| L(s) = 1 | − 4·11-s − 2·13-s − 2·17-s + 8·19-s + 4·23-s + 6·29-s − 2·37-s + 6·41-s − 4·43-s − 12·47-s − 7·49-s − 6·53-s − 12·59-s + 14·61-s + 12·67-s − 2·73-s − 8·79-s − 4·83-s − 2·89-s + 14·97-s + 14·101-s − 8·103-s − 12·107-s + 14·109-s + 6·113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 1.11·29-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.824·53-s − 1.56·59-s + 1.79·61-s + 1.46·67-s − 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.211·89-s + 1.42·97-s + 1.39·101-s − 0.788·103-s − 1.16·107-s + 1.34·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.642002060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642002060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.943184013171371967635015651652, −7.23872222303543421037760783803, −6.65901420442249702710943779508, −5.70302500098579482728439996542, −5.01031570670192541975545715163, −4.62415914961121111807651111727, −3.27047938063238426663065646264, −2.89629186967448122606704762228, −1.82943710133334756057271886720, −0.64036612837186325679111116880,
0.64036612837186325679111116880, 1.82943710133334756057271886720, 2.89629186967448122606704762228, 3.27047938063238426663065646264, 4.62415914961121111807651111727, 5.01031570670192541975545715163, 5.70302500098579482728439996542, 6.65901420442249702710943779508, 7.23872222303543421037760783803, 7.943184013171371967635015651652
