| L(s) = 1 | − 2-s − 7·4-s + 7·5-s + 10·7-s + 15·8-s − 7·10-s − 22·11-s − 10·14-s + 41·16-s − 37·17-s − 30·19-s − 49·20-s + 22·22-s + 162·23-s − 76·25-s − 70·28-s + 113·29-s − 196·31-s − 161·32-s + 37·34-s + 70·35-s − 13·37-s + 30·38-s + 105·40-s + 285·41-s − 246·43-s + 154·44-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.626·5-s + 0.539·7-s + 0.662·8-s − 0.221·10-s − 0.603·11-s − 0.190·14-s + 0.640·16-s − 0.527·17-s − 0.362·19-s − 0.547·20-s + 0.213·22-s + 1.46·23-s − 0.607·25-s − 0.472·28-s + 0.723·29-s − 1.13·31-s − 0.889·32-s + 0.186·34-s + 0.338·35-s − 0.0577·37-s + 0.128·38-s + 0.415·40-s + 1.08·41-s − 0.872·43-s + 0.527·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 37 T + p^{3} T^{2} \) |
| 19 | \( 1 + 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 113 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 13 T + p^{3} T^{2} \) |
| 41 | \( 1 - 285 T + p^{3} T^{2} \) |
| 43 | \( 1 + 246 T + p^{3} T^{2} \) |
| 47 | \( 1 + 462 T + p^{3} T^{2} \) |
| 53 | \( 1 - 537 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 + 635 T + p^{3} T^{2} \) |
| 67 | \( 1 + 202 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 - 805 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 - 518 T + p^{3} T^{2} \) |
| 89 | \( 1 - 194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1202 T + p^{3} T^{2} \) |
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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
−8.803756649813499893023455886656, −8.053417034673223902307939725107, −7.26599330844971751653116856537, −6.19448945249893137522473985733, −5.18865668346464858569492939138, −4.72256343699449993591989685845, −3.57650642168202236891237943258, −2.30233839815696874325824130612, −1.24403047013894068968848515494, 0,
1.24403047013894068968848515494, 2.30233839815696874325824130612, 3.57650642168202236891237943258, 4.72256343699449993591989685845, 5.18865668346464858569492939138, 6.19448945249893137522473985733, 7.26599330844971751653116856537, 8.053417034673223902307939725107, 8.803756649813499893023455886656
