L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s − 6·11-s + 2·13-s − 14-s + 16-s + 6·17-s − 7·19-s − 3·20-s + 6·22-s + 3·23-s + 4·25-s − 2·26-s + 28-s + 6·29-s + 2·31-s − 32-s − 6·34-s − 3·35-s + 2·37-s + 7·38-s + 3·40-s + 2·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.60·19-s − 0.670·20-s + 1.27·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.507·35-s + 0.328·37-s + 1.13·38-s + 0.474·40-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7505729247\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7505729247\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.978354774340963339289773651988, −8.666696633956550516520454458294, −8.094785783071409988385687751137, −7.73532298603754652567815778453, −6.73871282898076223759796265441, −5.56957044429086118267228114847, −4.59790954336953877055233514712, −3.48588234639422265078311386338, −2.44868872871059860732796494913, −0.71201387828374154761208002119,
0.71201387828374154761208002119, 2.44868872871059860732796494913, 3.48588234639422265078311386338, 4.59790954336953877055233514712, 5.56957044429086118267228114847, 6.73871282898076223759796265441, 7.73532298603754652567815778453, 8.094785783071409988385687751137, 8.666696633956550516520454458294, 9.978354774340963339289773651988