L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 6·11-s + 2·14-s + 16-s − 2·17-s − 4·19-s − 20-s + 6·22-s − 23-s + 25-s + 2·28-s + 10·29-s + 4·31-s + 32-s − 2·34-s − 2·35-s − 8·37-s − 4·38-s − 40-s + 8·41-s − 4·43-s + 6·44-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 + 1.80·11-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s + 1.27·22-s − 0.208·23-s + 1/5·25-s + 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 0.338·35-s − 1.31·37-s − 0.648·38-s − 0.158·40-s + 1.24·41-s − 0.609·43-s + 0.904·44-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
\(L(1)\) |
\(\approx\) |
\(3.113324368\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.113324368\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.849922240615540067173014466064, −8.459233800765295516939718761782, −7.41311638972761107890071624716, −6.61437950689788822205221644655, −6.09060882050277505424612314459, −4.79020188392287327724196525973, −4.33242784043968933860492555261, −3.51495836928776825881705604225, −2.27261498022819622696362541242, −1.15508694969457450243039729594,
1.15508694969457450243039729594, 2.27261498022819622696362541242, 3.51495836928776825881705604225, 4.33242784043968933860492555261, 4.79020188392287327724196525973, 6.09060882050277505424612314459, 6.61437950689788822205221644655, 7.41311638972761107890071624716, 8.459233800765295516939718761782, 8.849922240615540067173014466064