| L(s) = 1 | − 3-s + 7-s + 9-s − 3·11-s − 2·17-s + 8·19-s − 21-s − 23-s − 27-s − 9·29-s + 6·31-s + 3·33-s − 3·37-s − 8·41-s + 7·43-s − 8·47-s + 49-s + 2·51-s − 2·53-s − 8·57-s + 12·61-s + 63-s + 67-s + 69-s − 3·71-s − 2·73-s − 3·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.485·17-s + 1.83·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.67·29-s + 1.07·31-s + 0.522·33-s − 0.493·37-s − 1.24·41-s + 1.06·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 1.05·57-s + 1.53·61-s + 0.125·63-s + 0.122·67-s + 0.120·69-s − 0.356·71-s − 0.234·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.44974800818356008271444586557, −6.83264197806978042684580387229, −5.94306704479859832114389078068, −5.28820867636925972612572097648, −4.89908806771410433648024195261, −3.91605610478166170305428853570, −3.10440386692238502677243360582, −2.14947588932351565432975210898, −1.18281806632643958260268409135, 0,
1.18281806632643958260268409135, 2.14947588932351565432975210898, 3.10440386692238502677243360582, 3.91605610478166170305428853570, 4.89908806771410433648024195261, 5.28820867636925972612572097648, 5.94306704479859832114389078068, 6.83264197806978042684580387229, 7.44974800818356008271444586557
