| L(s) = 1 | + 2.28·2-s − 3.15·3-s + 3.22·4-s − 2.12·5-s − 7.19·6-s + 2.78·8-s + 6.92·9-s − 4.85·10-s + 0.308·11-s − 10.1·12-s + 6.69·15-s − 0.0699·16-s + 1.77·17-s + 15.8·18-s − 1.78·19-s − 6.84·20-s + 0.704·22-s + 1.15·23-s − 8.78·24-s − 0.484·25-s − 12.3·27-s + 2.01·29-s + 15.2·30-s + 4.60·31-s − 5.73·32-s − 0.971·33-s + 4.05·34-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 1.81·3-s + 1.61·4-s − 0.950·5-s − 2.93·6-s + 0.985·8-s + 2.30·9-s − 1.53·10-s + 0.0930·11-s − 2.92·12-s + 1.72·15-s − 0.0174·16-s + 0.430·17-s + 3.72·18-s − 0.408·19-s − 1.53·20-s + 0.150·22-s + 0.239·23-s − 1.79·24-s − 0.0968·25-s − 2.37·27-s + 0.373·29-s + 2.79·30-s + 0.827·31-s − 1.01·32-s − 0.169·33-s + 0.695·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 11 | \( 1 - 0.308T + 11T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
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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
−7.04668092327095934741830588220, −6.49720139082751716112669048055, −5.94220833815101006470826235502, −5.23451492816058157621861845751, −4.78562933453695970321159229358, −4.07015913207459448406906290814, −3.61251896242794407979096330499, −2.44231730706419414208011681800, −1.15334739931615011682030791655, 0,
1.15334739931615011682030791655, 2.44231730706419414208011681800, 3.61251896242794407979096330499, 4.07015913207459448406906290814, 4.78562933453695970321159229358, 5.23451492816058157621861845751, 5.94220833815101006470826235502, 6.49720139082751716112669048055, 7.04668092327095934741830588220
