L(s) = 1 | − 2·7-s − 11-s − 4·13-s + 6·17-s + 4·19-s + 6·23-s − 5·25-s − 6·29-s − 8·31-s − 10·37-s − 6·41-s − 8·43-s − 6·47-s − 3·49-s + 8·61-s + 4·67-s + 6·71-s + 2·73-s + 2·77-s − 14·79-s − 12·83-s + 6·89-s + 8·91-s + 14·97-s − 6·101-s + 4·103-s − 12·107-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s − 1.64·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 1.02·61-s + 0.488·67-s + 0.712·71-s + 0.234·73-s + 0.227·77-s − 1.57·79-s − 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s + 0.394·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.245228842316309160133095693960, −8.129852226719742477393784673308, −7.33998580522115612678977642043, −6.79963832425739493066244038791, −5.43952609375822489573657963774, −5.21184114914749280771382385734, −3.64282367595579751815074401037, −3.08131485219503756183708188012, −1.70794665150414314375514308074, 0,
1.70794665150414314375514308074, 3.08131485219503756183708188012, 3.64282367595579751815074401037, 5.21184114914749280771382385734, 5.43952609375822489573657963774, 6.79963832425739493066244038791, 7.33998580522115612678977642043, 8.129852226719742477393784673308, 9.245228842316309160133095693960