| L(s) = 1 | − 1.77·2-s − 203.·3-s − 508.·4-s + 1.13e3·5-s + 362.·6-s + 4.32e3·7-s + 1.81e3·8-s + 2.19e4·9-s − 2.02e3·10-s + 3.03e4·11-s + 1.03e5·12-s + 3.41e4·13-s − 7.69e3·14-s − 2.32e5·15-s + 2.57e5·16-s − 5.66e5·17-s − 3.89e4·18-s + 2.46e5·19-s − 5.79e5·20-s − 8.82e5·21-s − 5.40e4·22-s − 1.19e6·23-s − 3.70e5·24-s − 6.55e5·25-s − 6.07e4·26-s − 4.53e5·27-s − 2.20e6·28-s + ⋯ |
| L(s) = 1 | − 0.0786·2-s − 1.45·3-s − 0.993·4-s + 0.815·5-s + 0.114·6-s + 0.680·7-s + 0.156·8-s + 1.11·9-s − 0.0641·10-s + 0.625·11-s + 1.44·12-s + 0.331·13-s − 0.0535·14-s − 1.18·15-s + 0.981·16-s − 1.64·17-s − 0.0875·18-s + 0.434·19-s − 0.810·20-s − 0.989·21-s − 0.0492·22-s − 0.887·23-s − 0.227·24-s − 0.335·25-s − 0.0260·26-s − 0.164·27-s − 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 + 1.77T + 512T^{2} \) |
| 3 | \( 1 + 203.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.13e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.03e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.41e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.66e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.14e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.31e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.50e6T + 3.27e14T^{2} \) |
| 47 | \( 1 + 2.85e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.59e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.73e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.95e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.27e6T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.36e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.21e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.39e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.05e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.10e9T + 7.60e17T^{2} \) |
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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
−13.38196593093975822103777822610, −12.09288527087925795505529457778, −10.99400820774242247569184781593, −9.855919645960119559476312117478, −8.556600469492015958527622508886, −6.54913641848076058401277656682, −5.41818078912080184372645028598, −4.36935800238721591032898938427, −1.47027173487448859332596641769, 0,
1.47027173487448859332596641769, 4.36935800238721591032898938427, 5.41818078912080184372645028598, 6.54913641848076058401277656682, 8.556600469492015958527622508886, 9.855919645960119559476312117478, 10.99400820774242247569184781593, 12.09288527087925795505529457778, 13.38196593093975822103777822610
