L(s) = 1 | + (−0.939 − 0.342i)3-s + (−0.467 − 2.64i)5-s + (−0.480 − 0.277i)7-s + (0.766 + 0.642i)9-s + (−1.68 + 0.971i)11-s + (−2.00 − 5.51i)13-s + (−0.467 + 2.64i)15-s + (−2.18 + 1.83i)17-s + (3.22 + 2.93i)19-s + (0.356 + 0.424i)21-s + (−3.17 − 0.559i)23-s + (−2.09 + 0.764i)25-s + (−0.500 − 0.866i)27-s + (−0.952 + 1.13i)29-s + (−3.09 + 5.35i)31-s + ⋯ |
L(s) = 1 | + (−0.542 − 0.197i)3-s + (−0.208 − 1.18i)5-s + (−0.181 − 0.104i)7-s + (0.255 + 0.214i)9-s + (−0.507 + 0.293i)11-s + (−0.556 − 1.52i)13-s + (−0.120 + 0.683i)15-s + (−0.530 + 0.445i)17-s + (0.739 + 0.672i)19-s + (0.0778 + 0.0927i)21-s + (−0.662 − 0.116i)23-s + (−0.419 + 0.152i)25-s + (−0.0962 − 0.166i)27-s + (−0.176 + 0.210i)29-s + (−0.555 + 0.962i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0521377 + 0.264350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0521377 + 0.264350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-3.22 - 2.93i)T \) |
good | 5 | \( 1 + (0.467 + 2.64i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.480 + 0.277i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.68 - 0.971i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.00 + 5.51i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.18 - 1.83i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (3.17 + 0.559i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.952 - 1.13i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.09 - 5.35i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.18iT - 37T^{2} \) |
| 41 | \( 1 + (3.29 - 9.04i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.72 - 0.481i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.27 - 2.71i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (3.33 + 0.588i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (6.26 - 5.25i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.549 - 3.11i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.71 - 2.27i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.258 - 1.46i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.13 - 1.87i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (10.7 + 3.92i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (12.9 + 7.47i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.23 + 11.6i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.52 - 5.39i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−9.805072334092559681885689033294, −8.634815166067848691903920992232, −7.968123983806979676708933110262, −7.19165496370161031791785051773, −5.91490485556232051328453137893, −5.24315532307859037917287677083, −4.47091264728993364679231310494, −3.15217555400450377440004557146, −1.54200808530356081303967327304, −0.13382744391788218085938731834,
2.15585419445229383750450239847, 3.24584724294939994083975842389, 4.35415029511022226380791706783, 5.34550901226868204364271235167, 6.48834258281622261725018765036, 6.96259914101548439631987816537, 7.83348464340937165038956375657, 9.148383483476248426571625708099, 9.767296348385652584736411383502, 10.65899621919948599988817006208