| L(s) = 1 | + 2·5-s − 7-s − 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s − 8·55-s + 6·61-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s + 16·79-s + 8·83-s + 12·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + 0.878·83-s + 1.30·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874848804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874848804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.272203299109375537884659130698, −7.902596514850714806179317231692, −6.84383926850302326577754552348, −6.20669276463558444353009673367, −5.50140758689236696304146355325, −4.83297349612888492276603651901, −3.83348667018088193321396729321, −2.66451378084199854982327930143, −2.24055198785635289229441777506, −0.76540944404158607646425160976,
0.76540944404158607646425160976, 2.24055198785635289229441777506, 2.66451378084199854982327930143, 3.83348667018088193321396729321, 4.83297349612888492276603651901, 5.50140758689236696304146355325, 6.20669276463558444353009673367, 6.84383926850302326577754552348, 7.902596514850714806179317231692, 8.272203299109375537884659130698
