L(s) = 1 | + (−0.673 − 1.16i)2-s + (0.0923 − 0.160i)4-s + (−0.826 + 1.43i)5-s + (−1.20 − 2.08i)7-s − 2.94·8-s + 2.22·10-s + (−2.97 − 5.14i)11-s + (1.61 − 2.79i)13-s + (−1.62 + 2.81i)14-s + (1.79 + 3.11i)16-s + 3·17-s − 6.63·19-s + (0.152 + 0.264i)20-s + (−4.00 + 6.93i)22-s + (1.47 − 2.54i)23-s + ⋯ |
L(s) = 1 | + (−0.476 − 0.825i)2-s + (0.0461 − 0.0800i)4-s + (−0.369 + 0.640i)5-s + (−0.455 − 0.789i)7-s − 1.04·8-s + 0.704·10-s + (−0.896 − 1.55i)11-s + (0.447 − 0.775i)13-s + (−0.434 + 0.751i)14-s + (0.449 + 0.778i)16-s + 0.727·17-s − 1.52·19-s + (0.0341 + 0.0591i)20-s + (−0.853 + 1.47i)22-s + (0.306 − 0.531i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.115537 - 0.655246i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.115537 - 0.655246i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.673 + 1.16i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.826 - 1.43i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.20 + 2.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.97 + 5.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.61 + 2.79i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 + (-1.47 + 2.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.645 - 1.11i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.294 + 0.509i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.0418T + 37T^{2} \) |
| 41 | \( 1 + (-2.45 + 4.24i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.59 - 4.49i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.86 - 3.23i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + (-3.67 + 6.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.52 + 9.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.928 - 1.60i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + (-1.89 - 3.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.99 + 3.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + (0.130 + 0.225i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.19964486587913133270611216350, −10.71270262367944365464626137750, −10.24588821065945478743020071696, −8.852212638321943404216443099708, −7.930760320659186187490006535019, −6.60289198898296461745622061032, −5.64360835852003938982514155559, −3.63137902913137392021712943179, −2.78584564947636870525103061596, −0.60171043377104191197344935447,
2.46332813250541597831741007740, 4.25043740929435955733271032859, 5.57802418672570116619240814166, 6.69709544797994138387366683007, 7.64186901176447955143417151404, 8.588944161935841415200803896091, 9.272855415823705302506968119085, 10.39745704066383043092655668172, 11.95296208546410824492351527755, 12.37196222127075535943397243326