L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s − 7-s + 8-s + 6·9-s + 10-s + 3·12-s + 3·13-s − 14-s + 3·15-s + 16-s + 2·17-s + 6·18-s + 19-s + 20-s − 3·21-s − 23-s + 3·24-s + 25-s + 3·26-s + 9·27-s − 28-s − 6·29-s + 3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.866·12-s + 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.229·19-s + 0.223·20-s − 0.654·21-s − 0.208·23-s + 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s − 0.188·28-s − 1.11·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.565549594\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.565549594\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.78806356066214405674367595664, −7.14365186146004232143264336932, −6.48891791678400621217840638985, −5.68244030270100924074541534329, −4.90867289608240440082123498938, −3.78994372408229916887112487287, −3.63131466182685259719261454349, −2.74420485656560457116576733691, −2.08799418551487501004602516006, −1.22031585363749126776673308193,
1.22031585363749126776673308193, 2.08799418551487501004602516006, 2.74420485656560457116576733691, 3.63131466182685259719261454349, 3.78994372408229916887112487287, 4.90867289608240440082123498938, 5.68244030270100924074541534329, 6.48891791678400621217840638985, 7.14365186146004232143264336932, 7.78806356066214405674367595664