| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s + 3·11-s + 3·12-s + 16-s − 7·17-s − 6·18-s − 19-s − 3·22-s − 4·23-s − 3·24-s + 9·27-s + 4·29-s + 10·31-s − 32-s + 9·33-s + 7·34-s + 6·36-s − 12·37-s + 38-s + 5·41-s + 12·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s + 0.904·11-s + 0.866·12-s + 1/4·16-s − 1.69·17-s − 1.41·18-s − 0.229·19-s − 0.639·22-s − 0.834·23-s − 0.612·24-s + 1.73·27-s + 0.742·29-s + 1.79·31-s − 0.176·32-s + 1.56·33-s + 1.20·34-s + 36-s − 1.97·37-s + 0.162·38-s + 0.780·41-s + 1.82·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092827951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092827951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.058170497582267291450143216026, −7.22144624037514645175487134784, −6.73090491190586516053277585784, −6.05861195554381670015777386797, −4.65411356030144094955114104954, −4.08613504332076906770361629882, −3.34822874072490136199146020240, −2.38281164378354270639471982446, −2.04100819010582425076849582507, −0.886331692799814375620612397404,
0.886331692799814375620612397404, 2.04100819010582425076849582507, 2.38281164378354270639471982446, 3.34822874072490136199146020240, 4.08613504332076906770361629882, 4.65411356030144094955114104954, 6.05861195554381670015777386797, 6.73090491190586516053277585784, 7.22144624037514645175487134784, 8.058170497582267291450143216026
