| L(s) = 1 | + 5-s + 7-s + 4·11-s − 2·17-s + 6·19-s + 6·23-s + 25-s − 2·31-s + 35-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s − 10·53-s + 4·55-s + 4·59-s − 2·61-s − 12·67-s + 8·71-s + 8·73-s + 4·77-s + 8·79-s − 4·83-s − 2·85-s − 10·89-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.485·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.169·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s − 1.46·67-s + 0.949·71-s + 0.936·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.653782862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653782862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.297523715911639602296043348395, −7.39845228700372338746947075699, −6.84076816987454984441631920605, −6.09981769944242705978506745356, −5.28384527182480384396399060087, −4.63466967811446777264769027625, −3.69578381381220375969420298034, −2.88901897177821373074528250162, −1.77116371662178634425845624536, −0.960315949209506161907425934434,
0.960315949209506161907425934434, 1.77116371662178634425845624536, 2.88901897177821373074528250162, 3.69578381381220375969420298034, 4.63466967811446777264769027625, 5.28384527182480384396399060087, 6.09981769944242705978506745356, 6.84076816987454984441631920605, 7.39845228700372338746947075699, 8.297523715911639602296043348395
