| L(s) = 1 | − 2.34·2-s + 0.967·3-s + 3.50·4-s + 3.03·5-s − 2.26·6-s − 3.52·8-s − 2.06·9-s − 7.11·10-s + 1.10·11-s + 3.38·12-s + 6.46·13-s + 2.93·15-s + 1.27·16-s − 5.05·17-s + 4.84·18-s − 4.13·19-s + 10.6·20-s − 2.58·22-s + 4.02·23-s − 3.41·24-s + 4.19·25-s − 15.1·26-s − 4.89·27-s + 1.32·29-s − 6.88·30-s + 7.45·31-s + 4.07·32-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 0.558·3-s + 1.75·4-s + 1.35·5-s − 0.926·6-s − 1.24·8-s − 0.688·9-s − 2.24·10-s + 0.332·11-s + 0.978·12-s + 1.79·13-s + 0.757·15-s + 0.317·16-s − 1.22·17-s + 1.14·18-s − 0.948·19-s + 2.37·20-s − 0.552·22-s + 0.838·23-s − 0.696·24-s + 0.838·25-s − 2.97·26-s − 0.942·27-s + 0.246·29-s − 1.25·30-s + 1.33·31-s + 0.720·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267082560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267082560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 - 0.967T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 4.13T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 - 7.45T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.701T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−9.042566588719197623033361514900, −8.616818520805021940261231826760, −8.094434771773638893249139242686, −6.69623490248763781350643807670, −6.44840131529684816006995223874, −5.50187314562860245898868429802, −4.02809168859468465912638371740, −2.66916200616837178417111175150, −2.02825535827764658523529704803, −0.969255191205720454899997895051,
0.969255191205720454899997895051, 2.02825535827764658523529704803, 2.66916200616837178417111175150, 4.02809168859468465912638371740, 5.50187314562860245898868429802, 6.44840131529684816006995223874, 6.69623490248763781350643807670, 8.094434771773638893249139242686, 8.616818520805021940261231826760, 9.042566588719197623033361514900
