L(s) = 1 | + 7-s − 3·11-s + 13-s + 6·17-s − 4·19-s − 3·23-s − 3·29-s + 5·31-s − 2·37-s − 3·41-s + 43-s − 9·47-s − 6·49-s − 6·53-s + 3·59-s − 13·61-s + 7·67-s + 12·71-s + 10·73-s − 3·77-s + 11·79-s − 9·83-s − 6·89-s + 91-s − 11·97-s − 15·101-s + 7·103-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.625·23-s − 0.557·29-s + 0.898·31-s − 0.328·37-s − 0.468·41-s + 0.152·43-s − 1.31·47-s − 6/7·49-s − 0.824·53-s + 0.390·59-s − 1.66·61-s + 0.855·67-s + 1.42·71-s + 1.17·73-s − 0.341·77-s + 1.23·79-s − 0.987·83-s − 0.635·89-s + 0.104·91-s − 1.11·97-s − 1.49·101-s + 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−7.75314926578647491564094270446, −6.72133146228986384071249880729, −6.11060893927500307480708984924, −5.30857622281065952949184371760, −4.80169263581830045837240717679, −3.83890095450745910938201224805, −3.13075437159809575402702602064, −2.20057828002240470817538895608, −1.31330919352851385017845162380, 0,
1.31330919352851385017845162380, 2.20057828002240470817538895608, 3.13075437159809575402702602064, 3.83890095450745910938201224805, 4.80169263581830045837240717679, 5.30857622281065952949184371760, 6.11060893927500307480708984924, 6.72133146228986384071249880729, 7.75314926578647491564094270446