| L(s) = 1 | − 2.72·2-s − 3-s + 5.40·4-s − 1.58·5-s + 2.72·6-s + 7-s − 9.26·8-s + 9-s + 4.30·10-s − 5.54·11-s − 5.40·12-s − 2.72·14-s + 1.58·15-s + 14.4·16-s + 4.80·17-s − 2.72·18-s + 0.892·19-s − 8.55·20-s − 21-s + 15.0·22-s − 4.63·23-s + 9.26·24-s − 2.49·25-s − 27-s + 5.40·28-s − 1.88·29-s − 4.30·30-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.70·4-s − 0.707·5-s + 1.11·6-s + 0.377·7-s − 3.27·8-s + 0.333·9-s + 1.36·10-s − 1.67·11-s − 1.56·12-s − 0.727·14-s + 0.408·15-s + 3.60·16-s + 1.16·17-s − 0.641·18-s + 0.204·19-s − 1.91·20-s − 0.218·21-s + 3.21·22-s − 0.965·23-s + 1.89·24-s − 0.498·25-s − 0.192·27-s + 1.02·28-s − 0.349·29-s − 0.786·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 5.54T + 11T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 0.892T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 1.88T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 - 9.75T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 + 3.58T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 - 0.779T + 79T^{2} \) |
| 83 | \( 1 - 1.47T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 0.758T + 97T^{2} \) |
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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
−7.996689518153259695698910042843, −7.62818945666642796743229043756, −7.33605394701276059901678128399, −5.94795807081201436207776022302, −5.66853950612718078911442590270, −4.31485435281660103836297206937, −3.04424382300729983179753237378, −2.18588326086837732949402776631, −0.982033211614878504117878419597, 0,
0.982033211614878504117878419597, 2.18588326086837732949402776631, 3.04424382300729983179753237378, 4.31485435281660103836297206937, 5.66853950612718078911442590270, 5.94795807081201436207776022302, 7.33605394701276059901678128399, 7.62818945666642796743229043756, 7.996689518153259695698910042843
