L(s) = 1 | − 2.05·2-s + 2.25·3-s + 2.23·4-s − 5-s − 4.64·6-s + 1.74·7-s − 0.480·8-s + 2.09·9-s + 2.05·10-s + 0.110·11-s + 5.04·12-s + 2.92·13-s − 3.58·14-s − 2.25·15-s − 3.47·16-s + 7.82·17-s − 4.30·18-s − 4.46·19-s − 2.23·20-s + 3.93·21-s − 0.227·22-s + 2.62·23-s − 1.08·24-s + 25-s − 6.00·26-s − 2.04·27-s + 3.89·28-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.30·3-s + 1.11·4-s − 0.447·5-s − 1.89·6-s + 0.659·7-s − 0.169·8-s + 0.697·9-s + 0.650·10-s + 0.0333·11-s + 1.45·12-s + 0.809·13-s − 0.959·14-s − 0.582·15-s − 0.869·16-s + 1.89·17-s − 1.01·18-s − 1.02·19-s − 0.499·20-s + 0.858·21-s − 0.0485·22-s + 0.546·23-s − 0.221·24-s + 0.200·25-s − 1.17·26-s − 0.394·27-s + 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.430047736\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.430047736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 - 0.110T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 2.06T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 + 8.52T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 + 6.26T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 + 6.15T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 4.73T + 97T^{2} \) |
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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
−8.765747554620773309763700953723, −8.510781942492711539937230046484, −7.950087188688044683718869211289, −7.37099626932068529280253920709, −6.37948932275680126134489150135, −5.05101964237669324118634464040, −3.93229740813204553570530939651, −3.06665733815230100990406018272, −1.95734299173218721073578509866, −0.993488059474062711459693652512,
0.993488059474062711459693652512, 1.95734299173218721073578509866, 3.06665733815230100990406018272, 3.93229740813204553570530939651, 5.05101964237669324118634464040, 6.37948932275680126134489150135, 7.37099626932068529280253920709, 7.950087188688044683718869211289, 8.510781942492711539937230046484, 8.765747554620773309763700953723