L(s) = 1 | + (−0.225 − 1.71i)3-s + (−3.43 − 0.605i)5-s + (−1.29 + 2.24i)7-s + (−2.89 + 0.774i)9-s + 1.29i·11-s + (4.10 − 0.724i)13-s + (−0.265 + 6.03i)15-s + (2.73 + 0.483i)17-s + (3.61 + 2.44i)19-s + (4.15 + 1.72i)21-s + (0.674 + 0.803i)23-s + (6.74 + 2.45i)25-s + (1.98 + 4.80i)27-s + (−5.12 + 4.29i)29-s − 7.70i·31-s + ⋯ |
L(s) = 1 | + (−0.130 − 0.991i)3-s + (−1.53 − 0.270i)5-s + (−0.490 + 0.849i)7-s + (−0.966 + 0.258i)9-s + 0.390i·11-s + (1.13 − 0.200i)13-s + (−0.0686 + 1.55i)15-s + (0.664 + 0.117i)17-s + (0.828 + 0.560i)19-s + (0.905 + 0.375i)21-s + (0.140 + 0.167i)23-s + (1.34 + 0.490i)25-s + (0.381 + 0.924i)27-s + (−0.951 + 0.798i)29-s − 1.38i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.727441 + 0.261525i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.727441 + 0.261525i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.225 + 1.71i)T \) |
| 19 | \( 1 + (-3.61 - 2.44i)T \) |
good | 5 | \( 1 + (3.43 + 0.605i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.29 - 2.24i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 1.29iT - 11T^{2} \) |
| 13 | \( 1 + (-4.10 + 0.724i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.73 - 0.483i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.674 - 0.803i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (5.12 - 4.29i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + 7.70iT - 31T^{2} \) |
| 37 | \( 1 - 9.82iT - 37T^{2} \) |
| 41 | \( 1 + (0.914 - 0.332i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.91 - 3.28i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.18 + 2.60i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (7.28 + 2.65i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-6.50 - 5.46i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.51 - 8.60i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.54 - 12.4i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 0.429i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (9.18 + 7.70i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.35 - 0.592i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-14.9 - 8.63i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (12.9 - 10.8i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.27 - 11.7i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.89449445998067255772060535924, −9.540441284913028824813484310028, −8.549403911594146240228423212741, −7.948568747364020390302379088882, −7.24317865928535289866320305743, −6.15067228294585161669196859371, −5.32465847956042078992338473780, −3.86016821471870952073342996212, −2.95332590811291452521196669231, −1.22670510542795700612748156909,
0.49006073253520278289301152242, 3.34427573341314235907461898179, 3.62787320075905275566490665222, 4.61244171472848240293142387121, 5.82564026012441443690074246458, 6.97765456401269080740776317108, 7.79097194422718239211784857610, 8.702773886974155061226443693074, 9.559243415940643272910911134441, 10.63295820030298282071997417929