L(s) = 1 | − 2-s + 1.89·3-s + 4-s − 5-s − 1.89·6-s − 2.12·7-s − 8-s + 0.607·9-s + 10-s + 0.951·11-s + 1.89·12-s − 13-s + 2.12·14-s − 1.89·15-s + 16-s + 6.77·17-s − 0.607·18-s − 4.09·19-s − 20-s − 4.03·21-s − 0.951·22-s − 2.41·23-s − 1.89·24-s + 25-s + 26-s − 4.54·27-s − 2.12·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.09·3-s + 0.5·4-s − 0.447·5-s − 0.775·6-s − 0.803·7-s − 0.353·8-s + 0.202·9-s + 0.316·10-s + 0.286·11-s + 0.548·12-s − 0.277·13-s + 0.568·14-s − 0.490·15-s + 0.250·16-s + 1.64·17-s − 0.143·18-s − 0.938·19-s − 0.223·20-s − 0.881·21-s − 0.202·22-s − 0.504·23-s − 0.387·24-s + 0.200·25-s + 0.196·26-s − 0.874·27-s − 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 0.951T + 11T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 37 | \( 1 - 2.25T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 0.518T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−8.132429458462929604597865867013, −7.69656983213954640140746552504, −6.73186788373850722464493282486, −6.17422773495151283564817988684, −5.05802945486217142929541201521, −3.87869773717379800749590954708, −3.24230750396841995580840611253, −2.58887937754258001296456341099, −1.43749789558430310551429841497, 0,
1.43749789558430310551429841497, 2.58887937754258001296456341099, 3.24230750396841995580840611253, 3.87869773717379800749590954708, 5.05802945486217142929541201521, 6.17422773495151283564817988684, 6.73186788373850722464493282486, 7.69656983213954640140746552504, 8.132429458462929604597865867013