L(s) = 1 | + 1.70·2-s + 3.05·3-s + 0.906·4-s + 3.01·5-s + 5.21·6-s − 2.00·7-s − 1.86·8-s + 6.36·9-s + 5.13·10-s − 1.57·11-s + 2.77·12-s + 3.75·13-s − 3.41·14-s + 9.22·15-s − 4.99·16-s + 2.92·17-s + 10.8·18-s + 2.90·19-s + 2.73·20-s − 6.13·21-s − 2.68·22-s − 5.84·23-s − 5.70·24-s + 4.08·25-s + 6.40·26-s + 10.2·27-s − 1.81·28-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 1.76·3-s + 0.453·4-s + 1.34·5-s + 2.12·6-s − 0.757·7-s − 0.659·8-s + 2.12·9-s + 1.62·10-s − 0.474·11-s + 0.800·12-s + 1.04·13-s − 0.913·14-s + 2.38·15-s − 1.24·16-s + 0.708·17-s + 2.55·18-s + 0.666·19-s + 0.610·20-s − 1.33·21-s − 0.571·22-s − 1.21·23-s − 1.16·24-s + 0.816·25-s + 1.25·26-s + 1.97·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.526487750\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.526487750\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2011 | \( 1 - T \) |
good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 - 3.05T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 + 5.63T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 - 5.96T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 9.99T + 53T^{2} \) |
| 59 | \( 1 - 9.24T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 8.76T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−9.352577832475572839798820709455, −8.424962733983263106526126046360, −7.67000393798993380618496261577, −6.51965948755142025685219396150, −5.93341605975634485639924001831, −5.09320263488350603339328130531, −3.86830197174193303316282043069, −3.36063856310814260947661206821, −2.58879646926411983242145975756, −1.69203666221707070519160883290,
1.69203666221707070519160883290, 2.58879646926411983242145975756, 3.36063856310814260947661206821, 3.86830197174193303316282043069, 5.09320263488350603339328130531, 5.93341605975634485639924001831, 6.51965948755142025685219396150, 7.67000393798993380618496261577, 8.424962733983263106526126046360, 9.352577832475572839798820709455