| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·11-s + 2·13-s + 6·17-s − 4·19-s − 2·21-s + 23-s + 4·27-s − 6·29-s + 6·31-s − 8·33-s + 6·37-s − 4·39-s − 6·41-s + 2·43-s − 12·47-s + 49-s − 12·51-s − 2·53-s + 8·57-s + 4·59-s − 6·61-s + 63-s − 10·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s + 0.208·23-s + 0.769·27-s − 1.11·29-s + 1.07·31-s − 1.39·33-s + 0.986·37-s − 0.640·39-s − 0.937·41-s + 0.304·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.274·53-s + 1.05·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s − 1.22·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.82178747655631, −12.56023372495963, −11.96938961183265, −11.55241403653892, −11.45992098178784, −10.77266332040596, −10.45687774983234, −9.848264097683209, −9.468934184376649, −8.815615838631918, −8.425261926089922, −7.873829276690576, −7.395832307314620, −6.713284248539343, −6.220365984460976, −6.124325925361317, −5.417621582739471, −4.971312111796335, −4.446381850491340, −3.905179206422176, −3.354416025725440, −2.766618233920019, −1.811065459273189, −1.344925536211671, −0.8219977104240634, 0,
0.8219977104240634, 1.344925536211671, 1.811065459273189, 2.766618233920019, 3.354416025725440, 3.905179206422176, 4.446381850491340, 4.971312111796335, 5.417621582739471, 6.124325925361317, 6.220365984460976, 6.713284248539343, 7.395832307314620, 7.873829276690576, 8.425261926089922, 8.815615838631918, 9.468934184376649, 9.848264097683209, 10.45687774983234, 10.77266332040596, 11.45992098178784, 11.55241403653892, 11.96938961183265, 12.56023372495963, 12.82178747655631
