L(s) = 1 | + (−2.58 + 1.14i)2-s + (−5.56 + 2.30i)3-s + (5.35 − 5.94i)4-s + (6.28 − 15.1i)5-s + (11.7 − 12.3i)6-s + (−16.6 − 16.6i)7-s + (−7.02 + 21.5i)8-s + (6.60 − 6.60i)9-s + (1.19 + 46.4i)10-s + (3.11 + 1.28i)11-s + (−16.1 + 45.4i)12-s + (−28.8 − 69.7i)13-s + (62.1 + 23.9i)14-s + 98.9i·15-s + (−6.57 − 63.6i)16-s + 66.2i·17-s + ⋯ |
L(s) = 1 | + (−0.913 + 0.406i)2-s + (−1.07 + 0.443i)3-s + (0.669 − 0.742i)4-s + (0.562 − 1.35i)5-s + (0.798 − 0.841i)6-s + (−0.899 − 0.899i)7-s + (−0.310 + 0.950i)8-s + (0.244 − 0.244i)9-s + (0.0378 + 1.46i)10-s + (0.0853 + 0.0353i)11-s + (−0.388 + 1.09i)12-s + (−0.616 − 1.48i)13-s + (1.18 + 0.456i)14-s + 1.70i·15-s + (−0.102 − 0.994i)16-s + 0.944i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0675 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.279484 - 0.299052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.279484 - 0.299052i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.58 - 1.14i)T \) |
good | 3 | \( 1 + (5.56 - 2.30i)T + (19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (-6.28 + 15.1i)T + (-88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (16.6 + 16.6i)T + 343iT^{2} \) |
| 11 | \( 1 + (-3.11 - 1.28i)T + (941. + 941. i)T^{2} \) |
| 13 | \( 1 + (28.8 + 69.7i)T + (-1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 - 66.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (12.5 + 30.2i)T + (-4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (63.7 - 63.7i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + (-190. + 79.0i)T + (1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 - 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (46.0 - 111. i)T + (-3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (100. - 100. i)T - 6.89e4iT^{2} \) |
| 43 | \( 1 + (-27.5 - 11.4i)T + (5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 + 394. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-135. - 56.0i)T + (1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-297. + 717. i)T + (-1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (548. - 227. i)T + (1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-163. + 67.6i)T + (2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-194. - 194. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (-547. + 547. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 715. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-54.6 - 131. i)T + (-4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (-220. - 220. i)T + 7.04e5iT^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−16.42480611913736485951488044731, −15.47642041260745623710732032385, −13.44359336959243269346988454866, −12.15373597546187358334948435311, −10.44186313176056717721385336039, −9.841356099189015605302094136549, −8.199260236396395370744031860985, −6.27525712609495672268117744530, −5.07910840853422247328651475456, −0.54646206385723969630686662837,
2.59785318531544248620307171477, 6.27594806014611593905429482772, 6.91100024389589114025628247705, 9.235920667010862843645148683544, 10.36561505925168595109828836612, 11.63257285040211248085100419401, 12.34354248412924736066389495344, 14.21376215840643155876671061093, 15.88620827128411089611456973656, 16.93437013210689268226107475629