L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (3.11 − 1.79i)5-s + (−6.04 + 3.53i)7-s + 2.82i·8-s + (−2.54 + 4.40i)10-s + (−13.5 − 7.83i)11-s + 13.0·13-s + (4.89 − 8.60i)14-s + (−2.00 − 3.46i)16-s + (−22.9 − 13.2i)17-s + (6.5 + 11.2i)19-s − 7.18i·20-s + 22.1·22-s + (−17.8 + 10.2i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.622 − 0.359i)5-s + (−0.863 + 0.505i)7-s + 0.353i·8-s + (−0.254 + 0.440i)10-s + (−1.23 − 0.712i)11-s + 1.00·13-s + (0.349 − 0.614i)14-s + (−0.125 − 0.216i)16-s + (−1.34 − 0.778i)17-s + (0.342 + 0.592i)19-s − 0.359i·20-s + 1.00·22-s + (−0.774 + 0.447i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0331731 - 0.139745i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0331731 - 0.139745i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.04 - 3.53i)T \) |
good | 5 | \( 1 + (-3.11 + 1.79i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (13.5 + 7.83i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 + (22.9 + 13.2i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.5 - 11.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (17.8 - 10.2i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 9.48iT - 841T^{2} \) |
| 31 | \( 1 + (15.6 - 27.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (26.8 + 46.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 19.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 76.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-20.6 + 11.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (39.3 + 22.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (71.2 + 41.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (2.28 + 3.96i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.7 + 75.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 100. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (29.9 - 51.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-36.8 - 63.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 90.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (109. - 63.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 0.0827T + 9.40e3T^{2} \) |
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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
−10.59622579947631045386608294281, −9.673154463147019581240800498158, −8.934263352248787040769122564302, −8.167777869017238086401416373218, −6.91741351039156494583107826667, −5.91143436280194672958174125192, −5.27919586747687885304290418592, −3.39191598781618672069825421429, −1.98802159630345080740322983678, −0.07134758396239701447450004684,
1.93375269824777193735110132298, 3.10449272106525102221202156867, 4.45826103228241430539705827680, 6.08001399008992600216287713293, 6.79338646814985808909119071022, 7.914668082979105259859972322064, 8.904819774964423427436341233051, 9.955871760390825374042596452899, 10.41855769677750922186109823496, 11.20852305929741145062063446007