| L(s) = 1 | + 3-s + 5-s − 3·7-s − 2·9-s − 11-s − 6·13-s + 15-s − 7·17-s − 5·19-s − 3·21-s + 6·23-s + 25-s − 5·27-s + 5·29-s + 3·31-s − 33-s − 3·35-s + 3·37-s − 6·39-s + 2·41-s − 4·43-s − 2·45-s + 2·47-s + 2·49-s − 7·51-s − 53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.538·31-s − 0.174·33-s − 0.507·35-s + 0.493·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.291·47-s + 2/7·49-s − 0.980·51-s − 0.137·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.604100883180305119694719486737, −8.976631286949291998537604701701, −8.214855445870152436084038682621, −6.96290485918056996428743397373, −6.46925986382253407591418544135, −5.27646996011335233092203843669, −4.28393931774823338150480925867, −2.81267195278645779898796150454, −2.42545807821311073856045852532, 0,
2.42545807821311073856045852532, 2.81267195278645779898796150454, 4.28393931774823338150480925867, 5.27646996011335233092203843669, 6.46925986382253407591418544135, 6.96290485918056996428743397373, 8.214855445870152436084038682621, 8.976631286949291998537604701701, 9.604100883180305119694719486737
