| L(s) = 1 | + 2-s − 4-s − 3·5-s − 3·8-s − 3·10-s − 5·11-s + 6·13-s − 16-s + 6·17-s + 3·19-s + 3·20-s − 5·22-s − 23-s + 4·25-s + 6·26-s + 2·29-s − 3·31-s + 5·32-s + 6·34-s + 3·37-s + 3·38-s + 9·40-s + 9·41-s + 6·43-s + 5·44-s − 46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.34·5-s − 1.06·8-s − 0.948·10-s − 1.50·11-s + 1.66·13-s − 1/4·16-s + 1.45·17-s + 0.688·19-s + 0.670·20-s − 1.06·22-s − 0.208·23-s + 4/5·25-s + 1.17·26-s + 0.371·29-s − 0.538·31-s + 0.883·32-s + 1.02·34-s + 0.493·37-s + 0.486·38-s + 1.42·40-s + 1.40·41-s + 0.914·43-s + 0.753·44-s − 0.147·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353272036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353272036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.625164347701805988565846090482, −8.581778872452172552963636802734, −8.010418371423523710126781900939, −7.39691755528068449061259570564, −5.96466062289959274270681070046, −5.41389437444540368847999455753, −4.37621733692366770303800440112, −3.62230602066129289359790734359, −2.95504645610525949403863409430, −0.77022560857539424923599740084,
0.77022560857539424923599740084, 2.95504645610525949403863409430, 3.62230602066129289359790734359, 4.37621733692366770303800440112, 5.41389437444540368847999455753, 5.96466062289959274270681070046, 7.39691755528068449061259570564, 8.010418371423523710126781900939, 8.581778872452172552963636802734, 9.625164347701805988565846090482
