| L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.98 − 0.387i)5-s + (−5.04 − 2.91i)7-s − 2.82·8-s + (−4 + 5.83i)10-s + (−14.2 − 8.24i)11-s + (−7.14 + 4.12i)14-s + (−2.00 + 3.46i)16-s + 11.3·17-s + 12·19-s + (4.31 + 9.02i)20-s + (−20.1 + 11.6i)22-s + (12.0 + 20.8i)23-s + (24.6 + 3.86i)25-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.996 − 0.0775i)5-s + (−0.721 − 0.416i)7-s − 0.353·8-s + (−0.400 + 0.583i)10-s + (−1.29 − 0.749i)11-s + (−0.510 + 0.294i)14-s + (−0.125 + 0.216i)16-s + 0.665·17-s + 0.631·19-s + (0.215 + 0.451i)20-s + (−0.918 + 0.530i)22-s + (0.522 + 0.905i)23-s + (0.987 + 0.154i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6500535319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6500535319\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.98 + 0.387i)T \) |
| good | 7 | \( 1 + (5.04 + 2.91i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (14.2 + 8.24i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 11.3T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + (-12.0 - 20.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-16 - 27.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 23.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (49.9 - 28.8i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-35.3 - 20.4i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (17.6 - 30.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 67.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-14.2 + 8.24i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5.04 - 2.91i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 116. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-36 + 62.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (21.9 - 37.9i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-141. - 81.6i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.28162667693139687339479088364, −9.542984525919805124597692931814, −8.376856332044046493147665282526, −7.70223623449658372040521438564, −6.70057881716731183364851710686, −5.50211470880852479765392109869, −4.69573674169485759487427966573, −3.36431801859981159797849388747, −3.05933313280194389489404774109, −1.04097387343853413812455076275,
0.23397663589609460769690268594, 2.60217272322305084075786901877, 3.50868738563641401650157663758, 4.65360033322925092869548825298, 5.44247324089570654621718918584, 6.54142865868398262272350418344, 7.42923210865706912968348864766, 7.971986156774180970514669912957, 8.937217350543574779313266781846, 9.913245300391675637781695244727
