| L(s) = 1 | + (0.351 − 1.31i)2-s + (2.17 + 2.06i)3-s + (1.87 + 1.07i)4-s + (−4.97 − 0.457i)5-s + (3.47 − 2.12i)6-s + (6.91 − 1.11i)7-s + (5.90 − 5.90i)8-s + (0.437 + 8.98i)9-s + (−2.34 + 6.36i)10-s + (4.84 + 2.79i)11-s + (1.82 + 6.21i)12-s + (−4.06 + 4.06i)13-s + (0.963 − 9.44i)14-s + (−9.87 − 11.2i)15-s + (−1.34 − 2.33i)16-s + (−0.134 − 0.502i)17-s + ⋯ |
| L(s) = 1 | + (0.175 − 0.655i)2-s + (0.724 + 0.689i)3-s + (0.467 + 0.269i)4-s + (−0.995 − 0.0914i)5-s + (0.579 − 0.353i)6-s + (0.987 − 0.159i)7-s + (0.738 − 0.738i)8-s + (0.0486 + 0.998i)9-s + (−0.234 + 0.636i)10-s + (0.440 + 0.254i)11-s + (0.152 + 0.517i)12-s + (−0.312 + 0.312i)13-s + (0.0688 − 0.674i)14-s + (−0.658 − 0.753i)15-s + (−0.0843 − 0.146i)16-s + (−0.00792 − 0.0295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.92333 - 0.0693362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92333 - 0.0693362i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.17 - 2.06i)T \) |
| 5 | \( 1 + (4.97 + 0.457i)T \) |
| 7 | \( 1 + (-6.91 + 1.11i)T \) |
| good | 2 | \( 1 + (-0.351 + 1.31i)T + (-3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (-4.84 - 2.79i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.06 - 4.06i)T - 169iT^{2} \) |
| 17 | \( 1 + (0.134 + 0.502i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (16.5 + 28.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (23.3 + 6.26i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 20.4T + 841T^{2} \) |
| 31 | \( 1 + (36.5 + 21.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.91 + 0.781i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 59.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-37.6 - 37.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-52.4 - 14.0i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-3.55 - 13.2i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-27.5 - 15.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.67 + 4.43i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (12.9 + 48.5i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-18.8 - 70.5i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-1.26 + 0.730i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-73.9 - 73.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-61.8 + 35.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 13.0i)T + 9.40e3iT^{2} \) |
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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
−13.46428637655771541046824318306, −12.25545650383461952897925939593, −11.28308680752301059723229716714, −10.69449438593037581288349647335, −9.201727770508520161302336078510, −8.038956459386630610803238336046, −7.15255555349995043937262089421, −4.62817916019559686545102018615, −3.84110246460756984319083368472, −2.21701985785879642594976876527,
1.86493133926067346986119201176, 3.88474290218109919061466402082, 5.67434734081276302736410472578, 7.04870743412712459037917650222, 7.87155450589491671774420491847, 8.603014044379650718874412684386, 10.49841560872417590896809393591, 11.65626772645112497643457002592, 12.40498234028175072431046116841, 13.97408131519288736243972005936
