| L(s) = 1 | − 2·7-s − 3·11-s + 4·13-s + 6·17-s − 7·19-s + 6·23-s − 3·29-s + 5·31-s + 4·37-s − 3·41-s − 8·43-s − 3·49-s − 6·53-s + 3·59-s + 14·61-s − 2·67-s − 15·71-s + 10·73-s + 6·77-s + 8·79-s − 15·89-s − 8·91-s − 8·97-s + 3·101-s − 2·103-s − 6·107-s + 11·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.904·11-s + 1.10·13-s + 1.45·17-s − 1.60·19-s + 1.25·23-s − 0.557·29-s + 0.898·31-s + 0.657·37-s − 0.468·41-s − 1.21·43-s − 3/7·49-s − 0.824·53-s + 0.390·59-s + 1.79·61-s − 0.244·67-s − 1.78·71-s + 1.17·73-s + 0.683·77-s + 0.900·79-s − 1.58·89-s − 0.838·91-s − 0.812·97-s + 0.298·101-s − 0.197·103-s − 0.580·107-s + 1.05·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630872844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630872844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.011181851798971625882224844996, −7.01300921615128522375231496796, −6.46333903410091334635820289292, −5.78559950684525051625772718682, −5.11920029285336848124348032564, −4.23050829709861918795935458013, −3.36569975420881439816669763539, −2.86814411193379207615726215462, −1.74225366550401285340756691538, −0.63064144990947307827518761264,
0.63064144990947307827518761264, 1.74225366550401285340756691538, 2.86814411193379207615726215462, 3.36569975420881439816669763539, 4.23050829709861918795935458013, 5.11920029285336848124348032564, 5.78559950684525051625772718682, 6.46333903410091334635820289292, 7.01300921615128522375231496796, 8.011181851798971625882224844996
