| L(s) = 1 | + i·3-s − 5i·7-s − 9-s − 6·11-s + 3i·13-s + 2i·17-s − 19-s + 5·21-s + 2i·23-s − i·27-s + 6·29-s − 3·31-s − 6i·33-s − 6i·37-s − 3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.88i·7-s − 0.333·9-s − 1.80·11-s + 0.832i·13-s + 0.485i·17-s − 0.229·19-s + 1.09·21-s + 0.417i·23-s − 0.192i·27-s + 1.11·29-s − 0.538·31-s − 1.04i·33-s − 0.986i·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147841951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147841951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.251850449594290770023595522342, −7.69173816326860941358833451220, −7.10045582332478116726435216083, −6.27741211109226051045636000355, −5.31528355424822668119220824680, −4.57746334930610966332857877715, −4.02001639384621541705560289088, −3.20945375736697531201024546740, −2.15143293081397859888884334114, −0.817809241830656324683263937974,
0.39772422318481380045722269006, 2.01856566755962819780839909400, 2.65188150747992211289103483389, 3.15444996637308104280530847817, 4.79627571666541903189528960577, 5.32941339333530498454835308131, 5.85571954014255792015786742978, 6.63387266185230904261033029596, 7.61226632157604454082531512011, 8.216327510194061992988360882605
