| L(s) = 1 | − 2.60·2-s + 3-s + 4.78·4-s − 1.51·5-s − 2.60·6-s − 7-s − 7.26·8-s + 9-s + 3.94·10-s − 0.949·11-s + 4.78·12-s + 2.60·14-s − 1.51·15-s + 9.34·16-s + 5.20·17-s − 2.60·18-s + 5.53·19-s − 7.24·20-s − 21-s + 2.47·22-s − 3.94·23-s − 7.26·24-s − 2.70·25-s + 27-s − 4.78·28-s + 5.11·29-s + 3.94·30-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.577·3-s + 2.39·4-s − 0.676·5-s − 1.06·6-s − 0.377·7-s − 2.56·8-s + 0.333·9-s + 1.24·10-s − 0.286·11-s + 1.38·12-s + 0.696·14-s − 0.390·15-s + 2.33·16-s + 1.26·17-s − 0.614·18-s + 1.26·19-s − 1.62·20-s − 0.218·21-s + 0.527·22-s − 0.823·23-s − 1.48·24-s − 0.541·25-s + 0.192·27-s − 0.904·28-s + 0.949·29-s + 0.719·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7662105136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7662105136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 11 | \( 1 + 0.949T + 11T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 - 5.53T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 9.26T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.97T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 7.77T + 71T^{2} \) |
| 73 | \( 1 + 8.64T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.344529816922639385513499425722, −7.983733052759123011892113584098, −7.59098929784527413776515720191, −6.67958407653937350404730224688, −5.98682368587823767364976051815, −4.71381068168588428167364792663, −3.36239846502810836018919063723, −2.90550798809016232122660555028, −1.67948834385617012803870102819, −0.67235505030017457566014391051,
0.67235505030017457566014391051, 1.67948834385617012803870102819, 2.90550798809016232122660555028, 3.36239846502810836018919063723, 4.71381068168588428167364792663, 5.98682368587823767364976051815, 6.67958407653937350404730224688, 7.59098929784527413776515720191, 7.983733052759123011892113584098, 8.344529816922639385513499425722
