L(s) = 1 | − 3-s + 3·7-s + 9-s − 5·13-s + 4·17-s + 7·19-s − 3·21-s − 4·23-s − 27-s + 3·29-s − 3·31-s + 4·37-s + 5·39-s − 11·41-s + 43-s − 2·47-s + 2·49-s − 4·51-s − 10·53-s − 7·57-s + 4·59-s − 11·61-s + 3·63-s + 67-s + 4·69-s + 14·71-s + 3·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 1.38·13-s + 0.970·17-s + 1.60·19-s − 0.654·21-s − 0.834·23-s − 0.192·27-s + 0.557·29-s − 0.538·31-s + 0.657·37-s + 0.800·39-s − 1.71·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s − 0.560·51-s − 1.37·53-s − 0.927·57-s + 0.520·59-s − 1.40·61-s + 0.377·63-s + 0.122·67-s + 0.481·69-s + 1.66·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.877515684\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.877515684\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.40906037003660, −14.08008114401069, −13.62780459087264, −12.76597420263369, −12.23088252308026, −12.02583382674738, −11.33439137141287, −11.15699921161496, −10.19796208897208, −9.910351250186290, −9.526478638760298, −8.680041677867386, −7.937167788362951, −7.744644384855845, −7.157975081771095, −6.553960630254434, −5.691848283931110, −5.340311482324270, −4.823426334878694, −4.368266148602139, −3.428388654594990, −2.872378243589648, −1.905138496087442, −1.414948772393753, −0.5105473533061555,
0.5105473533061555, 1.414948772393753, 1.905138496087442, 2.872378243589648, 3.428388654594990, 4.368266148602139, 4.823426334878694, 5.340311482324270, 5.691848283931110, 6.553960630254434, 7.157975081771095, 7.744644384855845, 7.937167788362951, 8.680041677867386, 9.526478638760298, 9.910351250186290, 10.19796208897208, 11.15699921161496, 11.33439137141287, 12.02583382674738, 12.23088252308026, 12.76597420263369, 13.62780459087264, 14.08008114401069, 14.40906037003660