L(s) = 1 | − 5-s + 7-s − 3·11-s − 4·13-s − 6·17-s + 7·19-s − 6·23-s + 25-s − 3·29-s + 4·31-s − 35-s + 2·37-s + 6·41-s + 43-s + 12·47-s − 6·49-s + 3·55-s − 3·59-s + 61-s + 4·65-s − 5·67-s − 12·71-s − 16·73-s − 3·77-s + 10·79-s + 9·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.904·11-s − 1.10·13-s − 1.45·17-s + 1.60·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.152·43-s + 1.75·47-s − 6/7·49-s + 0.404·55-s − 0.390·59-s + 0.128·61-s + 0.496·65-s − 0.610·67-s − 1.42·71-s − 1.87·73-s − 0.341·77-s + 1.12·79-s + 0.987·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6731439179\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6731439179\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.40480776895459, −13.13513516798135, −12.32372727136077, −12.04812294779001, −11.66079236264340, −10.96945911064778, −10.74317359374065, −10.02614565117094, −9.638361051850985, −9.126385752140507, −8.552758980704715, −7.942456448340379, −7.536904365766406, −7.304174481663165, −6.570242168376437, −5.888720061825823, −5.429293206519788, −4.856590840963136, −4.335981225827289, −3.958216724404831, −2.940242222528139, −2.658957105869470, −2.009866570400865, −1.200080644930245, −0.2522569839927006,
0.2522569839927006, 1.200080644930245, 2.009866570400865, 2.658957105869470, 2.940242222528139, 3.958216724404831, 4.335981225827289, 4.856590840963136, 5.429293206519788, 5.888720061825823, 6.570242168376437, 7.304174481663165, 7.536904365766406, 7.942456448340379, 8.552758980704715, 9.126385752140507, 9.638361051850985, 10.02614565117094, 10.74317359374065, 10.96945911064778, 11.66079236264340, 12.04812294779001, 12.32372727136077, 13.13513516798135, 13.40480776895459