| L(s) = 1 | + (−1.66 − 1.10i)2-s + (−2.30 + 1.32i)3-s + (1.54 + 3.68i)4-s + 5.63·5-s + (5.30 + 0.337i)6-s + (7.32 + 4.22i)7-s + (1.51 − 7.85i)8-s + (−0.963 + 1.66i)9-s + (−9.37 − 6.23i)10-s + (−0.770 + 0.444i)11-s + (−8.46 − 6.44i)12-s + (8.44 + 9.88i)13-s + (−7.50 − 15.1i)14-s + (−12.9 + 7.48i)15-s + (−11.2 + 11.4i)16-s + (0.851 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.832 − 0.553i)2-s + (−0.767 + 0.443i)3-s + (0.386 + 0.922i)4-s + 1.12·5-s + (0.884 + 0.0562i)6-s + (1.04 + 0.603i)7-s + (0.189 − 0.981i)8-s + (−0.107 + 0.185i)9-s + (−0.937 − 0.623i)10-s + (−0.0700 + 0.0404i)11-s + (−0.705 − 0.536i)12-s + (0.649 + 0.760i)13-s + (−0.536 − 1.08i)14-s + (−0.864 + 0.499i)15-s + (−0.701 + 0.712i)16-s + (0.0501 − 0.0868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.776768 + 0.127404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.776768 + 0.127404i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.66 + 1.10i)T \) |
| 13 | \( 1 + (-8.44 - 9.88i)T \) |
| good | 3 | \( 1 + (2.30 - 1.32i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 - 5.63T + 25T^{2} \) |
| 7 | \( 1 + (-7.32 - 4.22i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (0.770 - 0.444i)T + (60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (-0.851 + 1.47i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (20.7 + 11.9i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-30.2 + 17.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (17.1 + 29.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 40.6iT - 961T^{2} \) |
| 37 | \( 1 + (11.7 + 20.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.35 - 16.2i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-19.0 - 10.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 28.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 98.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (73.3 + 42.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-44.3 + 76.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.7 + 9.66i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-29.5 - 17.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 - 99.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (17.9 + 31.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-17.0 + 29.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.59050910579627300267633595130, −14.06346420637958911615913303407, −12.71448727428774757300774301033, −11.28933839161840539660215132623, −10.81013994064902757762849561821, −9.427105782140507984206292838408, −8.384506326507763779041657278667, −6.40821465501683174974058369407, −4.86096181130668031838981666898, −2.07508826403861686497949312020,
1.38667338909644349178070548091, 5.37336192803822887666550688582, 6.31036933711663042340770388708, 7.71389091034995195822872714498, 9.103999377805990932180113981371, 10.54664842727591599834767537514, 11.24068762473581157762012917786, 12.98131689814564469042405447342, 14.20642940130958585974202215859, 15.19517778962187621363966626979
