| L(s) = 1 | + 3·7-s − 5·11-s − 4·13-s + 8·17-s − 2·19-s + 2·23-s + 6·29-s + 7·31-s + 6·37-s − 6·41-s − 2·43-s + 6·47-s + 2·49-s − 5·53-s + 4·59-s − 8·61-s − 10·67-s + 8·71-s − 73-s − 15·77-s − 16·79-s − 11·83-s + 6·89-s − 12·91-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.50·11-s − 1.10·13-s + 1.94·17-s − 0.458·19-s + 0.417·23-s + 1.11·29-s + 1.25·31-s + 0.986·37-s − 0.937·41-s − 0.304·43-s + 0.875·47-s + 2/7·49-s − 0.686·53-s + 0.520·59-s − 1.02·61-s − 1.22·67-s + 0.949·71-s − 0.117·73-s − 1.70·77-s − 1.80·79-s − 1.20·83-s + 0.635·89-s − 1.25·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.152212863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152212863\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−16.70043579387990, −15.86088682672610, −15.33781830112299, −14.81650459253272, −14.24888662045487, −13.85150883503672, −12.97559832248110, −12.51961654143163, −11.85911662995750, −11.45531433983895, −10.44960479564531, −10.28129278107669, −9.678677709420043, −8.656994324785518, −8.082711241691311, −7.714389692117788, −7.157120437254555, −6.120424572568034, −5.425216022094519, −4.856292659161429, −4.446941541448253, −3.115287765468090, −2.680266512845181, −1.697441514485627, −0.6920144180137361,
0.6920144180137361, 1.697441514485627, 2.680266512845181, 3.115287765468090, 4.446941541448253, 4.856292659161429, 5.425216022094519, 6.120424572568034, 7.157120437254555, 7.714389692117788, 8.082711241691311, 8.656994324785518, 9.678677709420043, 10.28129278107669, 10.44960479564531, 11.45531433983895, 11.85911662995750, 12.51961654143163, 12.97559832248110, 13.85150883503672, 14.24888662045487, 14.81650459253272, 15.33781830112299, 15.86088682672610, 16.70043579387990
