| L(s) = 1 | − 2-s + 2.33·3-s + 4-s + 4.19·5-s − 2.33·6-s + 1.15·7-s − 8-s + 2.42·9-s − 4.19·10-s + 11-s + 2.33·12-s − 3.78·13-s − 1.15·14-s + 9.78·15-s + 16-s + 5.30·17-s − 2.42·18-s + 4.46·19-s + 4.19·20-s + 2.68·21-s − 22-s + 8.27·23-s − 2.33·24-s + 12.6·25-s + 3.78·26-s − 1.32·27-s + 1.15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s + 1.87·5-s − 0.951·6-s + 0.435·7-s − 0.353·8-s + 0.809·9-s − 1.32·10-s + 0.301·11-s + 0.672·12-s − 1.05·13-s − 0.307·14-s + 2.52·15-s + 0.250·16-s + 1.28·17-s − 0.572·18-s + 1.02·19-s + 0.938·20-s + 0.585·21-s − 0.213·22-s + 1.72·23-s − 0.475·24-s + 2.52·25-s + 0.742·26-s − 0.255·27-s + 0.217·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.708956939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.708956939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 + 8.95T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 2.75T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 9.01T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 9.26T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 + 3.55T + 97T^{2} \) |
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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
−8.664739604193097205625247996139, −7.56480204296787687824498725288, −7.33587771799344202877031512919, −6.26109405219563504297848728195, −5.45108746981516829676277012984, −4.82951001549992062365217527958, −3.18560604658147637660655192282, −2.86816707092168237812613024672, −1.83206877233904724336797754286, −1.31499876850731018280596341499,
1.31499876850731018280596341499, 1.83206877233904724336797754286, 2.86816707092168237812613024672, 3.18560604658147637660655192282, 4.82951001549992062365217527958, 5.45108746981516829676277012984, 6.26109405219563504297848728195, 7.33587771799344202877031512919, 7.56480204296787687824498725288, 8.664739604193097205625247996139
