| L(s) = 1 | − 4·7-s − 4·11-s + 2·13-s + 6·17-s + 6·19-s + 8·23-s + 6·29-s − 2·31-s + 37-s − 10·43-s − 12·47-s + 9·49-s − 4·53-s + 4·59-s + 10·61-s − 4·67-s + 12·71-s + 10·73-s + 16·77-s − 10·79-s − 2·89-s − 8·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.37·19-s + 1.66·23-s + 1.11·29-s − 0.359·31-s + 0.164·37-s − 1.52·43-s − 1.75·47-s + 9/7·49-s − 0.549·53-s + 0.520·59-s + 1.28·61-s − 0.488·67-s + 1.42·71-s + 1.17·73-s + 1.82·77-s − 1.12·79-s − 0.211·89-s − 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.41436606501903, −13.25375732767287, −12.87227747751057, −12.31636207105901, −11.88562642742184, −11.23839361832293, −10.82749474401543, −10.06661473799618, −9.925631999544562, −9.528984862579101, −8.872517680840263, −8.238206761801786, −7.890691768345138, −7.222381299408995, −6.755491862090029, −6.365439979972248, −5.600043121895201, −5.212038994556284, −4.871364393289666, −3.776985243330643, −3.279596889342851, −3.072636264639358, −2.482407192109412, −1.363815273018851, −0.8529938363147937, 0,
0.8529938363147937, 1.363815273018851, 2.482407192109412, 3.072636264639358, 3.279596889342851, 3.776985243330643, 4.871364393289666, 5.212038994556284, 5.600043121895201, 6.365439979972248, 6.755491862090029, 7.222381299408995, 7.890691768345138, 8.238206761801786, 8.872517680840263, 9.528984862579101, 9.925631999544562, 10.06661473799618, 10.82749474401543, 11.23839361832293, 11.88562642742184, 12.31636207105901, 12.87227747751057, 13.25375732767287, 13.41436606501903
