| L(s) = 1 | + 2-s − 1.46·3-s + 4-s + 2.14·5-s − 1.46·6-s + 2.84·7-s + 8-s − 0.862·9-s + 2.14·10-s − 4.14·11-s − 1.46·12-s + 5.72·13-s + 2.84·14-s − 3.13·15-s + 16-s + 0.828·17-s − 0.862·18-s − 19-s + 2.14·20-s − 4.16·21-s − 4.14·22-s + 4.11·23-s − 1.46·24-s − 0.402·25-s + 5.72·26-s + 5.64·27-s + 2.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.844·3-s + 0.5·4-s + 0.958·5-s − 0.596·6-s + 1.07·7-s + 0.353·8-s − 0.287·9-s + 0.678·10-s − 1.24·11-s − 0.422·12-s + 1.58·13-s + 0.760·14-s − 0.809·15-s + 0.250·16-s + 0.200·17-s − 0.203·18-s − 0.229·19-s + 0.479·20-s − 0.907·21-s − 0.883·22-s + 0.857·23-s − 0.298·24-s − 0.0804·25-s + 1.12·26-s + 1.08·27-s + 0.537·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.430180788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.430180788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 + 0.741T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 0.0154T + 79T^{2} \) |
| 83 | \( 1 - 5.25T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 9.00T + 97T^{2} \) |
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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
−7.967361426339482163444432663674, −6.74217715791830328946530845776, −6.30075417046773193237007317989, −5.58768095731770946084402119721, −5.16237534190177365854296313081, −4.66184817816416869669320045040, −3.52405043420001991979971600856, −2.66809401701116827664512414800, −1.80458160152447498757900666323, −0.895640247847869999687669837710,
0.895640247847869999687669837710, 1.80458160152447498757900666323, 2.66809401701116827664512414800, 3.52405043420001991979971600856, 4.66184817816416869669320045040, 5.16237534190177365854296313081, 5.58768095731770946084402119721, 6.30075417046773193237007317989, 6.74217715791830328946530845776, 7.967361426339482163444432663674
