| L(s) = 1 | + 2-s + 1.42·3-s + 4-s + 5-s + 1.42·6-s − 7-s + 8-s − 0.966·9-s + 10-s + 1.42·12-s + 3.96·13-s − 14-s + 1.42·15-s + 16-s + 0.425·17-s − 0.966·18-s − 3.54·19-s + 20-s − 1.42·21-s + 2.57·23-s + 1.42·24-s + 25-s + 3.96·26-s − 5.65·27-s − 28-s + 2·29-s + 1.42·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.823·3-s + 0.5·4-s + 0.447·5-s + 0.582·6-s − 0.377·7-s + 0.353·8-s − 0.322·9-s + 0.316·10-s + 0.411·12-s + 1.10·13-s − 0.267·14-s + 0.368·15-s + 0.250·16-s + 0.103·17-s − 0.227·18-s − 0.812·19-s + 0.223·20-s − 0.311·21-s + 0.536·23-s + 0.291·24-s + 0.200·25-s + 0.777·26-s − 1.08·27-s − 0.188·28-s + 0.371·29-s + 0.260·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.906578575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.906578575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.42T + 3T^{2} \) |
| 13 | \( 1 - 3.96T + 13T^{2} \) |
| 17 | \( 1 - 0.425T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 2.57T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 - 4.85T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 + 1.57T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 - 0.722T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 5.81T + 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 - 1.88T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 - 1.54T + 79T^{2} \) |
| 83 | \( 1 - 0.574T + 83T^{2} \) |
| 89 | \( 1 + 2.54T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−7.82079352157366398392479157829, −6.97320080452922771966525236932, −6.27954092469827060366360430429, −5.81541156291993325210437001180, −4.99597607748053004288916937983, −4.05277052576165077742088783641, −3.50780969922096201518249327680, −2.69919525201860683473154942473, −2.13382650904045758320628807022, −0.947389074494029865234113993451,
0.947389074494029865234113993451, 2.13382650904045758320628807022, 2.69919525201860683473154942473, 3.50780969922096201518249327680, 4.05277052576165077742088783641, 4.99597607748053004288916937983, 5.81541156291993325210437001180, 6.27954092469827060366360430429, 6.97320080452922771966525236932, 7.82079352157366398392479157829
