L(s) = 1 | − 2-s + 2.19·3-s + 4-s + 2.87·5-s − 2.19·6-s + 2.16·7-s − 8-s + 1.82·9-s − 2.87·10-s + 5.37·11-s + 2.19·12-s − 3.75·13-s − 2.16·14-s + 6.32·15-s + 16-s + 17-s − 1.82·18-s − 4.39·19-s + 2.87·20-s + 4.74·21-s − 5.37·22-s + 0.898·23-s − 2.19·24-s + 3.28·25-s + 3.75·26-s − 2.58·27-s + 2.16·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.26·3-s + 0.5·4-s + 1.28·5-s − 0.896·6-s + 0.817·7-s − 0.353·8-s + 0.607·9-s − 0.910·10-s + 1.62·11-s + 0.633·12-s − 1.04·13-s − 0.577·14-s + 1.63·15-s + 0.250·16-s + 0.242·17-s − 0.429·18-s − 1.00·19-s + 0.643·20-s + 1.03·21-s − 1.14·22-s + 0.187·23-s − 0.448·24-s + 0.657·25-s + 0.735·26-s − 0.497·27-s + 0.408·28-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
\(L(1)\) |
\(\approx\) |
\(2.867885893\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.867885893\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 - 0.898T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 - 1.57T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 5.04T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−9.277662079388842653383515912204, −8.407363744722561358395250837977, −7.980934908843741975305273946717, −6.82432552475578126216537279582, −6.31376528759336440991860066475, −5.12454126875079687323619525631, −4.11881478591814544504657243718, −2.88126288524562872008966072231, −2.09346575180976178491275877194, −1.38212490822128384114065146964,
1.38212490822128384114065146964, 2.09346575180976178491275877194, 2.88126288524562872008966072231, 4.11881478591814544504657243718, 5.12454126875079687323619525631, 6.31376528759336440991860066475, 6.82432552475578126216537279582, 7.980934908843741975305273946717, 8.407363744722561358395250837977, 9.277662079388842653383515912204