| L(s) = 1 | + 1.30·2-s + 1.82·3-s − 0.294·4-s − 0.323·5-s + 2.37·6-s − 2.99·8-s + 0.314·9-s − 0.422·10-s − 4.02·11-s − 0.535·12-s − 2.56·13-s − 0.588·15-s − 3.32·16-s + 0.768·17-s + 0.411·18-s − 4.01·19-s + 0.0950·20-s − 5.25·22-s + 2.94·23-s − 5.45·24-s − 4.89·25-s − 3.34·26-s − 4.88·27-s − 2.90·29-s − 0.768·30-s − 0.256·31-s + 1.65·32-s + ⋯ |
| L(s) = 1 | + 0.923·2-s + 1.05·3-s − 0.147·4-s − 0.144·5-s + 0.970·6-s − 1.05·8-s + 0.104·9-s − 0.133·10-s − 1.21·11-s − 0.154·12-s − 0.710·13-s − 0.151·15-s − 0.831·16-s + 0.186·17-s + 0.0969·18-s − 0.921·19-s + 0.0212·20-s − 1.11·22-s + 0.614·23-s − 1.11·24-s − 0.979·25-s − 0.655·26-s − 0.940·27-s − 0.539·29-s − 0.140·30-s − 0.0461·31-s + 0.291·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 + 0.323T + 5T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 - 0.768T + 17T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 0.256T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 + 0.790T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 6.55T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 5.04T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−8.737185318975845644833447827561, −7.984279237592262791383850733447, −7.38919227166934588112430574220, −6.15484224003274146352796185080, −5.39729803040531390882078563962, −4.58922699385755973949344584742, −3.74543472215281827455551238522, −2.88916594961118213253952255553, −2.23614339626674393559044339824, 0,
2.23614339626674393559044339824, 2.88916594961118213253952255553, 3.74543472215281827455551238522, 4.58922699385755973949344584742, 5.39729803040531390882078563962, 6.15484224003274146352796185080, 7.38919227166934588112430574220, 7.984279237592262791383850733447, 8.737185318975845644833447827561
