L(s) = 1 | + (0.988 + 0.149i)2-s + (−0.930 + 0.634i)3-s + (0.955 + 0.294i)4-s + (−1.01 + 0.488i)6-s + (0.433 − 0.900i)7-s + (0.900 + 0.433i)8-s + (0.0983 − 0.250i)9-s + (−0.108 − 0.277i)11-s + (−1.07 + 0.332i)12-s + (1.23 − 1.54i)13-s + (0.563 − 0.826i)14-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.134 − 0.233i)18-s + (0.167 + 1.11i)21-s + (−0.0663 − 0.290i)22-s + ⋯ |
L(s) = 1 | + (0.988 + 0.149i)2-s + (−0.930 + 0.634i)3-s + (0.955 + 0.294i)4-s + (−1.01 + 0.488i)6-s + (0.433 − 0.900i)7-s + (0.900 + 0.433i)8-s + (0.0983 − 0.250i)9-s + (−0.108 − 0.277i)11-s + (−1.07 + 0.332i)12-s + (1.23 − 1.54i)13-s + (0.563 − 0.826i)14-s + (0.826 + 0.563i)16-s + (−0.733 − 0.680i)17-s + (0.134 − 0.233i)18-s + (0.167 + 1.11i)21-s + (−0.0663 − 0.290i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.905216520\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.905216520\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-0.433 + 0.900i)T \) |
| 17 | \( 1 + (0.733 + 0.680i)T \) |
good | 3 | \( 1 + (0.930 - 0.634i)T + (0.365 - 0.930i)T^{2} \) |
| 5 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 11 | \( 1 + (0.108 + 0.277i)T + (-0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 1.54i)T + (-0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.14 + 1.06i)T + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.433 - 0.751i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 61 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.443 - 1.94i)T + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.930 - 1.61i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.603 - 1.53i)T + (-0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.544854669448180681620023059006, −7.987088572267359132766412295056, −7.06414835668596702404265324653, −6.38842605655296625782685551100, −5.49396509038003610043160445917, −5.13040087743797182642496593630, −4.26752283547400761890879182960, −3.59343409956983852682922160116, −2.58933042287587042916739728316, −1.01103626063230301384191423904,
1.53432394649444765626658820204, 2.02493945710192283222924393453, 3.40992449925509977358027963309, 4.26945921441735921497853360271, 5.09732227874072791511161481757, 5.92850403891973733299358375918, 6.25880288436686877505895326910, 6.99323820144177553010275988145, 7.78917275938107207739490919930, 8.869588203443944849972650333485