L(s) = 1 | + (1.16 − 0.795i)2-s + (1.29 + 0.536i)3-s + (0.735 − 1.85i)4-s + (0.937 + 2.26i)5-s + (1.94 − 0.402i)6-s + (−0.707 + 0.707i)7-s + (−0.619 − 2.75i)8-s + (−0.730 − 0.730i)9-s + (2.89 + 1.90i)10-s + (−2.12 + 0.879i)11-s + (1.95 − 2.01i)12-s + (−0.267 + 0.644i)13-s + (−0.264 + 1.38i)14-s + 3.43i·15-s + (−2.91 − 2.73i)16-s − 2.83i·17-s + ⋯ |
L(s) = 1 | + (0.826 − 0.562i)2-s + (0.748 + 0.309i)3-s + (0.367 − 0.929i)4-s + (0.419 + 1.01i)5-s + (0.792 − 0.164i)6-s + (−0.267 + 0.267i)7-s + (−0.218 − 0.975i)8-s + (−0.243 − 0.243i)9-s + (0.916 + 0.601i)10-s + (−0.640 + 0.265i)11-s + (0.563 − 0.581i)12-s + (−0.0740 + 0.178i)13-s + (−0.0707 + 0.371i)14-s + 0.887i·15-s + (−0.729 − 0.683i)16-s − 0.687i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20584 - 0.392307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20584 - 0.392307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.795i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-1.29 - 0.536i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.937 - 2.26i)T + (-3.53 + 3.53i)T^{2} \) |
| 11 | \( 1 + (2.12 - 0.879i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.267 - 0.644i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 2.83iT - 17T^{2} \) |
| 19 | \( 1 + (0.445 - 1.07i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (4.15 + 4.15i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.37 - 1.39i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 + (-3.59 - 8.67i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.419 - 0.419i)T + 41iT^{2} \) |
| 43 | \( 1 + (6.63 - 2.74i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 + (-8.31 + 3.44i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (3.36 + 8.12i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-8.49 - 3.52i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-4.08 - 1.69i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (9.89 - 9.89i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.37 - 1.37i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.67iT - 79T^{2} \) |
| 83 | \( 1 + (-2.34 + 5.64i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (11.4 - 11.4i)T - 89iT^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−12.21615566076415315831159456582, −11.30621780436473353779613677342, −10.14410908465098124711890165906, −9.751468083392642528808479409507, −8.360707101118424746950357741013, −6.82915071499521567092636375108, −5.98258591785543977133084322564, −4.53882487445862753624073288441, −3.10629274548672198147374213982, −2.47215310939391468245211558139,
2.25417846368083165681867285238, 3.68298164016309748250272307555, 5.07648885848302870701364080603, 5.94090540311715409832129600422, 7.38003262430693106905142500713, 8.249628390533251956968130911128, 8.951773670018705533335317744546, 10.36932138603308859817712675631, 11.70867105293060212863811618571, 12.76943055913582738515416142734