L(s) = 1 | + (5.56 + 2.30i)3-s + (6.28 + 15.1i)5-s + (16.6 − 16.6i)7-s + (6.60 + 6.60i)9-s + (−3.11 + 1.28i)11-s + (−28.8 + 69.7i)13-s + 98.9i·15-s − 66.2i·17-s + (12.5 − 30.2i)19-s + (131. − 54.3i)21-s + (63.7 + 63.7i)23-s + (−102. + 102. i)25-s + (−40.7 − 98.3i)27-s + (190. + 79.0i)29-s − 123.·31-s + ⋯ |
L(s) = 1 | + (1.07 + 0.443i)3-s + (0.562 + 1.35i)5-s + (0.899 − 0.899i)7-s + (0.244 + 0.244i)9-s + (−0.0853 + 0.0353i)11-s + (−0.616 + 1.48i)13-s + 1.70i·15-s − 0.944i·17-s + (0.151 − 0.365i)19-s + (1.36 − 0.564i)21-s + (0.577 + 0.577i)23-s + (−0.818 + 0.818i)25-s + (−0.290 − 0.701i)27-s + (1.22 + 0.506i)29-s − 0.717·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.29466 + 1.11484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29466 + 1.11484i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
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
−13.65121415868600729157934674545, −11.71719816019690992390341320446, −10.82009392697927014589671309867, −9.836879785515965498782728498526, −8.958664689476609142774926816536, −7.49526129695723536271583944801, −6.77271856492342353415196865520, −4.78553638461705478455651034300, −3.39433716575468759670245980199, −2.14280278900131243648995711450,
1.44216417299400771096273970935, 2.73159970132421638010734824911, 4.83574632672533679814133032223, 5.76597392425619441733929809303, 7.85804337160910060046290951712, 8.405202192246651723288052965915, 9.168941639465349511822690085477, 10.49288092823060455795094880571, 12.14417074148669826432739858327, 12.78065158743296577069505335996