L(s) = 1 | + 3-s − 2·4-s + 9-s − 4·11-s − 2·12-s + 3·13-s + 4·16-s + 3·17-s − 7·19-s + 23-s − 5·25-s + 27-s − 29-s + 2·31-s − 4·33-s − 2·36-s + 37-s + 3·39-s − 6·41-s − 11·43-s + 8·44-s + 10·47-s + 4·48-s − 7·49-s + 3·51-s − 6·52-s − 6·53-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.832·13-s + 16-s + 0.727·17-s − 1.60·19-s + 0.208·23-s − 25-s + 0.192·27-s − 0.185·29-s + 0.359·31-s − 0.696·33-s − 1/3·36-s + 0.164·37-s + 0.480·39-s − 0.937·41-s − 1.67·43-s + 1.20·44-s + 1.45·47-s + 0.577·48-s − 49-s + 0.420·51-s − 0.832·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.523254846312252683150776584956, −8.284985457958284839376647745069, −7.47922549795812660000402102002, −6.29807289712189332348722636961, −5.47362976667824023781814191480, −4.59968017945003607154581953176, −3.79091691336369642691462072166, −2.91776941655104758368423858505, −1.62887664718372465237614143290, 0,
1.62887664718372465237614143290, 2.91776941655104758368423858505, 3.79091691336369642691462072166, 4.59968017945003607154581953176, 5.47362976667824023781814191480, 6.29807289712189332348722636961, 7.47922549795812660000402102002, 8.284985457958284839376647745069, 8.523254846312252683150776584956