| L(s) = 1 | + 3.33·2-s + 3·3-s + 3.11·4-s − 15.9·5-s + 10.0·6-s − 28.9·7-s − 16.2·8-s + 9·9-s − 53.2·10-s − 11·11-s + 9.34·12-s − 18.4·13-s − 96.6·14-s − 47.8·15-s − 79.2·16-s + 53.6·17-s + 30.0·18-s + 36.3·19-s − 49.7·20-s − 86.9·21-s − 36.6·22-s − 202.·23-s − 48.8·24-s + 129.·25-s − 61.3·26-s + 27·27-s − 90.3·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.389·4-s − 1.42·5-s + 0.680·6-s − 1.56·7-s − 0.719·8-s + 0.333·9-s − 1.68·10-s − 0.301·11-s + 0.224·12-s − 0.392·13-s − 1.84·14-s − 0.824·15-s − 1.23·16-s + 0.764·17-s + 0.392·18-s + 0.438·19-s − 0.555·20-s − 0.903·21-s − 0.355·22-s − 1.83·23-s − 0.415·24-s + 1.03·25-s − 0.462·26-s + 0.192·27-s − 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.175308953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175308953\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 3.33T + 8T^{2} \) |
| 5 | \( 1 + 15.9T + 125T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 13 | \( 1 + 18.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 202.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 32.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 220.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 791.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 618.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 20.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 800.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−8.763977462281534369737733820164, −7.79648524737105598012153364191, −7.28562604058398027068049774595, −6.28156754488935831778217564113, −5.54938914550154534908976133224, −4.43798388314712614239438770383, −3.65887392988442730137543798965, −3.40100862291857888606279311380, −2.41297288240597869431422587657, −0.37726603681653232189647084287,
0.37726603681653232189647084287, 2.41297288240597869431422587657, 3.40100862291857888606279311380, 3.65887392988442730137543798965, 4.43798388314712614239438770383, 5.54938914550154534908976133224, 6.28156754488935831778217564113, 7.28562604058398027068049774595, 7.79648524737105598012153364191, 8.763977462281534369737733820164
