L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s − 2·11-s + 2·13-s − 4·15-s − 2·17-s − 19-s + 2·21-s − 8·23-s − 25-s + 4·27-s + 8·31-s + 4·33-s − 2·35-s − 10·37-s − 4·39-s − 2·41-s + 10·43-s + 2·45-s + 47-s + 49-s + 4·51-s + 2·53-s − 4·55-s + 2·57-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s − 1.66·23-s − 1/5·25-s + 0.769·27-s + 1.43·31-s + 0.696·33-s − 0.338·35-s − 1.64·37-s − 0.640·39-s − 0.312·41-s + 1.52·43-s + 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.560·51-s + 0.274·53-s − 0.539·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8555892519\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8555892519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.19472252696530, −13.98093053168814, −13.56277197685187, −12.93392394525781, −12.36315199631839, −12.04055527840734, −11.40939475551292, −10.87179532271031, −10.33679476855670, −10.09796588646623, −9.455343009588534, −8.782257712828692, −8.253132923034096, −7.650269086328711, −6.752882684057071, −6.423026979481466, −5.931832066628459, −5.500239321372555, −4.967767395417872, −4.215564439130485, −3.611688335512336, −2.619837115489463, −2.155310638606013, −1.272266213367782, −0.3607783335072595,
0.3607783335072595, 1.272266213367782, 2.155310638606013, 2.619837115489463, 3.611688335512336, 4.215564439130485, 4.967767395417872, 5.500239321372555, 5.931832066628459, 6.423026979481466, 6.752882684057071, 7.650269086328711, 8.253132923034096, 8.782257712828692, 9.455343009588534, 10.09796588646623, 10.33679476855670, 10.87179532271031, 11.40939475551292, 12.04055527840734, 12.36315199631839, 12.93392394525781, 13.56277197685187, 13.98093053168814, 14.19472252696530