L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 3·7-s + 8-s + 6·9-s + 11-s − 3·12-s + 5·13-s − 3·14-s + 16-s + 4·17-s + 6·18-s − 8·19-s + 9·21-s + 22-s − 7·23-s − 3·24-s − 5·25-s + 5·26-s − 9·27-s − 3·28-s − 6·29-s + 2·31-s + 32-s − 3·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 1.13·7-s + 0.353·8-s + 2·9-s + 0.301·11-s − 0.866·12-s + 1.38·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 1.83·19-s + 1.96·21-s + 0.213·22-s − 1.45·23-s − 0.612·24-s − 25-s + 0.980·26-s − 1.73·27-s − 0.566·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.07242692069717643850111085072, −8.776639347270338139701112953897, −7.49583351412009405961962593268, −6.41172248240938673674332701340, −6.15204706147824673386685201199, −5.49162634992719629246345351544, −4.19021899042341320052142776997, −3.60928783415351637090625353762, −1.70548889593404928895143287271, 0,
1.70548889593404928895143287271, 3.60928783415351637090625353762, 4.19021899042341320052142776997, 5.49162634992719629246345351544, 6.15204706147824673386685201199, 6.41172248240938673674332701340, 7.49583351412009405961962593268, 8.776639347270338139701112953897, 10.07242692069717643850111085072