| L(s) = 1 | − 2.05·2-s + 1.62·3-s + 2.20·4-s − 4.18·5-s − 3.34·6-s − 0.421·8-s − 0.345·9-s + 8.57·10-s − 3.88·11-s + 3.59·12-s − 2.89·13-s − 6.81·15-s − 3.54·16-s − 2.83·17-s + 0.707·18-s − 5.96·19-s − 9.21·20-s + 7.96·22-s + 2.37·23-s − 0.686·24-s + 12.4·25-s + 5.94·26-s − 5.45·27-s − 5.37·29-s + 13.9·30-s + 3.48·31-s + 8.11·32-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 0.940·3-s + 1.10·4-s − 1.86·5-s − 1.36·6-s − 0.148·8-s − 0.115·9-s + 2.71·10-s − 1.17·11-s + 1.03·12-s − 0.804·13-s − 1.75·15-s − 0.886·16-s − 0.687·17-s + 0.166·18-s − 1.36·19-s − 2.06·20-s + 1.69·22-s + 0.494·23-s − 0.140·24-s + 2.49·25-s + 1.16·26-s − 1.04·27-s − 0.997·29-s + 2.55·30-s + 0.626·31-s + 1.43·32-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
| \(L(1)\) |
\(\approx\) |
\(0.2850904995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2850904995\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 3.48T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 0.0168T + 73T^{2} \) |
| 79 | \( 1 + 0.301T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.974213269513848777367204816810, −8.285583235314551298653026899287, −7.79075317724384455213904452786, −7.51559481694808734780107733166, −6.47488399362380566659414983049, −4.82918220528115058758246479604, −4.13931118655856687671244785075, −2.96446903679829706151231682661, −2.23866092074252673640138067723, −0.39370003238433975431005498443,
0.39370003238433975431005498443, 2.23866092074252673640138067723, 2.96446903679829706151231682661, 4.13931118655856687671244785075, 4.82918220528115058758246479604, 6.47488399362380566659414983049, 7.51559481694808734780107733166, 7.79075317724384455213904452786, 8.285583235314551298653026899287, 8.974213269513848777367204816810
