| L(s) = 1 | + 3·7-s − 3·11-s + 13-s + 3·17-s − 4·19-s + 3·23-s + 10·29-s − 6·31-s − 5·37-s + 5·41-s + 2·43-s − 2·47-s + 2·49-s − 11·53-s − 4·59-s − 61-s − 4·67-s + 3·71-s − 6·73-s − 9·77-s + 3·79-s + 16·83-s + 7·89-s + 3·91-s + 19·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s + 0.625·23-s + 1.85·29-s − 1.07·31-s − 0.821·37-s + 0.780·41-s + 0.304·43-s − 0.291·47-s + 2/7·49-s − 1.51·53-s − 0.520·59-s − 0.128·61-s − 0.488·67-s + 0.356·71-s − 0.702·73-s − 1.02·77-s + 0.337·79-s + 1.75·83-s + 0.741·89-s + 0.314·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.773599344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.773599344\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.10788810376025, −12.59885296741835, −12.26037786528553, −11.70630847640405, −11.12771415547824, −10.78650862013003, −10.42187815598774, −9.969346538531236, −9.168019477006178, −8.839618814685343, −8.300411098228561, −7.766101862714549, −7.634327793866784, −6.840853026204818, −6.305802807510145, −5.795374802510032, −5.171677548765789, −4.730833818756526, −4.465914019557590, −3.508408270558805, −3.148333418726823, −2.364917724874386, −1.870000080640515, −1.206038863256348, −0.4949504985960706,
0.4949504985960706, 1.206038863256348, 1.870000080640515, 2.364917724874386, 3.148333418726823, 3.508408270558805, 4.465914019557590, 4.730833818756526, 5.171677548765789, 5.795374802510032, 6.305802807510145, 6.840853026204818, 7.634327793866784, 7.766101862714549, 8.300411098228561, 8.839618814685343, 9.168019477006178, 9.969346538531236, 10.42187815598774, 10.78650862013003, 11.12771415547824, 11.70630847640405, 12.26037786528553, 12.59885296741835, 13.10788810376025
