| L(s) = 1 | − 5·7-s + 3·11-s − 13-s − 3·17-s − 4·19-s + 6·23-s + 9·29-s − 5·31-s − 2·37-s + 2·43-s − 9·47-s + 18·49-s − 9·53-s + 9·59-s + 61-s + 5·67-s + 14·73-s − 15·77-s + 16·79-s + 15·83-s + 6·89-s + 5·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.904·11-s − 0.277·13-s − 0.727·17-s − 0.917·19-s + 1.25·23-s + 1.67·29-s − 0.898·31-s − 0.328·37-s + 0.304·43-s − 1.31·47-s + 18/7·49-s − 1.23·53-s + 1.17·59-s + 0.128·61-s + 0.610·67-s + 1.63·73-s − 1.70·77-s + 1.80·79-s + 1.64·83-s + 0.635·89-s + 0.524·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520718969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520718969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.03554129951207, −12.72421198741360, −12.28484264689933, −11.85960656343306, −11.12497599607337, −10.80473602369601, −10.24404786171035, −9.713663894980344, −9.393801594234992, −8.871992753490416, −8.601610975231162, −7.812765632292146, −7.181163735185388, −6.601756296090522, −6.451829266732680, −6.183625063851971, −5.049534906879029, −4.956890299649504, −3.975173746795213, −3.678104062528048, −3.121213609797973, −2.521588879394229, −1.978648182425106, −0.9843020821504351, −0.4082036796554712,
0.4082036796554712, 0.9843020821504351, 1.978648182425106, 2.521588879394229, 3.121213609797973, 3.678104062528048, 3.975173746795213, 4.956890299649504, 5.049534906879029, 6.183625063851971, 6.451829266732680, 6.601756296090522, 7.181163735185388, 7.812765632292146, 8.601610975231162, 8.871992753490416, 9.393801594234992, 9.713663894980344, 10.24404786171035, 10.80473602369601, 11.12497599607337, 11.85960656343306, 12.28484264689933, 12.72421198741360, 13.03554129951207
