| L(s) = 1 | − 28·3-s + 49·7-s + 541·9-s − 577·11-s + 974·13-s − 1.70e3·17-s − 878·19-s − 1.37e3·21-s − 1.29e3·23-s − 8.34e3·27-s + 109·29-s − 2.92e3·31-s + 1.61e4·33-s + 1.41e4·37-s − 2.72e4·39-s + 1.34e4·41-s + 1.22e4·43-s + 1.69e4·47-s + 2.40e3·49-s + 4.76e4·51-s − 2.45e4·53-s + 2.45e4·57-s − 2.45e4·59-s − 5.08e3·61-s + 2.65e4·63-s + 5.85e4·67-s + 3.63e4·69-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 0.377·7-s + 2.22·9-s − 1.43·11-s + 1.59·13-s − 1.42·17-s − 0.557·19-s − 0.678·21-s − 0.511·23-s − 2.20·27-s + 0.0240·29-s − 0.546·31-s + 2.58·33-s + 1.70·37-s − 2.87·39-s + 1.25·41-s + 1.00·43-s + 1.12·47-s + 1/7·49-s + 2.56·51-s − 1.19·53-s + 1.00·57-s − 0.918·59-s − 0.174·61-s + 0.841·63-s + 1.59·67-s + 0.918·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 28 T + p^{5} T^{2} \) |
| 11 | \( 1 + 577 T + p^{5} T^{2} \) |
| 13 | \( 1 - 974 T + p^{5} T^{2} \) |
| 17 | \( 1 + 100 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 878 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1297 T + p^{5} T^{2} \) |
| 29 | \( 1 - 109 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2922 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14175 T + p^{5} T^{2} \) |
| 41 | \( 1 - 13464 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12229 T + p^{5} T^{2} \) |
| 47 | \( 1 - 16974 T + p^{5} T^{2} \) |
| 53 | \( 1 + 24506 T + p^{5} T^{2} \) |
| 59 | \( 1 + 24570 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5080 T + p^{5} T^{2} \) |
| 67 | \( 1 - 58575 T + p^{5} T^{2} \) |
| 71 | \( 1 + 20705 T + p^{5} T^{2} \) |
| 73 | \( 1 - 53540 T + p^{5} T^{2} \) |
| 79 | \( 1 + 101027 T + p^{5} T^{2} \) |
| 83 | \( 1 - 61638 T + p^{5} T^{2} \) |
| 89 | \( 1 - 47038 T + p^{5} T^{2} \) |
| 97 | \( 1 - 82 T + p^{5} T^{2} \) |
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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
−9.417547674239387726381737052432, −8.267090575214793157362771077183, −7.35786886290842914878624764325, −6.21285464308969869561597374205, −5.85826048058448876999644352255, −4.78566748007337376065461855286, −4.07920366453675505463295407310, −2.26766649707545221115016410084, −0.973485783525953530611383237650, 0,
0.973485783525953530611383237650, 2.26766649707545221115016410084, 4.07920366453675505463295407310, 4.78566748007337376065461855286, 5.85826048058448876999644352255, 6.21285464308969869561597374205, 7.35786886290842914878624764325, 8.267090575214793157362771077183, 9.417547674239387726381737052432
