| L(s) = 1 | − 3.62·2-s − 7.41·3-s + 5.10·4-s + 5·5-s + 26.8·6-s − 14.1·7-s + 10.4·8-s + 28.0·9-s − 18.1·10-s − 11·11-s − 37.8·12-s + 26.6·13-s + 51.0·14-s − 37.0·15-s − 78.7·16-s + 9.72·17-s − 101.·18-s + 19·19-s + 25.5·20-s + 104.·21-s + 39.8·22-s + 183.·23-s − 77.6·24-s + 25·25-s − 96.6·26-s − 7.50·27-s − 72.0·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 1.42·3-s + 0.638·4-s + 0.447·5-s + 1.82·6-s − 0.761·7-s + 0.462·8-s + 1.03·9-s − 0.572·10-s − 0.301·11-s − 0.911·12-s + 0.569·13-s + 0.974·14-s − 0.638·15-s − 1.23·16-s + 0.138·17-s − 1.32·18-s + 0.229·19-s + 0.285·20-s + 1.08·21-s + 0.385·22-s + 1.65·23-s − 0.660·24-s + 0.200·25-s − 0.728·26-s − 0.0534·27-s − 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4691299655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4691299655\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 3.62T + 8T^{2} \) |
| 3 | \( 1 + 7.41T + 27T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 13 | \( 1 - 26.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.72T + 4.91e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 85.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 700.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 193.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 222.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 983.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 519.T + 9.12e5T^{2} \) |
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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
−9.719860236095311773778667429949, −8.898600267804089550774190891028, −7.955397722655269558893546780920, −6.88929150231570382868032876698, −6.37675165244245488313506886879, −5.41069878076643431105860239466, −4.55717181167880812534327802439, −2.97616283242470935092982415351, −1.38426909710901267754676354929, −0.52420547083260003462181787732,
0.52420547083260003462181787732, 1.38426909710901267754676354929, 2.97616283242470935092982415351, 4.55717181167880812534327802439, 5.41069878076643431105860239466, 6.37675165244245488313506886879, 6.88929150231570382868032876698, 7.955397722655269558893546780920, 8.898600267804089550774190891028, 9.719860236095311773778667429949
