| L(s) = 1 | + 0.947·2-s − 2.80·3-s − 1.10·4-s − 2.22·5-s − 2.65·6-s + 3.55·7-s − 2.93·8-s + 4.87·9-s − 2.10·10-s − 11-s + 3.09·12-s + 4.58·13-s + 3.36·14-s + 6.24·15-s − 0.578·16-s − 4.15·17-s + 4.62·18-s + 5.33·19-s + 2.45·20-s − 9.96·21-s − 0.947·22-s − 1.26·23-s + 8.24·24-s − 0.0421·25-s + 4.34·26-s − 5.27·27-s − 3.91·28-s + ⋯ |
| L(s) = 1 | + 0.669·2-s − 1.62·3-s − 0.551·4-s − 0.995·5-s − 1.08·6-s + 1.34·7-s − 1.03·8-s + 1.62·9-s − 0.666·10-s − 0.301·11-s + 0.893·12-s + 1.27·13-s + 0.898·14-s + 1.61·15-s − 0.144·16-s − 1.00·17-s + 1.08·18-s + 1.22·19-s + 0.549·20-s − 2.17·21-s − 0.201·22-s − 0.264·23-s + 1.68·24-s − 0.00843·25-s + 0.851·26-s − 1.01·27-s − 0.739·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 0.947T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 + 0.756T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 0.369T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 0.580T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 - 3.39T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 0.615T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 0.678T + 97T^{2} \) |
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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
−9.008268000953897534413006029073, −8.241445827131739528016703719436, −7.43973102423515646217566544349, −6.35209406558486964294842951647, −5.61530629035900103500353907627, −4.83010791058802141767582115408, −4.40289502239981682916387074656, −3.41628144574589719117828471350, −1.34459200279062454922715944981, 0,
1.34459200279062454922715944981, 3.41628144574589719117828471350, 4.40289502239981682916387074656, 4.83010791058802141767582115408, 5.61530629035900103500353907627, 6.35209406558486964294842951647, 7.43973102423515646217566544349, 8.241445827131739528016703719436, 9.008268000953897534413006029073
