L(s) = 1 | + 7.49i·2-s − 14.5i·3-s − 40.1·4-s − 9.40i·5-s + 108.·6-s − 53.5i·7-s − 180. i·8-s − 129.·9-s + 70.4·10-s + 151.·11-s + 582. i·12-s − 319.·13-s + 401.·14-s − 136.·15-s + 713.·16-s + 32.2·17-s + ⋯ |
L(s) = 1 | + 1.87i·2-s − 1.61i·3-s − 2.50·4-s − 0.376i·5-s + 3.02·6-s − 1.09i·7-s − 2.82i·8-s − 1.60·9-s + 0.704·10-s + 1.24·11-s + 4.04i·12-s − 1.89·13-s + 2.04·14-s − 0.606·15-s + 2.78·16-s + 0.111·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.910207 - 0.358450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910207 - 0.358450i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-1.35e3 - 1.26e3i)T \) |
good | 2 | \( 1 - 7.49iT - 16T^{2} \) |
| 3 | \( 1 + 14.5iT - 81T^{2} \) |
| 5 | \( 1 + 9.40iT - 625T^{2} \) |
| 7 | \( 1 + 53.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 151.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 319.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 32.2T + 8.35e4T^{2} \) |
| 19 | \( 1 + 304. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 199.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 268. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 665.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.17e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 212.T + 2.82e6T^{2} \) |
| 47 | \( 1 + 3.10e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.66e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.74e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.88e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.09e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 5.84e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 663. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 6.32e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 8.35e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 4.25e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.58e3T + 8.85e7T^{2} \) |
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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
−14.81761843105858915231702981074, −14.05119709530819903126455625498, −13.19285402444284560807649657223, −12.11188949055336962393846516883, −9.514956026945666949651817206898, −8.095012586789821372641335223194, −7.15324638796110571059241292902, −6.52546708419537694418509711302, −4.72251665884262132833205219136, −0.66081432769580188627961834943,
2.58297413036754763427615799148, 3.96391021400871232148927834228, 5.21180956795999054692394872920, 8.856047613267616014365078790571, 9.642892147151646322229815737282, 10.42245338230509279328935417844, 11.69779987955800989150351380729, 12.32418339133577563517125872777, 14.38965795599064119859250067100, 14.85573042810230700479955601759