L(s) = 1 | + 2.51·2-s − 2.78·3-s + 4.34·4-s − 1.47·5-s − 7.01·6-s + 5.91·8-s + 4.75·9-s − 3.72·10-s − 4.83·11-s − 12.1·12-s + 4.02·13-s + 4.11·15-s + 6.20·16-s + 2.82·17-s + 11.9·18-s − 4.32·19-s − 6.42·20-s − 12.1·22-s − 7.27·23-s − 16.4·24-s − 2.81·25-s + 10.1·26-s − 4.89·27-s + 7.18·29-s + 10.3·30-s − 7.03·31-s + 3.81·32-s + ⋯ |
L(s) = 1 | + 1.78·2-s − 1.60·3-s + 2.17·4-s − 0.660·5-s − 2.86·6-s + 2.09·8-s + 1.58·9-s − 1.17·10-s − 1.45·11-s − 3.49·12-s + 1.11·13-s + 1.06·15-s + 1.55·16-s + 0.684·17-s + 2.82·18-s − 0.992·19-s − 1.43·20-s − 2.59·22-s − 1.51·23-s − 3.36·24-s − 0.563·25-s + 1.98·26-s − 0.941·27-s + 1.33·29-s + 1.89·30-s − 1.26·31-s + 0.673·32-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 + 7.03T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 43 | \( 1 - 0.174T + 43T^{2} \) |
| 47 | \( 1 + 5.63T + 47T^{2} \) |
| 53 | \( 1 - 7.57T + 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 6.42T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.428706235210771058993509006400, −7.59328103120221972408000628700, −6.76464964183745229511528212193, −5.92599684638573936033534878470, −5.63195426685967677294935149440, −4.75246630560029441568542260541, −4.09243636093052358145647214447, −3.22241097685389183434689785122, −1.83111405119626231341421797784, 0,
1.83111405119626231341421797784, 3.22241097685389183434689785122, 4.09243636093052358145647214447, 4.75246630560029441568542260541, 5.63195426685967677294935149440, 5.92599684638573936033534878470, 6.76464964183745229511528212193, 7.59328103120221972408000628700, 8.428706235210771058993509006400