| L(s) = 1 | + 0.0693·2-s − 4.10·3-s − 3.99·4-s − 0.284·6-s − 7·7-s − 0.554·8-s + 7.81·9-s + 16.3·12-s − 25.5·13-s − 0.485·14-s + 15.9·16-s + 31.0·17-s + 0.542·18-s − 17.3·19-s + 28.7·21-s + 27.0·23-s + 2.27·24-s + 25·25-s − 1.76·26-s + 4.86·27-s + 27.9·28-s + 3.32·32-s + 2.15·34-s − 31.2·36-s − 26.6·37-s − 1.20·38-s + 104.·39-s + ⋯ |
| L(s) = 1 | + 0.0346·2-s − 1.36·3-s − 0.998·4-s − 0.0474·6-s − 7-s − 0.0693·8-s + 0.868·9-s + 1.36·12-s − 1.96·13-s − 0.0346·14-s + 0.996·16-s + 1.82·17-s + 0.0301·18-s − 0.911·19-s + 1.36·21-s + 1.17·23-s + 0.0947·24-s + 25-s − 0.0680·26-s + 0.180·27-s + 0.998·28-s + 0.103·32-s + 0.0634·34-s − 0.867·36-s − 0.719·37-s − 0.0316·38-s + 2.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4559029599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4559029599\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 - 0.0693T + 4T^{2} \) |
| 3 | \( 1 + 4.10T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 25.5T + 169T^{2} \) |
| 17 | \( 1 - 31.0T + 289T^{2} \) |
| 19 | \( 1 + 17.3T + 361T^{2} \) |
| 23 | \( 1 - 27.0T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 26.6T + 1.36e3T^{2} \) |
| 43 | \( 1 + 72.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 23.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 177.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 95.9T + 9.40e3T^{2} \) |
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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
−11.83942720411717743870067613728, −10.41210876599540241461403358896, −9.968181078040890186603489244601, −8.969156834686473878443193785857, −7.52537404923291605190572421083, −6.49932360745098238835953774378, −5.35667300140548979292496920336, −4.78019594529829108896004563810, −3.19822775608545646370322877835, −0.57778164861474880804161102853,
0.57778164861474880804161102853, 3.19822775608545646370322877835, 4.78019594529829108896004563810, 5.35667300140548979292496920336, 6.49932360745098238835953774378, 7.52537404923291605190572421083, 8.969156834686473878443193785857, 9.968181078040890186603489244601, 10.41210876599540241461403358896, 11.83942720411717743870067613728
