| L(s) = 1 | − 3-s + 2i·7-s + 9-s + 4i·11-s + 6i·17-s − 4i·19-s − 2i·21-s + 4i·23-s − 27-s − 6i·29-s − 10·31-s − 4i·33-s − 4·37-s + 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755i·7-s + 0.333·9-s + 1.20i·11-s + 1.45i·17-s − 0.917i·19-s − 0.436i·21-s + 0.834i·23-s − 0.192·27-s − 1.11i·29-s − 1.79·31-s − 0.696i·33-s − 0.657·37-s + 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6567916599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6567916599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.355738761642335657850371875855, −8.634135121749319929154757407694, −7.61557598846984265057822489301, −7.05159310559630881198887940022, −6.02876562090140838611465707651, −5.53595924609118622895828984315, −4.56513399708495759407429599375, −3.80336272057689396977373197147, −2.44464457843524711260500178520, −1.58127164751329578205446041613,
0.24663664178368590765317991110, 1.36093322532895384015085682539, 2.87314781687839045679315728058, 3.75787166219974200743834835114, 4.66758607753109548995196162392, 5.55473086636024242305198686806, 6.18914512511484198721034028399, 7.18604710501407784642102430824, 7.62119837464683328652284447261, 8.717655758100331891481308842799
