| L(s) = 1 | + 0.0716·2-s + 2.79·3-s − 1.99·4-s + 2.81·5-s + 0.200·6-s + 0.519·7-s − 0.286·8-s + 4.80·9-s + 0.201·10-s − 3.70·11-s − 5.57·12-s + 5.65·13-s + 0.0372·14-s + 7.86·15-s + 3.96·16-s + 2.63·17-s + 0.344·18-s + 0.800·19-s − 5.62·20-s + 1.45·21-s − 0.265·22-s − 5.44·23-s − 0.799·24-s + 2.93·25-s + 0.405·26-s + 5.03·27-s − 1.03·28-s + ⋯ |
| L(s) = 1 | + 0.0506·2-s + 1.61·3-s − 0.997·4-s + 1.26·5-s + 0.0817·6-s + 0.196·7-s − 0.101·8-s + 1.60·9-s + 0.0638·10-s − 1.11·11-s − 1.60·12-s + 1.56·13-s + 0.00994·14-s + 2.03·15-s + 0.992·16-s + 0.637·17-s + 0.0811·18-s + 0.183·19-s − 1.25·20-s + 0.316·21-s − 0.0566·22-s − 1.13·23-s − 0.163·24-s + 0.587·25-s + 0.0795·26-s + 0.968·27-s − 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.601317473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.601317473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0716T + 2T^{2} \) |
| 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 - 0.519T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 0.800T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 8.34T + 59T^{2} \) |
| 61 | \( 1 + 0.267T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 + 6.57T + 97T^{2} \) |
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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
−8.732046511837228552840295396754, −8.217364618273681034222727099419, −7.85453748061987531837130833151, −6.46769470775212379352566540666, −5.69051071064644794722942230459, −4.88246232163783822498340000220, −3.85814075269515255721611964106, −3.14498435671611929561653944482, −2.21987199972470170392928293856, −1.22999893625631400726520042887,
1.22999893625631400726520042887, 2.21987199972470170392928293856, 3.14498435671611929561653944482, 3.85814075269515255721611964106, 4.88246232163783822498340000220, 5.69051071064644794722942230459, 6.46769470775212379352566540666, 7.85453748061987531837130833151, 8.217364618273681034222727099419, 8.732046511837228552840295396754
