| L(s) = 1 | + (−1.93 + 1.11i)5-s + (−3.63 + 6.28i)7-s + (6.50 + 3.75i)11-s + (−1.46 − 2.54i)13-s − 1.90i·17-s + 7.38·19-s + (−30.6 + 17.6i)23-s + (2.5 − 4.33i)25-s + (14.2 + 8.19i)29-s + (13.3 + 23.0i)31-s − 16.2i·35-s − 44.6·37-s + (−14.8 + 8.58i)41-s + (20.5 − 35.6i)43-s + (36.4 + 21.0i)47-s + ⋯ |
| L(s) = 1 | + (−0.387 + 0.223i)5-s + (−0.518 + 0.898i)7-s + (0.591 + 0.341i)11-s + (−0.112 − 0.195i)13-s − 0.111i·17-s + 0.388·19-s + (−1.33 + 0.769i)23-s + (0.100 − 0.173i)25-s + (0.489 + 0.282i)29-s + (0.429 + 0.744i)31-s − 0.463i·35-s − 1.20·37-s + (−0.362 + 0.209i)41-s + (0.478 − 0.827i)43-s + (0.776 + 0.448i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3299388093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3299388093\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| good | 7 | \( 1 + (3.63 - 6.28i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.50 - 3.75i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.46 + 2.54i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 - 7.38T + 361T^{2} \) |
| 23 | \( 1 + (30.6 - 17.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.2 - 8.19i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-13.3 - 23.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 44.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (14.8 - 8.58i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-20.5 + 35.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-36.4 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (4.12 - 2.37i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-38.4 + 66.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (41.9 + 72.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 23.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 103.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-40.2 + 69.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (17.1 + 9.87i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 29.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-47.7 + 82.7i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.245931129667595470581727097517, −8.664102038080019442827717844373, −7.73479964367364515704207752553, −7.01518852421742240908738668092, −6.16790233389298716944285300333, −5.47840960359613260195346265987, −4.44559107682602877529135989267, −3.49920771882006084646294085693, −2.68665577184729242000749999364, −1.51950185850907530298569015230,
0.088727267943870389112419685118, 1.13228146569088812937731239053, 2.50499093809267244213382239524, 3.76732368637349087446102835101, 4.10581049995808373338927939951, 5.23056968386465117679043880912, 6.27216919805286293753413016179, 6.85430401896561791155136025626, 7.71162093409621653755421813634, 8.429421370855948532700201319687
