| L(s) = 1 | + 2.15·2-s + 2.23·3-s + 2.65·4-s + 3.33·5-s + 4.82·6-s − 7-s + 1.40·8-s + 2.00·9-s + 7.18·10-s + 2.87·11-s + 5.93·12-s − 2.61·13-s − 2.15·14-s + 7.45·15-s − 2.27·16-s − 0.0349·17-s + 4.33·18-s − 4.10·19-s + 8.83·20-s − 2.23·21-s + 6.21·22-s − 5.60·23-s + 3.14·24-s + 6.09·25-s − 5.64·26-s − 2.21·27-s − 2.65·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.29·3-s + 1.32·4-s + 1.48·5-s + 1.97·6-s − 0.377·7-s + 0.497·8-s + 0.669·9-s + 2.27·10-s + 0.868·11-s + 1.71·12-s − 0.725·13-s − 0.576·14-s + 1.92·15-s − 0.567·16-s − 0.00847·17-s + 1.02·18-s − 0.941·19-s + 1.97·20-s − 0.488·21-s + 1.32·22-s − 1.16·23-s + 0.642·24-s + 1.21·25-s − 1.10·26-s − 0.426·27-s − 0.501·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.836322559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.836322559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 227 | \( 1 - T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 0.0349T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 - 0.381T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 2.20T + 43T^{2} \) |
| 47 | \( 1 - 2.71T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 - 0.0500T + 59T^{2} \) |
| 61 | \( 1 + 2.33T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.340092907863176074008559650838, −8.864344284637880730879384054737, −7.69163870285436945816060648647, −6.61060450935244733898434851594, −6.11731135693948862967962926499, −5.26594872190731723715366041913, −4.20805016453367378068946696039, −3.47758532623766791432983188750, −2.41633734484524119394957306116, −2.01059852256431531292789966983,
2.01059852256431531292789966983, 2.41633734484524119394957306116, 3.47758532623766791432983188750, 4.20805016453367378068946696039, 5.26594872190731723715366041913, 6.11731135693948862967962926499, 6.61060450935244733898434851594, 7.69163870285436945816060648647, 8.864344284637880730879384054737, 9.340092907863176074008559650838
