| L(s) = 1 | − 3.55·2-s + 3.83·3-s + 4.65·4-s + 5·5-s − 13.6·6-s − 6.66·7-s + 11.9·8-s − 12.2·9-s − 17.7·10-s + 11·11-s + 17.8·12-s − 52.2·13-s + 23.6·14-s + 19.1·15-s − 79.5·16-s + 85.6·17-s + 43.5·18-s + 19·19-s + 23.2·20-s − 25.5·21-s − 39.1·22-s + 64.2·23-s + 45.7·24-s + 25·25-s + 185.·26-s − 150.·27-s − 30.9·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.738·3-s + 0.581·4-s + 0.447·5-s − 0.929·6-s − 0.359·7-s + 0.526·8-s − 0.453·9-s − 0.562·10-s + 0.301·11-s + 0.429·12-s − 1.11·13-s + 0.452·14-s + 0.330·15-s − 1.24·16-s + 1.22·17-s + 0.570·18-s + 0.229·19-s + 0.260·20-s − 0.265·21-s − 0.379·22-s + 0.582·23-s + 0.388·24-s + 0.200·25-s + 1.40·26-s − 1.07·27-s − 0.209·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 3.55T + 8T^{2} \) |
| 3 | \( 1 - 3.83T + 27T^{2} \) |
| 7 | \( 1 + 6.66T + 343T^{2} \) |
| 13 | \( 1 + 52.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 85.6T + 4.91e3T^{2} \) |
| 23 | \( 1 - 64.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 18.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 98.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 748.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 585.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 894.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 356.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 10.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−9.051959462858194021905012917514, −8.552161302027454882485237262445, −7.62912468940268489745561931535, −7.04493662137125197874396427729, −5.81423230466096543376890226508, −4.78713531729860430092362533319, −3.37867854065190929725609930700, −2.45179393517441800142471393421, −1.31399443633970114171411257933, 0,
1.31399443633970114171411257933, 2.45179393517441800142471393421, 3.37867854065190929725609930700, 4.78713531729860430092362533319, 5.81423230466096543376890226508, 7.04493662137125197874396427729, 7.62912468940268489745561931535, 8.552161302027454882485237262445, 9.051959462858194021905012917514
