| L(s) = 1 | + (−0.340 − 0.590i)5-s + (−2.79 − 4.84i)7-s + (−7.44 − 12.9i)11-s − 2.36i·13-s + (8.72 − 15.1i)17-s + (−17.2 − 8.02i)19-s + 10.4·23-s + (12.2 − 21.2i)25-s + (37.6 + 21.7i)29-s + (28.9 + 16.7i)31-s + (−1.90 + 3.29i)35-s − 30.3i·37-s + (−48.7 + 28.1i)41-s − 62.4·43-s + (−43.3 + 75.0i)47-s + ⋯ |
| L(s) = 1 | + (−0.0681 − 0.118i)5-s + (−0.399 − 0.691i)7-s + (−0.677 − 1.17i)11-s − 0.182i·13-s + (0.513 − 0.889i)17-s + (−0.906 − 0.422i)19-s + 0.454·23-s + (0.490 − 0.849i)25-s + (1.29 + 0.749i)29-s + (0.934 + 0.539i)31-s + (−0.0544 + 0.0942i)35-s − 0.819i·37-s + (−1.18 + 0.686i)41-s − 1.45·43-s + (−0.921 + 1.59i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2052 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2052 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5925148890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5925148890\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (17.2 + 8.02i)T \) |
| good | 5 | \( 1 + (0.340 + 0.590i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2.79 + 4.84i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (7.44 + 12.9i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 2.36iT - 169T^{2} \) |
| 17 | \( 1 + (-8.72 + 15.1i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 - 10.4T + 529T^{2} \) |
| 29 | \( 1 + (-37.6 - 21.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-28.9 - 16.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 30.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (48.7 - 28.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 62.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (43.3 - 75.0i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (43.7 - 25.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (13.9 - 8.02i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-0.219 + 0.380i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 41.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (62.9 + 36.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-38.3 + 66.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-53.7 - 93.0i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-122. + 70.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 171. iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.406894097380053908789760251348, −7.926521139397168457660977192535, −6.78882727490484098210382136449, −6.37053540341078588607022782997, −5.16063314802052924610749080167, −4.58704738560187636050039488203, −3.28778889868421698536595528284, −2.80486980717055985564825239152, −1.10775243681661977582653005452, −0.16012744839233370537596669018,
1.61540011449032639989988870895, 2.55593390742790062825759303017, 3.51959500220294052367849033038, 4.61980799065760340146490912339, 5.31199326768833485902996124158, 6.37124320428645362275616547472, 6.85802064079484947121290238670, 8.010117248787793839997194250620, 8.437425778893329666700534219393, 9.440834894271367569561012988955
