L(s) = 1 | + (−0.857 + 1.12i)2-s + (−0.530 − 1.92i)4-s + (−0.604 + 0.349i)5-s + (1.16 − 2.37i)7-s + (2.62 + 1.05i)8-s + (0.125 − 0.979i)10-s + (1.27 − 2.21i)11-s + 1.88·13-s + (1.67 + 3.34i)14-s + (−3.43 + 2.04i)16-s + (3.44 + 1.98i)17-s + (6.11 − 3.52i)19-s + (0.994 + 0.980i)20-s + (1.39 + 3.32i)22-s + (−2.01 − 3.48i)23-s + ⋯ |
L(s) = 1 | + (−0.606 + 0.795i)2-s + (−0.265 − 0.964i)4-s + (−0.270 + 0.156i)5-s + (0.439 − 0.898i)7-s + (0.927 + 0.373i)8-s + (0.0397 − 0.309i)10-s + (0.384 − 0.666i)11-s + 0.521·13-s + (0.448 + 0.893i)14-s + (−0.859 + 0.511i)16-s + (0.834 + 0.481i)17-s + (1.40 − 0.809i)19-s + (0.222 + 0.219i)20-s + (0.296 + 0.709i)22-s + (−0.419 − 0.727i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.943507 + 0.0860364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.943507 + 0.0860364i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.857 - 1.12i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.16 + 2.37i)T \) |
good | 5 | \( 1 + (0.604 - 0.349i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.27 + 2.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 + (-3.44 - 1.98i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.11 + 3.52i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.01 + 3.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.86iT - 29T^{2} \) |
| 31 | \( 1 + (-0.815 - 0.470i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.74 - 6.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 3.97iT - 43T^{2} \) |
| 47 | \( 1 + (4.45 + 7.71i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.458 - 0.264i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.65 - 11.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.18 + 8.97i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.35 + 1.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.7 - 6.76i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + (10.2 - 5.93i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−11.77869504048454445233750947418, −10.93786317268426021970380791966, −10.11159390979817274411749657423, −9.022119801429092544670867209044, −8.022733570848361877239153035738, −7.28211795969962747577546665272, −6.20170330409886904020102271945, −5.02060778938155254701134424015, −3.63619957167902880557338250339, −1.12927318895203837137729564340,
1.56981199857865061283647040091, 3.12860782209932450207143080456, 4.45889888571321694799330175836, 5.83686594931863441061942665006, 7.54183538576819843170734150886, 8.162123058783152843614219223838, 9.369988915104405850470978833739, 9.899513518110394502080584472319, 11.28969263117928607280734042022, 11.90331151259065005924672372868