| L(s) = 1 | + 3.14·2-s − 2.81·3-s + 1.89·4-s + 19.3·5-s − 8.86·6-s + 15.5·7-s − 19.1·8-s − 19.0·9-s + 61.0·10-s + 65.6·11-s − 5.35·12-s − 6.62·13-s + 48.8·14-s − 54.6·15-s − 75.5·16-s + 17·17-s − 59.9·18-s + 46.1·19-s + 36.8·20-s − 43.7·21-s + 206.·22-s + 13.0·23-s + 54.0·24-s + 251.·25-s − 20.8·26-s + 129.·27-s + 29.4·28-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 0.542·3-s + 0.237·4-s + 1.73·5-s − 0.603·6-s + 0.837·7-s − 0.848·8-s − 0.705·9-s + 1.92·10-s + 1.79·11-s − 0.128·12-s − 0.141·13-s + 0.932·14-s − 0.940·15-s − 1.18·16-s + 0.242·17-s − 0.785·18-s + 0.557·19-s + 0.411·20-s − 0.454·21-s + 2.00·22-s + 0.118·23-s + 0.459·24-s + 2.00·25-s − 0.157·26-s + 0.925·27-s + 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.354216949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.354216949\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - 17T \) |
| 43 | \( 1 - 43T \) |
| good | 2 | \( 1 - 3.14T + 8T^{2} \) |
| 3 | \( 1 + 2.81T + 27T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 - 65.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.62T + 2.19e3T^{2} \) |
| 19 | \( 1 - 46.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 46.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 134.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 43.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 224.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 877.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 811.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 274.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 458.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 233.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 471.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 71.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 171.T + 9.12e5T^{2} \) |
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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
−9.898087910325711235722209065137, −9.277030365900313292270643224598, −8.485342147893298611125900155824, −6.78905132651481947249414610901, −6.13702649825589599725601001611, −5.45478912505118174812951785340, −4.83429975045762405832742927790, −3.60268887583264220525984907130, −2.31275705803384043377480235743, −1.13872844602964034393808862569,
1.13872844602964034393808862569, 2.31275705803384043377480235743, 3.60268887583264220525984907130, 4.83429975045762405832742927790, 5.45478912505118174812951785340, 6.13702649825589599725601001611, 6.78905132651481947249414610901, 8.485342147893298611125900155824, 9.277030365900313292270643224598, 9.898087910325711235722209065137
