| L(s) = 1 | − 3.63·2-s + 3·3-s + 5.18·4-s − 21.1·5-s − 10.8·6-s + 10.2·8-s + 9·9-s + 76.6·10-s + 11·11-s + 15.5·12-s − 87.3·13-s − 63.3·15-s − 78.6·16-s + 28.2·17-s − 32.6·18-s + 97.9·19-s − 109.·20-s − 39.9·22-s − 112.·23-s + 30.6·24-s + 320.·25-s + 317.·26-s + 27·27-s + 14.9·29-s + 229.·30-s − 138.·31-s + 203.·32-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.647·4-s − 1.88·5-s − 0.741·6-s + 0.452·8-s + 0.333·9-s + 2.42·10-s + 0.301·11-s + 0.373·12-s − 1.86·13-s − 1.09·15-s − 1.22·16-s + 0.403·17-s − 0.427·18-s + 1.18·19-s − 1.22·20-s − 0.387·22-s − 1.01·23-s + 0.261·24-s + 2.56·25-s + 2.39·26-s + 0.192·27-s + 0.0955·29-s + 1.39·30-s − 0.802·31-s + 1.12·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.63T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 13 | \( 1 + 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 554.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−8.507126646509247077700875738500, −7.81738443498023077731730396322, −7.51242210372315931697660397898, −6.88664980059442956062835391572, −5.11008558763318098568649113075, −4.28620412685939943946780208570, −3.44626770361049781170092358181, −2.31877099508548332754164536001, −0.897222116034224600577558247717, 0,
0.897222116034224600577558247717, 2.31877099508548332754164536001, 3.44626770361049781170092358181, 4.28620412685939943946780208570, 5.11008558763318098568649113075, 6.88664980059442956062835391572, 7.51242210372315931697660397898, 7.81738443498023077731730396322, 8.507126646509247077700875738500
