + 11T^{2} \) |
| 13 | \( 1 - 0.869T + 13T^{2} \) |
| 17 | \( 1 - 0.498T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.36T + 47T^{2} \) |
| 53 | \( 1 + 7.53T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 - 8.78T + 73T^{2} \) |
| 79 | \( 1 + 0.541T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.641104038831044405818131199736, −8.490150394410519168707695652195, −7.23752296547079954049966966561, −6.55148268725148205705027740947, −5.68559972404170478623235329055, −5.29376329201043853464960404932, −4.21107902375552685857557457988, −3.61681780487672379605228489726, −2.03543401419923863021312601202, 0,
2.03543401419923863021312601202, 3.61681780487672379605228489726, 4.21107902375552685857557457988, 5.29376329201043853464960404932, 5.68559972404170478623235329055, 6.55148268725148205705027740947, 7.23752296547079954049966966561, 8.490150394410519168707695652195, 9.641104038831044405818131199736
