L(s) = 1 | + 2.28·2-s + 0.625·3-s + 3.22·4-s − 3.41·5-s + 1.43·6-s − 3.04·7-s + 2.79·8-s − 2.60·9-s − 7.80·10-s + 11-s + 2.01·12-s − 0.0108·13-s − 6.95·14-s − 2.13·15-s − 0.0590·16-s + 4.75·17-s − 5.96·18-s − 6.00·19-s − 11.0·20-s − 1.90·21-s + 2.28·22-s − 6.25·23-s + 1.74·24-s + 6.65·25-s − 0.0247·26-s − 3.51·27-s − 9.80·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 0.361·3-s + 1.61·4-s − 1.52·5-s + 0.584·6-s − 1.15·7-s + 0.988·8-s − 0.869·9-s − 2.46·10-s + 0.301·11-s + 0.582·12-s − 0.00300·13-s − 1.85·14-s − 0.551·15-s − 0.0147·16-s + 1.15·17-s − 1.40·18-s − 1.37·19-s − 2.46·20-s − 0.415·21-s + 0.487·22-s − 1.30·23-s + 0.357·24-s + 1.33·25-s − 0.00485·26-s − 0.675·27-s − 1.85·28-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 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 0.625T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 13 | \( 1 + 0.0108T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 0.876T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 0.594T + 67T^{2} \) |
| 71 | \( 1 + 9.32T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 8.92T + 79T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 5.22T + 97T^{2} \) |
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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
−8.935642441801921464095706583791, −8.191645034833578210092698863000, −7.32696327673519036896702583438, −6.42506999119704936483660933175, −5.81338287747293283851290894881, −4.67240995641887845987794665778, −3.69621751567879016709372118293, −3.48818335254007163228164724642, −2.45231956548808721105571933351, 0,
2.45231956548808721105571933351, 3.48818335254007163228164724642, 3.69621751567879016709372118293, 4.67240995641887845987794665778, 5.81338287747293283851290894881, 6.42506999119704936483660933175, 7.32696327673519036896702583438, 8.191645034833578210092698863000, 8.935642441801921464095706583791