L(s) = 1 | − 2.67·2-s − 1.23·3-s + 5.14·4-s + 0.526·5-s + 3.28·6-s − 0.672·7-s − 8.41·8-s − 1.48·9-s − 1.40·10-s − 0.433·11-s − 6.33·12-s − 2.99·13-s + 1.79·14-s − 0.648·15-s + 12.1·16-s − 1.32·17-s + 3.97·18-s + 5.07·19-s + 2.71·20-s + 0.827·21-s + 1.15·22-s − 8.17·23-s + 10.3·24-s − 4.72·25-s + 8.01·26-s + 5.51·27-s − 3.46·28-s + ⋯ |
L(s) = 1 | − 1.89·2-s − 0.710·3-s + 2.57·4-s + 0.235·5-s + 1.34·6-s − 0.254·7-s − 2.97·8-s − 0.495·9-s − 0.445·10-s − 0.130·11-s − 1.82·12-s − 0.831·13-s + 0.480·14-s − 0.167·15-s + 3.04·16-s − 0.321·17-s + 0.936·18-s + 1.16·19-s + 0.606·20-s + 0.180·21-s + 0.246·22-s − 1.70·23-s + 2.11·24-s − 0.944·25-s + 1.57·26-s + 1.06·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 0.526T + 5T^{2} \) |
| 7 | \( 1 + 0.672T + 7T^{2} \) |
| 11 | \( 1 + 0.433T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 - 0.599T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.60T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 - 0.165T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 - 8.04T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.80478845699361347556954681207, −7.24153153830585997197791328799, −6.45107454668652955739604983500, −5.86170678259436884474868381256, −5.25291513250012828104543382149, −3.86583497716930418132478768924, −2.67119974417095194043267488844, −2.12751503440132671214744483011, −0.884274808256386615505066384241, 0,
0.884274808256386615505066384241, 2.12751503440132671214744483011, 2.67119974417095194043267488844, 3.86583497716930418132478768924, 5.25291513250012828104543382149, 5.86170678259436884474868381256, 6.45107454668652955739604983500, 7.24153153830585997197791328799, 7.80478845699361347556954681207