L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s − 2·9-s + 3·10-s + 12-s − 14-s − 3·15-s + 16-s − 6·17-s + 2·18-s − 4·19-s − 3·20-s + 21-s + 3·23-s − 24-s + 4·25-s − 5·27-s + 28-s − 6·29-s + 3·30-s + 10·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.917·19-s − 0.670·20-s + 0.218·21-s + 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.962·27-s + 0.188·28-s − 1.11·29-s + 0.547·30-s + 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.97894934351938, −12.32078896883477, −11.82967097076430, −11.56954885756172, −11.16682097157430, −10.66365016414280, −10.34941773110919, −9.594690777576369, −9.029625024518801, −8.680096247747065, −8.427378255534384, −7.968405973851659, −7.524540683608801, −7.028403426933977, −6.492794988053119, −6.128015080321571, −5.210006550306666, −4.813010291987377, −4.205705431800609, −3.701273228294005, −3.216359865230996, −2.594429976494216, −2.098314363437013, −1.483947812921115, −0.5058049001254996, 0,
0.5058049001254996, 1.483947812921115, 2.098314363437013, 2.594429976494216, 3.216359865230996, 3.701273228294005, 4.205705431800609, 4.813010291987377, 5.210006550306666, 6.128015080321571, 6.492794988053119, 7.028403426933977, 7.524540683608801, 7.968405973851659, 8.427378255534384, 8.680096247747065, 9.029625024518801, 9.594690777576369, 10.34941773110919, 10.66365016414280, 11.16682097157430, 11.56954885756172, 11.82967097076430, 12.32078896883477, 12.97894934351938