| L(s) = 1 | + 2.57·2-s − 0.468·3-s + 4.62·4-s − 2.24·5-s − 1.20·6-s + 6.76·8-s − 2.78·9-s − 5.77·10-s + 5.76·11-s − 2.16·12-s + 0.0725·13-s + 1.04·15-s + 8.15·16-s + 5.01·17-s − 7.15·18-s + 5.06·19-s − 10.3·20-s + 14.8·22-s + 1.90·23-s − 3.16·24-s + 0.0276·25-s + 0.186·26-s + 2.70·27-s + 6.90·29-s + 2.70·30-s − 4.57·31-s + 7.46·32-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.270·3-s + 2.31·4-s − 1.00·5-s − 0.492·6-s + 2.39·8-s − 0.926·9-s − 1.82·10-s + 1.73·11-s − 0.625·12-s + 0.0201·13-s + 0.271·15-s + 2.03·16-s + 1.21·17-s − 1.68·18-s + 1.16·19-s − 2.31·20-s + 3.16·22-s + 0.398·23-s − 0.646·24-s + 0.00552·25-s + 0.0366·26-s + 0.520·27-s + 1.28·29-s + 0.493·30-s − 0.821·31-s + 1.31·32-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
| \(L(1)\) |
\(\approx\) |
\(4.600285620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.600285620\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.468T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 0.0725T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 2.59T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 3.95T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 4.84T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.062897421162630755943568160095, −8.140773129774607924049186594731, −7.16875247743853564734242227546, −6.66884123640607924444076225226, −5.66892979305142919863225005724, −5.19782887323166692965987076395, −4.05843132419946966281918452770, −3.60860272190391817974338224598, −2.81621036995446768520012674575, −1.22127545248555124903326398244,
1.22127545248555124903326398244, 2.81621036995446768520012674575, 3.60860272190391817974338224598, 4.05843132419946966281918452770, 5.19782887323166692965987076395, 5.66892979305142919863225005724, 6.66884123640607924444076225226, 7.16875247743853564734242227546, 8.140773129774607924049186594731, 9.062897421162630755943568160095
