L(s) = 1 | + 2·3-s − 7-s + 9-s + 11-s + 4·13-s + 6·19-s − 2·21-s + 3·23-s − 4·27-s − 3·29-s + 2·33-s + 9·37-s + 8·39-s + 2·41-s + 9·43-s − 6·47-s + 49-s + 6·53-s + 12·57-s + 8·59-s − 10·61-s − 63-s − 67-s + 6·69-s − 7·71-s − 2·73-s − 77-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.37·19-s − 0.436·21-s + 0.625·23-s − 0.769·27-s − 0.557·29-s + 0.348·33-s + 1.47·37-s + 1.28·39-s + 0.312·41-s + 1.37·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 1.58·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s − 0.122·67-s + 0.722·69-s − 0.830·71-s − 0.234·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.578870203\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.578870203\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.234557757554469281635943708049, −8.980836199915903149934567896828, −7.959877599159570385491536329376, −7.38322340640296597043186225521, −6.30014882094802389859152162655, −5.48537791264114755262362840859, −4.14978299323561255021731698576, −3.37604612672851245413577971235, −2.60034004996090891588435847585, −1.20821614349589040787995759307,
1.20821614349589040787995759307, 2.60034004996090891588435847585, 3.37604612672851245413577971235, 4.14978299323561255021731698576, 5.48537791264114755262362840859, 6.30014882094802389859152162655, 7.38322340640296597043186225521, 7.959877599159570385491536329376, 8.980836199915903149934567896828, 9.234557757554469281635943708049