L(s) = 1 | + (−0.951 − 0.309i)3-s + 0.547i·7-s + (0.809 + 0.587i)9-s + (1.08 − 0.786i)11-s + (−0.177 + 0.244i)13-s + (−3.84 + 1.24i)17-s + (1.74 + 5.35i)19-s + (0.169 − 0.520i)21-s + (0.144 + 0.198i)23-s + (−0.587 − 0.809i)27-s + (0.423 − 1.30i)29-s + (−1.09 − 3.36i)31-s + (−1.27 + 0.413i)33-s + (−1.28 + 1.76i)37-s + (0.244 − 0.177i)39-s + ⋯ |
L(s) = 1 | + (−0.549 − 0.178i)3-s + 0.206i·7-s + (0.269 + 0.195i)9-s + (0.326 − 0.237i)11-s + (−0.0492 + 0.0677i)13-s + (−0.932 + 0.302i)17-s + (0.399 + 1.22i)19-s + (0.0369 − 0.113i)21-s + (0.0300 + 0.0413i)23-s + (−0.113 − 0.155i)27-s + (0.0785 − 0.241i)29-s + (−0.196 − 0.604i)31-s + (−0.221 + 0.0719i)33-s + (−0.211 + 0.290i)37-s + (0.0391 − 0.0284i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.104064476\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.104064476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.547iT - 7T^{2} \) |
| 11 | \( 1 + (-1.08 + 0.786i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.177 - 0.244i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.84 - 1.24i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.74 - 5.35i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.144 - 0.198i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.423 + 1.30i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.09 + 3.36i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.28 - 1.76i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.93 - 5.76i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.35iT - 43T^{2} \) |
| 47 | \( 1 + (9.96 + 3.23i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.31 - 2.37i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.35 - 2.44i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.67 - 1.22i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (8.07 - 2.62i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.83 - 8.73i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.43 - 8.86i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.69 - 14.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.32 - 2.05i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.02 + 2.92i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.86 - 1.25i)T + (78.4 + 57.0i)T^{2} \) |
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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
−9.731744744078145300020017329554, −8.804936573915509649434054890167, −8.025813086803578080299038471548, −7.16634478507224068512627655087, −6.24312787392144727646394099185, −5.71333863834984598664339469530, −4.61485559219459604467599958903, −3.76824559398552422265856545933, −2.45815323032545369072651974594, −1.22247608778768302451289120762,
0.51930849746290762327023029818, 2.05835743937421501844697264945, 3.32922845373948877336185213964, 4.42303233105138081251743213256, 5.06863712167759041932529755208, 6.07680713305704950266156870035, 6.95607985470564064500192165230, 7.46354525545262900498917772382, 8.824298560021959116300130709455, 9.196717748550365864815493595206