| L(s) = 1 | − 3-s − 2·4-s + 5-s − 2·7-s − 2·9-s + 2·12-s − 15-s + 4·16-s + 4·17-s + 2·19-s − 2·20-s + 2·21-s + 7·23-s − 4·25-s + 5·27-s + 4·28-s + 2·29-s + 3·31-s − 2·35-s + 4·36-s + 11·37-s + 10·41-s + 4·43-s − 2·45-s + 4·47-s − 4·48-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.577·12-s − 0.258·15-s + 16-s + 0.970·17-s + 0.458·19-s − 0.447·20-s + 0.436·21-s + 1.45·23-s − 4/5·25-s + 0.962·27-s + 0.755·28-s + 0.371·29-s + 0.538·31-s − 0.338·35-s + 2/3·36-s + 1.80·37-s + 1.56·41-s + 0.609·43-s − 0.298·45-s + 0.583·47-s − 0.577·48-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.348538799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.348538799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.76316631622863, −14.94824611728600, −14.35985781750716, −14.12319521382479, −13.27582993240180, −13.03970698645994, −12.42838271886004, −11.83492291050138, −11.26343466909198, −10.56906760511715, −10.01806688137886, −9.412848065430039, −9.184137808911452, −8.358875470806486, −7.774067233996502, −7.099669272288214, −6.120516762743990, −5.909599720858237, −5.290495566635343, −4.643508500724299, −3.913408912165806, −3.073904492524855, −2.614756117824672, −1.131151685974114, −0.6026577540543438,
0.6026577540543438, 1.131151685974114, 2.614756117824672, 3.073904492524855, 3.913408912165806, 4.643508500724299, 5.290495566635343, 5.909599720858237, 6.120516762743990, 7.099669272288214, 7.774067233996502, 8.358875470806486, 9.184137808911452, 9.412848065430039, 10.01806688137886, 10.56906760511715, 11.26343466909198, 11.83492291050138, 12.42838271886004, 13.03970698645994, 13.27582993240180, 14.12319521382479, 14.35985781750716, 14.94824611728600, 15.76316631622863
