L(s) = 1 | + (1.29 − 0.577i)2-s + (1.33 − 1.49i)4-s + (−3.03 − 1.75i)5-s + (−0.151 − 2.64i)7-s + (0.859 − 2.69i)8-s + (−4.93 − 0.509i)10-s + (−2.81 + 1.62i)11-s + 2.17i·13-s + (−1.72 − 3.32i)14-s + (−0.446 − 3.97i)16-s + (−4.04 + 2.33i)17-s + (−0.0375 + 0.0650i)19-s + (−6.66 + 2.19i)20-s + (−2.70 + 3.73i)22-s + (−2.40 − 1.38i)23-s + ⋯ |
L(s) = 1 | + (0.912 − 0.408i)2-s + (0.666 − 0.745i)4-s + (−1.35 − 0.784i)5-s + (−0.0571 − 0.998i)7-s + (0.303 − 0.952i)8-s + (−1.56 − 0.161i)10-s + (−0.850 + 0.490i)11-s + 0.603i·13-s + (−0.459 − 0.888i)14-s + (−0.111 − 0.993i)16-s + (−0.980 + 0.566i)17-s + (−0.00861 + 0.0149i)19-s + (−1.48 + 0.489i)20-s + (−0.575 + 0.795i)22-s + (−0.501 − 0.289i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.172647 - 1.38483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.172647 - 1.38483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 + 0.577i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.151 + 2.64i)T \) |
good | 5 | \( 1 + (3.03 + 1.75i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.81 - 1.62i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.17iT - 13T^{2} \) |
| 17 | \( 1 + (4.04 - 2.33i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0375 - 0.0650i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.40 + 1.38i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 + (3.66 + 6.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.08 + 8.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.20iT - 41T^{2} \) |
| 43 | \( 1 + 1.49iT - 43T^{2} \) |
| 47 | \( 1 + (0.225 - 0.391i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.69 - 2.93i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.23 + 10.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-12.7 - 7.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.05 + 4.65i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (0.852 - 0.492i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.09 + 1.78i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.350T + 83T^{2} \) |
| 89 | \( 1 + (-3.06 - 1.76i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.05iT - 97T^{2} \) |
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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
−10.29449099331943281265026074462, −9.162876629036310657004944141588, −8.004080014467641389401702481276, −7.34887215894141379546899996580, −6.39975017361594055734725301105, −5.03902987312058476915587606841, −4.25018451026691459097511395295, −3.82002971492677118152489928451, −2.20440563210365695748977553348, −0.50182159665087640710050150220,
2.68647809237697786136493251252, 3.16973850227194822284455809803, 4.42970470352013400395777322929, 5.32152694188115778105764382707, 6.37889927902482986626011636892, 7.13743463686500752893953534398, 8.160183585414454920539908102111, 8.472051888296070232965345169971, 10.11046445129524785041442326101, 11.18689652423198733846064927419