| L(s) = 1 | + 9·3-s − 5·5-s + 7·7-s + 54·9-s + 55·11-s + 69·13-s − 45·15-s + 113·17-s − 126·19-s + 63·21-s + 102·23-s + 25·25-s + 243·27-s + 81·29-s − 176·31-s + 495·33-s − 35·35-s − 254·37-s + 621·39-s − 184·41-s − 230·43-s − 270·45-s + 187·47-s + 49·49-s + 1.01e3·51-s + 488·53-s − 275·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 1.47·13-s − 0.774·15-s + 1.61·17-s − 1.52·19-s + 0.654·21-s + 0.924·23-s + 1/5·25-s + 1.73·27-s + 0.518·29-s − 1.01·31-s + 2.61·33-s − 0.169·35-s − 1.12·37-s + 2.54·39-s − 0.700·41-s − 0.815·43-s − 0.894·45-s + 0.580·47-s + 1/7·49-s + 2.79·51-s + 1.26·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.735524666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.735524666\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 69 T + p^{3} T^{2} \) |
| 17 | \( 1 - 113 T + p^{3} T^{2} \) |
| 19 | \( 1 + 126 T + p^{3} T^{2} \) |
| 23 | \( 1 - 102 T + p^{3} T^{2} \) |
| 29 | \( 1 - 81 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 184 T + p^{3} T^{2} \) |
| 43 | \( 1 + 230 T + p^{3} T^{2} \) |
| 47 | \( 1 - 187 T + p^{3} T^{2} \) |
| 53 | \( 1 - 488 T + p^{3} T^{2} \) |
| 59 | \( 1 - 388 T + p^{3} T^{2} \) |
| 61 | \( 1 - 728 T + p^{3} T^{2} \) |
| 67 | \( 1 + 96 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 994 T + p^{3} T^{2} \) |
| 79 | \( 1 + 337 T + p^{3} T^{2} \) |
| 83 | \( 1 - 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 884 T + p^{3} T^{2} \) |
| 97 | \( 1 + 451 T + p^{3} T^{2} \) |
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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
−8.507556770264154708580813005267, −8.346327070094946883703471023522, −7.20613193615654818814716441322, −6.69386956690329443140595429760, −5.49126652773867455688371001737, −4.11391681283706557104227766312, −3.79125428671953315412981563848, −3.03230054913908409176426792859, −1.76379068488396042402757031122, −1.12488843612381535006087033778,
1.12488843612381535006087033778, 1.76379068488396042402757031122, 3.03230054913908409176426792859, 3.79125428671953315412981563848, 4.11391681283706557104227766312, 5.49126652773867455688371001737, 6.69386956690329443140595429760, 7.20613193615654818814716441322, 8.346327070094946883703471023522, 8.507556770264154708580813005267
