| L(s) = 1 | + 2-s + 0.285·3-s + 4-s − 4.07·5-s + 0.285·6-s + 3.91·7-s + 8-s − 2.91·9-s − 4.07·10-s − 4.78·11-s + 0.285·12-s − 1.31·13-s + 3.91·14-s − 1.16·15-s + 16-s + 17-s − 2.91·18-s + 6.42·19-s − 4.07·20-s + 1.12·21-s − 4.78·22-s + 9.01·23-s + 0.285·24-s + 11.6·25-s − 1.31·26-s − 1.69·27-s + 3.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.165·3-s + 0.5·4-s − 1.82·5-s + 0.116·6-s + 1.48·7-s + 0.353·8-s − 0.972·9-s − 1.28·10-s − 1.44·11-s + 0.0825·12-s − 0.365·13-s + 1.04·14-s − 0.301·15-s + 0.250·16-s + 0.242·17-s − 0.687·18-s + 1.47·19-s − 0.911·20-s + 0.244·21-s − 1.01·22-s + 1.87·23-s + 0.0583·24-s + 2.32·25-s − 0.258·26-s − 0.325·27-s + 0.740·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174271007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174271007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.285T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 19 | \( 1 - 6.42T + 19T^{2} \) |
| 23 | \( 1 - 9.01T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.60T + 41T^{2} \) |
| 43 | \( 1 + 7.85T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.92T + 53T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 - 0.330T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−8.734968965945095455134485815160, −8.162843038134970564538254136310, −7.63112733870439294440255607245, −7.14380887346702363732036540568, −5.60370908371840379092835874841, −4.92470956056206129830112413290, −4.47363698446606457108303037149, −3.16242294725974556974407677756, −2.74115316242248129219958845270, −0.888163991284624211738354956984,
0.888163991284624211738354956984, 2.74115316242248129219958845270, 3.16242294725974556974407677756, 4.47363698446606457108303037149, 4.92470956056206129830112413290, 5.60370908371840379092835874841, 7.14380887346702363732036540568, 7.63112733870439294440255607245, 8.162843038134970564538254136310, 8.734968965945095455134485815160
