| L(s) = 1 | + 5-s + 4·7-s + 2·13-s − 17-s + 23-s + 25-s + 6·29-s + 31-s + 4·35-s + 4·37-s − 2·41-s − 2·43-s − 9·47-s + 9·49-s − 9·53-s + 6·59-s + 61-s + 2·65-s + 4·67-s + 4·73-s − 11·79-s − 8·83-s − 85-s + 16·89-s + 8·91-s + 16·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.554·13-s − 0.242·17-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.179·31-s + 0.676·35-s + 0.657·37-s − 0.312·41-s − 0.304·43-s − 1.31·47-s + 9/7·49-s − 1.23·53-s + 0.781·59-s + 0.128·61-s + 0.248·65-s + 0.488·67-s + 0.468·73-s − 1.23·79-s − 0.878·83-s − 0.108·85-s + 1.69·89-s + 0.838·91-s + 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743260904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743260904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.62590103703211, −14.19759009093535, −13.77906162115702, −13.08400840295343, −12.80720351283547, −11.78539840047306, −11.70238851982347, −11.00640212691648, −10.63317368938173, −9.985664030797192, −9.457052527864999, −8.698495593936553, −8.355209484355466, −7.903252703280549, −7.218316541390013, −6.532096800106645, −6.070702195201692, −5.304579210217449, −4.811090371248972, −4.402531079546154, −3.534713397386285, −2.806732314608132, −2.008623995278295, −1.485045517991586, −0.7279114553245109,
0.7279114553245109, 1.485045517991586, 2.008623995278295, 2.806732314608132, 3.534713397386285, 4.402531079546154, 4.811090371248972, 5.304579210217449, 6.070702195201692, 6.532096800106645, 7.218316541390013, 7.903252703280549, 8.355209484355466, 8.698495593936553, 9.457052527864999, 9.985664030797192, 10.63317368938173, 11.00640212691648, 11.70238851982347, 11.78539840047306, 12.80720351283547, 13.08400840295343, 13.77906162115702, 14.19759009093535, 14.62590103703211
