L(s) = 1 | + (1.72 + 0.101i)3-s + (−1.24 − 2.15i)5-s + (0.909 − 1.57i)7-s + (2.97 + 0.350i)9-s + (0.598 − 1.03i)11-s + (−2.83 − 4.90i)13-s + (−1.93 − 3.84i)15-s − 5.30·17-s − 4.55·19-s + (1.73 − 2.63i)21-s + (−2.01 − 3.48i)23-s + (−0.589 + 1.02i)25-s + (5.11 + 0.909i)27-s + (−3.01 + 5.22i)29-s + (2.81 + 4.87i)31-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0585i)3-s + (−0.555 − 0.962i)5-s + (0.343 − 0.595i)7-s + (0.993 + 0.116i)9-s + (0.180 − 0.312i)11-s + (−0.785 − 1.36i)13-s + (−0.498 − 0.993i)15-s − 1.28·17-s − 1.04·19-s + (0.377 − 0.574i)21-s + (−0.419 − 0.727i)23-s + (−0.117 + 0.204i)25-s + (0.984 + 0.174i)27-s + (−0.559 + 0.969i)29-s + (0.505 + 0.876i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773161143\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773161143\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 - 0.101i)T \) |
good | 5 | \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.909 + 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.598 + 1.03i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.83 + 4.90i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 + (2.01 + 3.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.01 - 5.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.81 - 4.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + (-4.57 - 7.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.99 + 6.91i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.39 + 2.41i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.01 + 6.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.91 + 11.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + (-4.36 + 7.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.89 + 15.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 0.455T + 89T^{2} \) |
| 97 | \( 1 + (1.01 - 1.76i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.303099422338192081571306559150, −8.640445834707326751513002917321, −8.054755628670047813365354432511, −7.39255852619366808248729560830, −6.31181909119764397861383617272, −4.79669751382185208904069491923, −4.44146550438797347858454505818, −3.32841793648078109578457529473, −2.14980703466880795462637261607, −0.65719488674408006815123577991,
2.10063026408653145058470019787, 2.53586110975415561907995716339, 4.04623804272207889610852785697, 4.39671703762363778664966036866, 6.06621670048997680426224114746, 6.97367216810498069665544012211, 7.50390373580347253630819443344, 8.421135777467439638639693198317, 9.227736327444064579656206442384, 9.786186893795050427740081938441