L(s) = 1 | + (0.729 + 1.21i)2-s + (−1.08 − 1.35i)3-s + (−0.934 + 1.76i)4-s + (−0.281 − 0.366i)5-s + (0.847 − 2.29i)6-s + (−5.02 − 1.34i)7-s + (−2.82 + 0.158i)8-s + (−0.655 + 2.92i)9-s + (0.238 − 0.608i)10-s + (−2.81 + 0.370i)11-s + (3.40 − 0.650i)12-s + (−0.181 + 1.38i)13-s + (−2.03 − 7.06i)14-s + (−0.191 + 0.777i)15-s + (−2.25 − 3.30i)16-s + 0.539·17-s + ⋯ |
L(s) = 1 | + (0.516 + 0.856i)2-s + (−0.625 − 0.780i)3-s + (−0.467 + 0.884i)4-s + (−0.125 − 0.163i)5-s + (0.345 − 0.938i)6-s + (−1.89 − 0.508i)7-s + (−0.998 + 0.0559i)8-s + (−0.218 + 0.975i)9-s + (0.0754 − 0.192i)10-s + (−0.849 + 0.111i)11-s + (0.982 − 0.187i)12-s + (−0.0503 + 0.382i)13-s + (−0.544 − 1.88i)14-s + (−0.0493 + 0.200i)15-s + (−0.563 − 0.826i)16-s + 0.130·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0105472 - 0.0408188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0105472 - 0.0408188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.729 - 1.21i)T \) |
| 3 | \( 1 + (1.08 + 1.35i)T \) |
good | 5 | \( 1 + (0.281 + 0.366i)T + (-1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (5.02 + 1.34i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.81 - 0.370i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (0.181 - 1.38i)T + (-12.5 - 3.36i)T^{2} \) |
| 17 | \( 1 - 0.539T + 17T^{2} \) |
| 19 | \( 1 + (-1.93 - 4.66i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.658 + 2.45i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (4.86 + 3.72i)T + (7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (3.42 + 1.97i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.96 - 2.88i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (10.9 - 2.93i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.840 - 6.38i)T + (-41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (-5.80 + 3.35i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.36 + 1.80i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (9.11 - 6.99i)T + (15.2 - 56.9i)T^{2} \) |
| 61 | \( 1 + (5.40 - 7.03i)T + (-15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (0.00279 - 0.0212i)T + (-64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (5.54 + 5.54i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.845 - 0.845i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.626 + 1.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.453 + 0.347i)T + (21.4 + 80.1i)T^{2} \) |
| 89 | \( 1 + (-1.11 + 1.11i)T - 89iT^{2} \) |
| 97 | \( 1 + (0.924 + 1.60i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.56629901219367257832737125135, −11.90052365585758195938953049724, −10.42779885354183292602714637378, −9.499006328451792245806382140649, −8.086253270732789930498783096783, −7.29181827664409511940646430253, −6.37732832828649740437897964507, −5.74322259332282971974799455877, −4.30555778531221177082264392072, −2.92802513362939496426168926215,
0.02691361843433561430451226780, 2.94169878063341322521587016340, 3.59074633608443688003460898298, 5.17886734556346583201125226653, 5.83699195040551727510271601759, 6.98662273558625548289170545173, 9.096807119734220396192231442303, 9.537967508688134972695616682143, 10.48897334976478116551761887986, 11.15785088404601223919501209387