L(s) = 1 | + (0.156 + 1.99i)2-s + (−3.95 + 0.624i)4-s − 5.18·5-s − 9.10i·7-s + (−1.86 − 7.77i)8-s + (−0.812 − 10.3i)10-s + 9.19i·11-s − 3.83·13-s + (18.1 − 1.42i)14-s + (15.2 − 4.93i)16-s + 7.85·17-s − 4.35i·19-s + (20.4 − 3.24i)20-s + (−18.3 + 1.44i)22-s + 8.61i·23-s + ⋯ |
L(s) = 1 | + (0.0783 + 0.996i)2-s + (−0.987 + 0.156i)4-s − 1.03·5-s − 1.30i·7-s + (−0.233 − 0.972i)8-s + (−0.0812 − 1.03i)10-s + 0.835i·11-s − 0.294·13-s + (1.29 − 0.101i)14-s + (0.951 − 0.308i)16-s + 0.461·17-s − 0.229i·19-s + (1.02 − 0.162i)20-s + (−0.833 + 0.0654i)22-s + 0.374i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.096058966\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096058966\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.156 - 1.99i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 5.18T + 25T^{2} \) |
| 7 | \( 1 + 9.10iT - 49T^{2} \) |
| 11 | \( 1 - 9.19iT - 121T^{2} \) |
| 13 | \( 1 + 3.83T + 169T^{2} \) |
| 17 | \( 1 - 7.85T + 289T^{2} \) |
| 23 | \( 1 - 8.61iT - 529T^{2} \) |
| 29 | \( 1 - 30.9T + 841T^{2} \) |
| 31 | \( 1 - 43.4iT - 961T^{2} \) |
| 37 | \( 1 - 59.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.02T + 1.68e3T^{2} \) |
| 43 | \( 1 - 7.62iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 45.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 65.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 59.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 67.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 81.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 2.49iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 116.T + 9.40e3T^{2} \) |
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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
−10.29651193340992743425423243356, −9.642989694886775246642463406989, −8.472253605086614995881775788184, −7.58173894587713623339410437718, −7.28332585071028088385775768668, −6.29107295897963740319991164788, −4.84015095539925051256316608303, −4.29473747057315228918626035278, −3.30540185212847036165099740409, −0.897227239840434587163424530142,
0.55062564974443163481647408537, 2.30465971375416846308685144687, 3.25088977163712182841422597749, 4.25577389562324226504217899287, 5.33001935302425679202806169173, 6.21806809529782429532869774401, 7.88814738644772554617062321447, 8.355248089087107779325023287409, 9.259677010354234266574028698487, 10.08114106760071396620499876634