| L(s) = 1 | + 1.39i·3-s + (−2.13 − 2.13i)7-s + 1.05·9-s + (−2.17 − 2.17i)11-s − 1.54·13-s + (3.86 + 3.86i)17-s + (0.0136 + 0.0136i)19-s + (2.97 − 2.97i)21-s + (−3.15 + 3.15i)23-s + 5.65i·27-s + (−3.33 + 3.33i)29-s + 8.92i·31-s + (3.02 − 3.02i)33-s − 7.24·37-s − 2.15i·39-s + ⋯ |
| L(s) = 1 | + 0.804i·3-s + (−0.806 − 0.806i)7-s + 0.353·9-s + (−0.654 − 0.654i)11-s − 0.428·13-s + (0.937 + 0.937i)17-s + (0.00313 + 0.00313i)19-s + (0.648 − 0.648i)21-s + (−0.657 + 0.657i)23-s + 1.08i·27-s + (−0.619 + 0.619i)29-s + 1.60i·31-s + (0.526 − 0.526i)33-s − 1.19·37-s − 0.345i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7722790292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7722790292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.39iT - 3T^{2} \) |
| 7 | \( 1 + (2.13 + 2.13i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.17 + 2.17i)T + 11iT^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + (-3.86 - 3.86i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0136 - 0.0136i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.15 - 3.15i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.33 - 3.33i)T - 29iT^{2} \) |
| 31 | \( 1 - 8.92iT - 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 + (3.34 - 3.34i)T - 47iT^{2} \) |
| 53 | \( 1 + 7.30iT - 53T^{2} \) |
| 59 | \( 1 + (3.52 - 3.52i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.41 - 1.41i)T + 61iT^{2} \) |
| 67 | \( 1 - 0.748T + 67T^{2} \) |
| 71 | \( 1 - 0.269T + 71T^{2} \) |
| 73 | \( 1 + (-0.811 - 0.811i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + (6.33 + 6.33i)T + 97iT^{2} \) |
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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
−10.01991606562300845276015136074, −9.093557474860275259381200505104, −8.141057009766067319311510333823, −7.35270369328505010046474758263, −6.52571760528471283959692358308, −5.51973132370818857626730011631, −4.74237613077095774425209717415, −3.59636096507967960762874447329, −3.27382579167972147498940420714, −1.47023198439760154996746171153,
0.29296722827250779643333556890, 1.99733882144724421838382646298, 2.68464326682058879838215139678, 3.94398343507601315571244830622, 5.11787213125309911184302966184, 5.88824149023536636852776487365, 6.76825520733300670868851513983, 7.48298449834150514123192052848, 8.057508812903822301260535389873, 9.227617696604776970236306269412
