L(s) = 1 | + (1.24 + 0.673i)2-s + (1.09 + 1.67i)4-s + (1.11 − 2.70i)5-s + (1.57 − 1.57i)7-s + (0.232 + 2.81i)8-s + (3.21 − 2.60i)10-s + (1.00 − 2.41i)11-s + (−6.48 + 2.68i)13-s + (3.01 − 0.896i)14-s + (−1.60 + 3.66i)16-s − 0.520·17-s + (2.95 + 7.14i)19-s + (5.75 − 1.08i)20-s + (2.87 − 2.33i)22-s + (1.35 − 1.35i)23-s + ⋯ |
L(s) = 1 | + (0.879 + 0.476i)2-s + (0.546 + 0.837i)4-s + (0.500 − 1.20i)5-s + (0.593 − 0.593i)7-s + (0.0823 + 0.996i)8-s + (1.01 − 0.824i)10-s + (0.301 − 0.729i)11-s + (−1.79 + 0.745i)13-s + (0.805 − 0.239i)14-s + (−0.401 + 0.915i)16-s − 0.126·17-s + (0.678 + 1.63i)19-s + (1.28 − 0.241i)20-s + (0.612 − 0.497i)22-s + (0.282 − 0.282i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.22589 + 0.270520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22589 + 0.270520i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 - 0.673i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.11 + 2.70i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.57 + 1.57i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.00 + 2.41i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (6.48 - 2.68i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 0.520T + 17T^{2} \) |
| 19 | \( 1 + (-2.95 - 7.14i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 1.35i)T - 23iT^{2} \) |
| 29 | \( 1 + (5.31 - 2.20i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 1.54iT - 31T^{2} \) |
| 37 | \( 1 + (3.79 + 1.57i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (1.08 + 1.08i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.71 + 1.12i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + (-3.70 - 1.53i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (3.19 + 1.32i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (3.78 + 9.12i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 4.53i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.83 - 6.83i)T + 71iT^{2} \) |
| 73 | \( 1 + (-2.94 + 2.94i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.79T + 79T^{2} \) |
| 83 | \( 1 + (13.9 - 5.79i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (7.09 - 7.09i)T - 89iT^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−12.19411291262118875422040053405, −11.22049295326007914225463326368, −9.890473981845491579289495007260, −8.820668067879722269560868205603, −7.85632012133259147344232287526, −6.89765923183909353660777114069, −5.50154352064201928686322866180, −4.90092776687140332075303590814, −3.75447653745978775766897524278, −1.83490494024512346110651977321,
2.20526491410463918858127644412, 2.95943246518352128220267562024, 4.70593402562860941742098640385, 5.51497093057877368773835352070, 6.81868059540340921173066137787, 7.46656762538692360263974522882, 9.397591398315869332175203719149, 10.03087550588354164698120586594, 11.04993062399712250714129426962, 11.73720002143617947137058200449