| L(s) = 1 | + 2.79·2-s + 3-s + 5.80·4-s + 1.86·5-s + 2.79·6-s − 7-s + 10.6·8-s + 9-s + 5.19·10-s + 1.50·11-s + 5.80·12-s + 4.85·13-s − 2.79·14-s + 1.86·15-s + 18.1·16-s − 6.69·17-s + 2.79·18-s − 6.70·19-s + 10.8·20-s − 21-s + 4.20·22-s + 2.91·23-s + 10.6·24-s − 1.53·25-s + 13.5·26-s + 27-s − 5.80·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.90·4-s + 0.832·5-s + 1.14·6-s − 0.377·7-s + 3.76·8-s + 0.333·9-s + 1.64·10-s + 0.454·11-s + 1.67·12-s + 1.34·13-s − 0.746·14-s + 0.480·15-s + 4.52·16-s − 1.62·17-s + 0.658·18-s − 1.53·19-s + 2.41·20-s − 0.218·21-s + 0.897·22-s + 0.606·23-s + 2.17·24-s − 0.307·25-s + 2.65·26-s + 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.22900856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.22900856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 + 1.93T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.441578692020419048870319220917, −7.27381940846948916255477067639, −6.51076777581502345790911764467, −6.30218224466080368535565737992, −5.42965911859254139715261316199, −4.54470104348489186066631222122, −3.82364950214654470392477347412, −3.27511364386882985104450170568, −2.07520107207440592049727856894, −1.79914341205501113352645174339,
1.79914341205501113352645174339, 2.07520107207440592049727856894, 3.27511364386882985104450170568, 3.82364950214654470392477347412, 4.54470104348489186066631222122, 5.42965911859254139715261316199, 6.30218224466080368535565737992, 6.51076777581502345790911764467, 7.27381940846948916255477067639, 8.441578692020419048870319220917
