| L(s) = 1 | − 5-s + 11-s − 5·13-s − 2·17-s − 6·19-s − 2·23-s + 25-s + 4·29-s + 9·31-s + 6·37-s + 12·41-s + 43-s − 2·47-s − 6·53-s − 55-s − 59-s + 6·61-s + 5·65-s + 2·67-s − 3·71-s − 3·73-s − 6·79-s + 7·83-s + 2·85-s − 89-s + 6·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.38·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 1/5·25-s + 0.742·29-s + 1.61·31-s + 0.986·37-s + 1.87·41-s + 0.152·43-s − 0.291·47-s − 0.824·53-s − 0.134·55-s − 0.130·59-s + 0.768·61-s + 0.620·65-s + 0.244·67-s − 0.356·71-s − 0.351·73-s − 0.675·79-s + 0.768·83-s + 0.216·85-s − 0.105·89-s + 0.615·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354809095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354809095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−13.89531804400910, −13.11290914296475, −12.84563821567143, −12.25087979137978, −11.92045580232605, −11.36572379112311, −10.85718328572693, −10.34395213979685, −9.798255338976690, −9.410633800854633, −8.735555865603809, −8.260948221520860, −7.794667259393180, −7.283394419967760, −6.611555985062021, −6.292832912965111, −5.666535618128327, −4.762796817431895, −4.481209270454438, −4.103508637855360, −3.193155876863188, −2.501326688127945, −2.224801039639480, −1.157248770381432, −0.3927951890436051,
0.3927951890436051, 1.157248770381432, 2.224801039639480, 2.501326688127945, 3.193155876863188, 4.103508637855360, 4.481209270454438, 4.762796817431895, 5.666535618128327, 6.292832912965111, 6.611555985062021, 7.283394419967760, 7.794667259393180, 8.260948221520860, 8.735555865603809, 9.410633800854633, 9.798255338976690, 10.34395213979685, 10.85718328572693, 11.36572379112311, 11.92045580232605, 12.25087979137978, 12.84563821567143, 13.11290914296475, 13.89531804400910
