L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 2·11-s − 2·13-s − 2·14-s + 16-s + 8·19-s − 20-s − 2·22-s + 23-s + 25-s − 2·26-s − 2·28-s + 10·29-s + 8·31-s + 32-s + 2·35-s + 8·37-s + 8·38-s − 40-s + 6·41-s + 12·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.83·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.377·28-s + 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.338·35-s + 1.31·37-s + 1.29·38-s − 0.158·40-s + 0.937·41-s + 1.82·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.317444424\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.317444424\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.283348689935136161569513079241, −8.040277062943398585197403562446, −7.57644602657472282166780449391, −6.66130389553996223625924184752, −5.95275754254527558239009243866, −4.97067027675431769602986746565, −4.34546863210538081926021268093, −3.08125036920220032976699659501, −2.74454852585105864639857770974, −0.917064197038992630100605264992,
0.917064197038992630100605264992, 2.74454852585105864639857770974, 3.08125036920220032976699659501, 4.34546863210538081926021268093, 4.97067027675431769602986746565, 5.95275754254527558239009243866, 6.66130389553996223625924184752, 7.57644602657472282166780449391, 8.040277062943398585197403562446, 9.283348689935136161569513079241