| L(s) = 1 | + 2·2-s − 5.77·3-s + 4·4-s + 3.54·5-s − 11.5·6-s + 8·8-s + 6.31·9-s + 7.08·10-s − 6·11-s − 23.0·12-s − 21.4·13-s − 20.4·15-s + 16·16-s + 7.36·17-s + 12.6·18-s + 93.4·19-s + 14.1·20-s − 12·22-s + 23·23-s − 46.1·24-s − 112.·25-s − 42.9·26-s + 119.·27-s + 112.·29-s − 40.9·30-s − 286.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.316·5-s − 0.785·6-s + 0.353·8-s + 0.233·9-s + 0.224·10-s − 0.164·11-s − 0.555·12-s − 0.458·13-s − 0.352·15-s + 0.250·16-s + 0.105·17-s + 0.165·18-s + 1.12·19-s + 0.158·20-s − 0.116·22-s + 0.208·23-s − 0.392·24-s − 0.899·25-s − 0.324·26-s + 0.850·27-s + 0.720·29-s − 0.248·30-s − 1.65·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.77T + 27T^{2} \) |
| 5 | \( 1 - 3.54T + 125T^{2} \) |
| 11 | \( 1 + 6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.36T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 286.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 59.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 62.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 507.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 536.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 187.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 49.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 778.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 289.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 50.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 807.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 946.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 715.T + 9.12e5T^{2} \) |
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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
−8.090536348939550719534202170578, −7.23751385200936398460061222504, −6.57579829511820589130392500583, −5.54919293971300598348733980328, −5.44465835973939616448873339390, −4.45672545316847206766154441900, −3.41980830657220066875069720155, −2.40673171734908573687266624004, −1.21576741934918205902719968133, 0,
1.21576741934918205902719968133, 2.40673171734908573687266624004, 3.41980830657220066875069720155, 4.45672545316847206766154441900, 5.44465835973939616448873339390, 5.54919293971300598348733980328, 6.57579829511820589130392500583, 7.23751385200936398460061222504, 8.090536348939550719534202170578
