| L(s) = 1 | + 4·7-s + 11-s − 13-s + 3·17-s + 2·19-s + 7·23-s + 29-s − 9·31-s + 2·37-s + 10·41-s − 9·43-s − 3·47-s + 9·49-s + 8·53-s + 12·59-s + 8·61-s + 4·67-s − 2·71-s − 14·73-s + 4·77-s − 11·79-s − 4·83-s − 18·89-s − 4·91-s + 2·97-s + 15·101-s + 10·103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 1.45·23-s + 0.185·29-s − 1.61·31-s + 0.328·37-s + 1.56·41-s − 1.37·43-s − 0.437·47-s + 9/7·49-s + 1.09·53-s + 1.56·59-s + 1.02·61-s + 0.488·67-s − 0.237·71-s − 1.63·73-s + 0.455·77-s − 1.23·79-s − 0.439·83-s − 1.90·89-s − 0.419·91-s + 0.203·97-s + 1.49·101-s + 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.676820997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.676820997\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.250753059198372621842863007400, −7.30273175928844977872341062142, −7.10633763081876884008193767755, −5.77929108363887044685246092335, −5.28336656851119826969081029703, −4.60019010793796062254390640988, −3.76308178513999314928544722270, −2.78422602132970708019809056525, −1.76162707944215562858517118786, −0.951765719262476622955965515874,
0.951765719262476622955965515874, 1.76162707944215562858517118786, 2.78422602132970708019809056525, 3.76308178513999314928544722270, 4.60019010793796062254390640988, 5.28336656851119826969081029703, 5.77929108363887044685246092335, 7.10633763081876884008193767755, 7.30273175928844977872341062142, 8.250753059198372621842863007400
