| L(s) = 1 | + 1.20·2-s − 2.59·3-s − 0.545·4-s + 2.91·5-s − 3.12·6-s + 2.87·7-s − 3.06·8-s + 3.72·9-s + 3.52·10-s − 1.69·11-s + 1.41·12-s − 2.86·13-s + 3.46·14-s − 7.56·15-s − 2.61·16-s − 6.32·17-s + 4.49·18-s + 0.209·19-s − 1.59·20-s − 7.45·21-s − 2.04·22-s + 1.70·23-s + 7.96·24-s + 3.52·25-s − 3.46·26-s − 1.87·27-s − 1.56·28-s + ⋯ |
| L(s) = 1 | + 0.852·2-s − 1.49·3-s − 0.272·4-s + 1.30·5-s − 1.27·6-s + 1.08·7-s − 1.08·8-s + 1.24·9-s + 1.11·10-s − 0.511·11-s + 0.408·12-s − 0.795·13-s + 0.927·14-s − 1.95·15-s − 0.653·16-s − 1.53·17-s + 1.05·18-s + 0.0481·19-s − 0.355·20-s − 1.62·21-s − 0.435·22-s + 0.355·23-s + 1.62·24-s + 0.704·25-s − 0.678·26-s − 0.361·27-s − 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2011 | \( 1 + T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 0.209T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.91T + 71T^{2} \) |
| 73 | \( 1 + 0.119T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 + 0.303T + 83T^{2} \) |
| 89 | \( 1 - 2.99T + 89T^{2} \) |
| 97 | \( 1 + 0.230T + 97T^{2} \) |
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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
−9.001190683570909230992590120447, −7.84994563634699237612672144739, −6.69752760121233186719965918011, −6.13351569609291945145434721680, −5.33208415696002783306772708090, −4.94243398175650052130069507814, −4.37401645069746464618858927030, −2.71750341541323562153110302235, −1.64410416553037196493642548558, 0,
1.64410416553037196493642548558, 2.71750341541323562153110302235, 4.37401645069746464618858927030, 4.94243398175650052130069507814, 5.33208415696002783306772708090, 6.13351569609291945145434721680, 6.69752760121233186719965918011, 7.84994563634699237612672144739, 9.001190683570909230992590120447
