| L(s) = 1 | + (0.848 − 0.184i)3-s + (−1.60 − 1.56i)5-s + (−0.374 + 0.204i)7-s + (−2.04 + 0.932i)9-s + (−2.23 − 1.94i)11-s + (−0.424 + 0.777i)13-s + (−1.64 − 1.03i)15-s + (1.07 + 0.804i)17-s + (−0.658 + 4.58i)19-s + (−0.279 + 0.242i)21-s + (−4.18 − 2.34i)23-s + (0.121 + 4.99i)25-s + (−3.64 + 2.73i)27-s + (−5.23 + 0.752i)29-s + (−3.96 + 2.54i)31-s + ⋯ |
| L(s) = 1 | + (0.490 − 0.106i)3-s + (−0.715 − 0.698i)5-s + (−0.141 + 0.0772i)7-s + (−0.680 + 0.310i)9-s + (−0.675 − 0.585i)11-s + (−0.117 + 0.215i)13-s + (−0.425 − 0.265i)15-s + (0.260 + 0.195i)17-s + (−0.151 + 1.05i)19-s + (−0.0610 + 0.0529i)21-s + (−0.872 − 0.488i)23-s + (0.0242 + 0.999i)25-s + (−0.701 + 0.525i)27-s + (−0.971 + 0.139i)29-s + (−0.711 + 0.457i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00169388 + 0.00963305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00169388 + 0.00963305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.60 + 1.56i)T \) |
| 23 | \( 1 + (4.18 + 2.34i)T \) |
| good | 3 | \( 1 + (-0.848 + 0.184i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (0.374 - 0.204i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (2.23 + 1.94i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.424 - 0.777i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 0.804i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.658 - 4.58i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (5.23 - 0.752i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.96 - 2.54i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.91 + 1.83i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-4.09 + 8.95i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.0259 - 0.119i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-6.05 - 6.05i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.21 + 5.88i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.934 - 3.18i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.18 + 4.95i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (8.31 + 0.594i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-0.788 - 0.910i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (4.30 + 5.74i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-15.6 + 4.58i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.58 - 9.62i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (5.80 + 3.72i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.07 - 10.9i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.57871747450898752895680270661, −9.371654100900844285195802723214, −8.679408017582564076565385865569, −7.995160432856633280968291855021, −7.42949771440089604423256161731, −5.95393526453323688390953230217, −5.29768075840356029852445261395, −4.05132603225740606690625417033, −3.20790255471888911963483059364, −1.88948905194769988317966867557,
0.00400853128741618450002292921, 2.33299807497415441941184328494, 3.20082594856882851315783332479, 4.10254477843146195166003611539, 5.30152354286464443697495876709, 6.35516601686976040987058458872, 7.40054033354201518713462307450, 7.87141978787706088667342622054, 8.876215893755098097733181151928, 9.700693346531401021077699853612
