| L(s) = 1 | − 2·3-s + 4·7-s + 9-s + 11-s − 4·13-s + 4·19-s − 8·21-s + 6·23-s + 4·27-s + 6·29-s + 8·31-s − 2·33-s + 2·37-s + 8·39-s + 6·41-s + 8·43-s − 6·47-s + 9·49-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s + 4·63-s − 10·67-s − 12·69-s − 12·71-s + 16·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.917·19-s − 1.74·21-s + 1.25·23-s + 0.769·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.503·63-s − 1.22·67-s − 1.44·69-s − 1.42·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973609022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973609022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.98093957325750, −15.23082998923882, −14.70641077075693, −14.25656462965305, −13.77848182359225, −12.93332217295843, −12.22427758829659, −11.89206025473415, −11.44888992060401, −10.91896297531145, −10.45246601285317, −9.704178301614957, −9.109818127381130, −8.300669707012618, −7.824186659177893, −7.148157785050184, −6.568236062854672, −5.816615190347757, −5.169352789610878, −4.760941600787582, −4.343150592297687, −3.052979957060714, −2.395515499714491, −1.256139726602068, −0.7402801383816275,
0.7402801383816275, 1.256139726602068, 2.395515499714491, 3.052979957060714, 4.343150592297687, 4.760941600787582, 5.169352789610878, 5.816615190347757, 6.568236062854672, 7.148157785050184, 7.824186659177893, 8.300669707012618, 9.109818127381130, 9.704178301614957, 10.45246601285317, 10.91896297531145, 11.44888992060401, 11.89206025473415, 12.22427758829659, 12.93332217295843, 13.77848182359225, 14.25656462965305, 14.70641077075693, 15.23082998923882, 15.98093957325750
