| L(s) = 1 | + 2·2-s − 8.27·3-s + 4·4-s − 16.5·6-s + 22.8·7-s + 8·8-s + 41.4·9-s − 0.987·11-s − 33.0·12-s − 4.22·13-s + 45.7·14-s + 16·16-s + 73.8·17-s + 82.8·18-s + 71.1·19-s − 189.·21-s − 1.97·22-s + 23·23-s − 66.1·24-s − 8.44·26-s − 119.·27-s + 91.4·28-s + 27.8·29-s − 136.·31-s + 32·32-s + 8.17·33-s + 147.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.59·3-s + 0.5·4-s − 1.12·6-s + 1.23·7-s + 0.353·8-s + 1.53·9-s − 0.0270·11-s − 0.795·12-s − 0.0900·13-s + 0.872·14-s + 0.250·16-s + 1.05·17-s + 1.08·18-s + 0.859·19-s − 1.96·21-s − 0.0191·22-s + 0.208·23-s − 0.562·24-s − 0.0637·26-s − 0.849·27-s + 0.617·28-s + 0.178·29-s − 0.792·31-s + 0.176·32-s + 0.0431·33-s + 0.745·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.383198455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.383198455\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 8.27T + 27T^{2} \) |
| 7 | \( 1 - 22.8T + 343T^{2} \) |
| 11 | \( 1 + 0.987T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.22T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 54.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 428.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 164.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 188.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 932.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 900.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 956.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 628.T + 9.12e5T^{2} \) |
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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
−9.780421028175904092815261067072, −8.405580366124389436459868004928, −7.46166044377749870673521548900, −6.78788795449526997105230346273, −5.64995237733993653043920041603, −5.28780854767617493973328524920, −4.58602833351075500895889568851, −3.43558220590874235892924173526, −1.80814195157938397723932525605, −0.822113403014396558207343522542,
0.822113403014396558207343522542, 1.80814195157938397723932525605, 3.43558220590874235892924173526, 4.58602833351075500895889568851, 5.28780854767617493973328524920, 5.64995237733993653043920041603, 6.78788795449526997105230346273, 7.46166044377749870673521548900, 8.405580366124389436459868004928, 9.780421028175904092815261067072
