L(s) = 1 | + (1.41 + 1.41i)3-s + (2.44 + 2.44i)7-s + 1.00i·9-s − 3.46·11-s + (−2.82 + 2.82i)13-s + (−4.89 + 4.89i)17-s − 3.46i·19-s + 6.92i·21-s + (−2.44 + 2.44i)23-s + (2.82 − 2.82i)27-s − 6.92·29-s + 8i·31-s + (−4.89 − 4.89i)33-s + (5.65 + 5.65i)37-s − 8.00·39-s + ⋯ |
L(s) = 1 | + (0.816 + 0.816i)3-s + (0.925 + 0.925i)7-s + 0.333i·9-s − 1.04·11-s + (−0.784 + 0.784i)13-s + (−1.18 + 1.18i)17-s − 0.794i·19-s + 1.51i·21-s + (−0.510 + 0.510i)23-s + (0.544 − 0.544i)27-s − 1.28·29-s + 1.43i·31-s + (−0.852 − 0.852i)33-s + (0.929 + 0.929i)37-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.767746242\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.767746242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.41 - 1.41i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + (2.82 - 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.89 - 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 - 2.44i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + (-5.65 - 5.65i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (1.41 + 1.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.44 - 2.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (4.24 + 4.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \) |
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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
−9.548272194337666667389728990405, −8.857618593462556912402646369631, −8.397452112796354318938813937382, −7.55023451100202566315766167387, −6.47885336620460060871831267205, −5.35392575290532782623154800273, −4.69192388739954689084644332210, −3.86349655577462485443856435177, −2.62498223754788556946633957783, −1.99674856173114966470937690343,
0.56671114249244282836212119350, 2.10319284910965956844446777323, 2.62578357731562846369020398923, 4.03773102240531091406943995849, 4.88992805881955242960827917432, 5.82610831821539943444365956989, 7.14821659464360267549213022569, 7.73243612055357711623956249370, 7.87338092728483146215711134545, 8.963220461073299015431294177364