| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s + 11-s + 12-s + 6·13-s + 3·14-s + 16-s + 7·17-s + 2·18-s + 5·19-s − 3·21-s − 22-s + 6·23-s − 24-s − 6·26-s − 5·27-s − 3·28-s + 5·29-s − 3·31-s − 32-s + 33-s − 7·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s + 0.471·18-s + 1.14·19-s − 0.654·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 1.17·26-s − 0.962·27-s − 0.566·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216011166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216011166\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−10.62761614296171945475675158355, −9.696869758195084295021589593937, −9.051313158030191365780373658814, −8.319735201663290337491520314526, −7.35838560113530362823348866724, −6.32452856749704006618306134382, −5.51294812234663797155432190910, −3.46996281594193181675719351221, −3.07694177217990984214438626883, −1.14442070453994046878747912184,
1.14442070453994046878747912184, 3.07694177217990984214438626883, 3.46996281594193181675719351221, 5.51294812234663797155432190910, 6.32452856749704006618306134382, 7.35838560113530362823348866724, 8.319735201663290337491520314526, 9.051313158030191365780373658814, 9.696869758195084295021589593937, 10.62761614296171945475675158355
