| L(s) = 1 | + 5.37·2-s + 9.47·3-s + 20.8·4-s + 7.47·5-s + 50.9·6-s − 7·7-s + 69.0·8-s + 62.8·9-s + 40.1·10-s − 61.2·11-s + 197.·12-s + 54.6·13-s − 37.6·14-s + 70.8·15-s + 204.·16-s + 337.·18-s − 17.4·19-s + 155.·20-s − 66.3·21-s − 328.·22-s − 97.0·23-s + 654.·24-s − 69.1·25-s + 293.·26-s + 339.·27-s − 146.·28-s + 78.3·29-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 1.82·3-s + 2.60·4-s + 0.668·5-s + 3.46·6-s − 0.377·7-s + 3.05·8-s + 2.32·9-s + 1.26·10-s − 1.67·11-s + 4.75·12-s + 1.16·13-s − 0.717·14-s + 1.21·15-s + 3.19·16-s + 4.42·18-s − 0.210·19-s + 1.74·20-s − 0.689·21-s − 3.18·22-s − 0.879·23-s + 5.57·24-s − 0.552·25-s + 2.21·26-s + 2.42·27-s − 0.985·28-s + 0.501·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.17905891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.17905891\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 5.37T + 8T^{2} \) |
| 3 | \( 1 - 9.47T + 27T^{2} \) |
| 5 | \( 1 - 7.47T + 125T^{2} \) |
| 11 | \( 1 + 61.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 17.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 78.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 87.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 52.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 728.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 349.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 407.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 316.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 473.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−8.511188544081870648075106641883, −7.905575232880569386487721686605, −7.15221434593350338690398357311, −6.19801088913724372294085095474, −5.54086411967462343818375724351, −4.46752254295082195363193750326, −3.75005150223370677588107295410, −2.96327093807643449007924740365, −2.40965831027910329086692103274, −1.63136273797830932722201391941,
1.63136273797830932722201391941, 2.40965831027910329086692103274, 2.96327093807643449007924740365, 3.75005150223370677588107295410, 4.46752254295082195363193750326, 5.54086411967462343818375724351, 6.19801088913724372294085095474, 7.15221434593350338690398357311, 7.905575232880569386487721686605, 8.511188544081870648075106641883
