| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 2·11-s − 3·13-s + 14-s + 16-s + 17-s + 20-s − 2·22-s + 3·23-s + 25-s + 3·26-s − 28-s + 3·29-s − 31-s − 32-s − 34-s − 35-s − 4·37-s − 40-s + 10·41-s + 43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.223·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s + 0.588·26-s − 0.188·28-s + 0.557·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s − 0.169·35-s − 0.657·37-s − 0.158·40-s + 1.56·41-s + 0.152·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287227801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287227801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.201083011878925058568508602928, −8.677223137643702261252067771868, −7.57168811898548713089886538984, −7.00089154525091260537202626506, −6.16514613608994129930558514176, −5.35265791560478832880711336693, −4.24407251799714467038164439819, −3.07253292824470384429806850260, −2.15129871944255289534696773590, −0.859351320544949077152406649964,
0.859351320544949077152406649964, 2.15129871944255289534696773590, 3.07253292824470384429806850260, 4.24407251799714467038164439819, 5.35265791560478832880711336693, 6.16514613608994129930558514176, 7.00089154525091260537202626506, 7.57168811898548713089886538984, 8.677223137643702261252067771868, 9.201083011878925058568508602928
