| L(s) = 1 | + 0.931·2-s + 3-s − 1.13·4-s + 1.13·5-s + 0.931·6-s − 2.87·7-s − 2.91·8-s + 9-s + 1.06·10-s + 11-s − 1.13·12-s + 1.40·13-s − 2.67·14-s + 1.13·15-s − 0.453·16-s − 0.846·17-s + 0.931·18-s + 4.27·19-s − 1.28·20-s − 2.87·21-s + 0.931·22-s + 5.20·23-s − 2.91·24-s − 3.70·25-s + 1.30·26-s + 27-s + 3.25·28-s + ⋯ |
| L(s) = 1 | + 0.658·2-s + 0.577·3-s − 0.566·4-s + 0.509·5-s + 0.380·6-s − 1.08·7-s − 1.03·8-s + 0.333·9-s + 0.335·10-s + 0.301·11-s − 0.326·12-s + 0.389·13-s − 0.715·14-s + 0.293·15-s − 0.113·16-s − 0.205·17-s + 0.219·18-s + 0.981·19-s − 0.288·20-s − 0.627·21-s + 0.198·22-s + 1.08·23-s − 0.595·24-s − 0.740·25-s + 0.256·26-s + 0.192·27-s + 0.614·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527858040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527858040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.931T + 2T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 + 0.846T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 0.166T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 - 0.378T + 47T^{2} \) |
| 53 | \( 1 + 0.923T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.212395864689445386302948081968, −8.584687769822883273065907368489, −7.58810451696662356121550221950, −6.49385405907633592585683470544, −6.05763891573124581204314823108, −5.01308144702236229830583295123, −4.18781744487296383877420358449, −3.25476527489992079364641018936, −2.68769632727280908357087957327, −0.969714227629449261241428208165,
0.969714227629449261241428208165, 2.68769632727280908357087957327, 3.25476527489992079364641018936, 4.18781744487296383877420358449, 5.01308144702236229830583295123, 6.05763891573124581204314823108, 6.49385405907633592585683470544, 7.58810451696662356121550221950, 8.584687769822883273065907368489, 9.212395864689445386302948081968
