| L(s) = 1 | − 2·2-s − 3.40·3-s + 4·4-s − 0.220·5-s + 6.80·6-s − 8·8-s − 15.4·9-s + 0.441·10-s + 49.1·11-s − 13.6·12-s + 37.0·13-s + 0.750·15-s + 16·16-s − 42.4·17-s + 30.8·18-s − 1.64·19-s − 0.882·20-s − 98.2·22-s + 23·23-s + 27.2·24-s − 124.·25-s − 74.1·26-s + 144.·27-s + 42.1·29-s − 1.50·30-s − 141.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.654·3-s + 0.5·4-s − 0.0197·5-s + 0.462·6-s − 0.353·8-s − 0.571·9-s + 0.0139·10-s + 1.34·11-s − 0.327·12-s + 0.791·13-s + 0.0129·15-s + 0.250·16-s − 0.605·17-s + 0.404·18-s − 0.0198·19-s − 0.00986·20-s − 0.952·22-s + 0.208·23-s + 0.231·24-s − 0.999·25-s − 0.559·26-s + 1.02·27-s + 0.269·29-s − 0.00913·30-s − 0.821·31-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 | \( 1 + 3.40T + 27T^{2} \) |
| 5 | \( 1 + 0.220T + 125T^{2} \) |
| 11 | \( 1 - 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.64T + 6.85e3T^{2} \) |
| 29 | \( 1 - 42.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 211.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 221.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 118.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 434.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 69.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 103.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 8.86T + 3.00e5T^{2} \) |
| 71 | \( 1 + 172.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 291.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 258.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 26.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 35.9T + 9.12e5T^{2} \) |
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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
−8.597528386458488892690196767003, −7.49655019230472612352120198525, −6.63693450666549920657941165580, −6.14063543752360815535948511718, −5.34055040206580622205022260391, −4.16734854506378300448245620529, −3.32260495254504794781919859686, −2.03085937311638423851543153164, −1.05411275825287858249394115683, 0,
1.05411275825287858249394115683, 2.03085937311638423851543153164, 3.32260495254504794781919859686, 4.16734854506378300448245620529, 5.34055040206580622205022260391, 6.14063543752360815535948511718, 6.63693450666549920657941165580, 7.49655019230472612352120198525, 8.597528386458488892690196767003
