L(s) = 1 | − 2-s − 3-s + 4-s − 4.27·5-s + 6-s + 7-s − 8-s + 9-s + 4.27·10-s − 2.27·11-s − 12-s − 13-s − 14-s + 4.27·15-s + 16-s + 0.274·17-s − 18-s − 2.27·19-s − 4.27·20-s − 21-s + 2.27·22-s + 2.27·23-s + 24-s + 13.2·25-s + 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.91·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.35·10-s − 0.685·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1.10·15-s + 0.250·16-s + 0.0666·17-s − 0.235·18-s − 0.521·19-s − 0.955·20-s − 0.218·21-s + 0.485·22-s + 0.474·23-s + 0.204·24-s + 2.65·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5044367619\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5044367619\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−10.78868508975187744268244702582, −10.21964007909320570154373489479, −8.742751029092682789085905086951, −8.118377903121081729655039409972, −7.39387180731048144549132281965, −6.56221015366234753717816573741, −5.05266373541693740873949136430, −4.19611378843197887720672721530, −2.83488970863995323177745930789, −0.71120220211023209380963073748,
0.71120220211023209380963073748, 2.83488970863995323177745930789, 4.19611378843197887720672721530, 5.05266373541693740873949136430, 6.56221015366234753717816573741, 7.39387180731048144549132281965, 8.118377903121081729655039409972, 8.742751029092682789085905086951, 10.21964007909320570154373489479, 10.78868508975187744268244702582