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
−10.65899621919948599988817006208, −9.767296348385652584736411383502, −9.148383483476248426571625708099, −7.83348464340937165038956375657, −6.96259914101548439631987816537, −6.48834258281622261725018765036, −5.34550901226868204364271235167, −4.35415029511022226380791706783, −3.24584724294939994083975842389, −2.15585419445229383750450239847,
0.13382744391788218085938731834, 1.54200808530356081303967327304, 3.15217555400450377440004557146, 4.47091264728993364679231310494, 5.24315532307859037917287677083, 5.91490485556232051328453137893, 7.19165496370161031791785051773, 7.968123983806979676708933110262, 8.634815166067848691903920992232, 9.805072334092559681885689033294