L(s) = 1 | − 2-s + 3·3-s + 4-s − 5-s − 3·6-s + 7-s − 8-s + 6·9-s + 10-s + 2·11-s + 3·12-s + 13-s − 14-s − 3·15-s + 16-s + 7·17-s − 6·18-s − 7·19-s − 20-s + 3·21-s − 2·22-s − 6·23-s − 3·24-s + 25-s − 26-s + 9·27-s + 28-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 + 0.603·11-s + 0.866·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s − 1.41·18-s − 1.60·19-s − 0.223·20-s + 0.654·21-s − 0.426·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s − 0.196·26-s + 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.96133194807210, −14.50881600668776, −14.18075653608249, −13.74177417666003, −12.91165893469087, −12.34566936965330, −12.13555096018476, −11.24943092441434, −10.59438014204763, −10.26150835496013, −9.563778571613425, −9.003913217933762, −8.777287582000617, −7.971322622988686, −7.840306883857479, −7.363458644263749, −6.518052922277940, −6.016997205961726, −5.044439956491742, −4.187445049186543, −3.724650703235708, −3.259276943259768, −2.450318296270152, −1.744075304281378, −1.332135496410880, 0,
1.332135496410880, 1.744075304281378, 2.450318296270152, 3.259276943259768, 3.724650703235708, 4.187445049186543, 5.044439956491742, 6.016997205961726, 6.518052922277940, 7.363458644263749, 7.840306883857479, 7.971322622988686, 8.777287582000617, 9.003913217933762, 9.563778571613425, 10.26150835496013, 10.59438014204763, 11.24943092441434, 12.13555096018476, 12.34566936965330, 12.91165893469087, 13.74177417666003, 14.18075653608249, 14.50881600668776, 14.96133194807210