L(s) = 1 | + (1.43 + 2.49i)3-s + (0.5 − 0.866i)5-s + (−2.64 + 4.58i)9-s + (−0.732 − 1.26i)11-s − 2.22·13-s + 2.87·15-s + (3.63 + 6.29i)17-s + (−2.74 + 4.75i)19-s + (−1.25 + 2.17i)23-s + (−0.499 − 0.866i)25-s − 6.60·27-s − 7.12·29-s + (−3.25 − 5.64i)31-s + (2.11 − 3.65i)33-s + (−3.45 + 5.97i)37-s + ⋯ |
L(s) = 1 | + (0.831 + 1.43i)3-s + (0.223 − 0.387i)5-s + (−0.882 + 1.52i)9-s + (−0.220 − 0.382i)11-s − 0.616·13-s + 0.743·15-s + (0.881 + 1.52i)17-s + (−0.629 + 1.09i)19-s + (−0.262 + 0.454i)23-s + (−0.0999 − 0.173i)25-s − 1.27·27-s − 1.32·29-s + (−0.585 − 1.01i)31-s + (0.367 − 0.636i)33-s + (−0.567 + 0.982i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.858999835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.858999835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.43 - 2.49i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (0.732 + 1.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + (-3.63 - 6.29i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.74 - 4.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + (3.25 + 5.64i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.45 - 5.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + (4.18 - 7.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.50 - 6.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.53 - 7.85i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.60 + 9.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.20 - 5.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + (-5.26 - 9.11i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.09 - 8.82i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + (-4.91 + 8.51i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−9.667035199134589925463668829131, −8.749033719349428661541614552428, −8.178418086882929996700673352276, −7.50327837793285318162755531849, −5.87222003672502167205443699102, −5.55972354923698549158449667511, −4.27004472867601051292334015579, −3.90197087621052174601491268932, −2.89502611656109036023381537583, −1.75193383268404646405830099782,
0.56365724433097074802322251908, 2.03350911094861567683376027530, 2.56183932602477452443813896045, 3.51415206954428380749684823314, 4.90931086256637607199112421468, 5.82050271476485924295853350166, 7.00035501567745752147911545773, 7.15608854950149485769783020872, 7.85438680069631822599867255241, 8.868563072835522965689784928186