| L(s) = 1 | + 2.73·2-s − 2.52·3-s + 5.48·4-s + 1.62·5-s − 6.89·6-s + 9.52·8-s + 3.36·9-s + 4.45·10-s − 1.21·11-s − 13.8·12-s + 3.49·13-s − 4.10·15-s + 15.0·16-s − 3.59·17-s + 9.19·18-s + 0.898·19-s + 8.92·20-s − 3.32·22-s + 8.47·23-s − 24.0·24-s − 2.35·25-s + 9.55·26-s − 0.915·27-s − 7.30·29-s − 11.2·30-s + 10.0·31-s + 22.1·32-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 1.45·3-s + 2.74·4-s + 0.727·5-s − 2.81·6-s + 3.36·8-s + 1.12·9-s + 1.40·10-s − 0.367·11-s − 3.99·12-s + 0.969·13-s − 1.06·15-s + 3.76·16-s − 0.871·17-s + 2.16·18-s + 0.206·19-s + 1.99·20-s − 0.709·22-s + 1.76·23-s − 4.90·24-s − 0.470·25-s + 1.87·26-s − 0.176·27-s − 1.35·29-s − 2.05·30-s + 1.80·31-s + 3.92·32-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
| \(L(1)\) |
\(\approx\) |
\(4.616746352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.616746352\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 - 0.898T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.346T + 37T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 + 0.195T + 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 0.144T + 61T^{2} \) |
| 67 | \( 1 + 2.58T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−9.408521058405786468859962732817, −8.006488697624458557849466408976, −6.81313813693087232044111400278, −6.51364160952434413441227886185, −5.75206907056994323897463401419, −5.20321549743310703805260770418, −4.59285407083412328614226546892, −3.56324358687804245010206711972, −2.46926047171997146639078818484, −1.29702480152933380300248328224,
1.29702480152933380300248328224, 2.46926047171997146639078818484, 3.56324358687804245010206711972, 4.59285407083412328614226546892, 5.20321549743310703805260770418, 5.75206907056994323897463401419, 6.51364160952434413441227886185, 6.81313813693087232044111400278, 8.006488697624458557849466408976, 9.408521058405786468859962732817
