L(s) = 1 | − 2.50·3-s + 2.36i·5-s + i·7-s + 3.25·9-s + i·11-s + (0.178 + 3.60i)13-s − 5.92i·15-s − 2.97·17-s − 0.164i·19-s − 2.50i·21-s − 8.05·23-s − 0.613·25-s − 0.649·27-s − 4.68·29-s − 1.30i·31-s + ⋯ |
L(s) = 1 | − 1.44·3-s + 1.05i·5-s + 0.377i·7-s + 1.08·9-s + 0.301i·11-s + (0.0495 + 0.998i)13-s − 1.53i·15-s − 0.720·17-s − 0.0377i·19-s − 0.545i·21-s − 1.67·23-s − 0.122·25-s − 0.124·27-s − 0.870·29-s − 0.234i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0495 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02606514161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02606514161\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.178 - 3.60i)T \) |
good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 2.36iT - 5T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 0.164iT - 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 + 1.30iT - 31T^{2} \) |
| 37 | \( 1 + 1.83iT - 37T^{2} \) |
| 41 | \( 1 - 4.62iT - 41T^{2} \) |
| 43 | \( 1 + 7.77T + 43T^{2} \) |
| 47 | \( 1 + 7.81iT - 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 7.01iT - 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 + 3.03iT - 67T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 - 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 11.4iT - 83T^{2} \) |
| 89 | \( 1 - 18.0iT - 89T^{2} \) |
| 97 | \( 1 + 4.72iT - 97T^{2} \) |
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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
−8.212247477898736492965813603309, −7.07614563890967158127037503714, −6.83190197661481751138623164926, −6.03389842848003124522500254200, −5.52777970522861691654907709379, −4.51400676972737683128264000120, −3.86168282894036078186272026013, −2.58099446108863873203704355634, −1.71183051197886866932108124884, −0.01258819015431853501179374113,
0.78131231653728150804258235457, 1.87136090724412353735112407119, 3.38563471060042588908031367331, 4.38702089167551371307146251618, 4.93398681337310507908010136657, 5.72170743509252405940447650486, 6.12782058589972600005598397527, 7.06028232424605206680547734547, 7.912203778295018739353408538562, 8.561239639205019121915356830992