L(s) = 1 | + 2·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s + 19-s − 2·23-s − 5·25-s + 2·29-s + 8·31-s − 10·37-s + 6·41-s + 4·43-s + 2·47-s − 3·49-s + 2·53-s − 12·59-s − 8·61-s − 6·63-s − 4·67-s − 12·71-s + 10·73-s − 8·77-s − 4·79-s + 9·81-s − 4·83-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s − 0.417·23-s − 25-s + 0.371·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s − 1.02·61-s − 0.755·63-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.911·77-s − 0.450·79-s + 81-s − 0.439·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.000555338118287366802045004402, −7.43056536951446491133299904047, −6.27312342430781451668950357662, −5.69507693605672396917445129315, −5.06378007645009501123963447883, −4.23077688008870743761779697587, −3.17610016756607359733112130734, −2.49617533529582800680760769255, −1.39427811435276876748283416588, 0,
1.39427811435276876748283416588, 2.49617533529582800680760769255, 3.17610016756607359733112130734, 4.23077688008870743761779697587, 5.06378007645009501123963447883, 5.69507693605672396917445129315, 6.27312342430781451668950357662, 7.43056536951446491133299904047, 8.000555338118287366802045004402