L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 3·11-s + 2·13-s + 16-s + 6·17-s + 2·19-s + 3·20-s − 3·22-s − 6·23-s + 4·25-s + 2·26-s + 9·29-s − 7·31-s + 32-s + 6·34-s − 10·37-s + 2·38-s + 3·40-s − 4·43-s − 3·44-s − 6·46-s + 12·47-s + 4·50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 0.904·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.458·19-s + 0.670·20-s − 0.639·22-s − 1.25·23-s + 4/5·25-s + 0.392·26-s + 1.67·29-s − 1.25·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s + 0.324·38-s + 0.474·40-s − 0.609·43-s − 0.452·44-s − 0.884·46-s + 1.75·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.017796288\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.017796288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−10.26872983008473533451567565203, −9.521561835609810671858571335000, −8.388380094054281186601061743616, −7.49053770083860455326674259398, −6.40196296984473786838190696766, −5.62582070914781549215591544481, −5.13267260304391847265082866628, −3.69829470531395738272962092099, −2.65191870625457881403725845166, −1.52906039199785314571915081352,
1.52906039199785314571915081352, 2.65191870625457881403725845166, 3.69829470531395738272962092099, 5.13267260304391847265082866628, 5.62582070914781549215591544481, 6.40196296984473786838190696766, 7.49053770083860455326674259398, 8.388380094054281186601061743616, 9.521561835609810671858571335000, 10.26872983008473533451567565203