| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 11-s − 12-s + 2·13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 22-s + 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 6·29-s + 30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331886409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331886409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.263245916652137648768056548346, −7.35284289134810156595966275878, −6.73034022983676842963480004857, −6.17979730105811009829056357854, −5.24782021011330946478114343156, −4.78445616198338088336599939909, −3.47722324865831194243662083729, −2.72558452751296671636591477988, −1.57147525857787416140079301837, −0.75049248480518745009477310682,
0.75049248480518745009477310682, 1.57147525857787416140079301837, 2.72558452751296671636591477988, 3.47722324865831194243662083729, 4.78445616198338088336599939909, 5.24782021011330946478114343156, 6.17979730105811009829056357854, 6.73034022983676842963480004857, 7.35284289134810156595966275878, 8.263245916652137648768056548346
