| L(s) = 1 | + 3-s − 2·5-s + 4·7-s − 2·9-s + 2·11-s + 7·13-s − 2·15-s − 4·17-s + 6·19-s + 4·21-s + 23-s − 25-s − 5·27-s + 5·29-s − 3·31-s + 2·33-s − 8·35-s + 2·37-s + 7·39-s − 9·41-s − 8·43-s + 4·45-s + 47-s + 9·49-s − 4·51-s − 6·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s − 2/3·9-s + 0.603·11-s + 1.94·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.872·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s + 0.928·29-s − 0.538·31-s + 0.348·33-s − 1.35·35-s + 0.328·37-s + 1.12·39-s − 1.40·41-s − 1.21·43-s + 0.596·45-s + 0.145·47-s + 9/7·49-s − 0.560·51-s − 0.824·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650535973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650535973\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−11.42430187210008812945880438281, −10.86848721677029243426021484334, −9.254384528219192456945387156073, −8.329813902784183039465685329406, −8.114123921695714436821685463650, −6.78162209635099188733794145481, −5.45154558642829652829288266961, −4.23360018955841801936580827802, −3.28529126527781208900067174562, −1.50730411079549891694764955601,
1.50730411079549891694764955601, 3.28529126527781208900067174562, 4.23360018955841801936580827802, 5.45154558642829652829288266961, 6.78162209635099188733794145481, 8.114123921695714436821685463650, 8.329813902784183039465685329406, 9.254384528219192456945387156073, 10.86848721677029243426021484334, 11.42430187210008812945880438281
