| L(s) = 1 | + 5-s − 2·7-s − 4·11-s + 2·13-s − 2·17-s + 8·19-s − 23-s + 25-s − 4·29-s − 8·31-s − 2·35-s + 8·37-s − 4·41-s + 10·43-s − 8·47-s − 3·49-s + 6·53-s − 4·55-s − 10·59-s + 10·61-s + 2·65-s − 2·67-s − 2·71-s + 2·73-s + 8·77-s − 16·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.742·29-s − 1.43·31-s − 0.338·35-s + 1.31·37-s − 0.624·41-s + 1.52·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s − 1.30·59-s + 1.28·61-s + 0.248·65-s − 0.244·67-s − 0.237·71-s + 0.234·73-s + 0.911·77-s − 1.80·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640784174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640784174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.07013538549001, −15.53699416785484, −14.76390409782493, −14.26807719936421, −13.47780572854988, −13.21108377584068, −12.81528447616240, −12.06217913313078, −11.30252922963780, −10.93768865857472, −10.18976355527110, −9.670008630913798, −9.245784504780637, −8.573161569938928, −7.659009405160264, −7.436363342170250, −6.557621290039100, −5.872718942747041, −5.453189459550875, −4.767972112722010, −3.785334940801869, −3.177096355080034, −2.513988239382549, −1.629668433372799, −0.5414298489615494,
0.5414298489615494, 1.629668433372799, 2.513988239382549, 3.177096355080034, 3.785334940801869, 4.767972112722010, 5.453189459550875, 5.872718942747041, 6.557621290039100, 7.436363342170250, 7.659009405160264, 8.573161569938928, 9.245784504780637, 9.670008630913798, 10.18976355527110, 10.93768865857472, 11.30252922963780, 12.06217913313078, 12.81528447616240, 13.21108377584068, 13.47780572854988, 14.26807719936421, 14.76390409782493, 15.53699416785484, 16.07013538549001
