L(s) = 1 | + 1.69·2-s + (−1.29 + 1.15i)3-s + 0.888·4-s + (−1.79 − 3.10i)5-s + (−2.19 + 1.95i)6-s − 1.88·8-s + (0.349 − 2.97i)9-s + (−3.04 − 5.28i)10-s + (1.40 − 2.43i)11-s + (−1.15 + 1.02i)12-s + (0.5 − 0.866i)13-s + (5.89 + 1.95i)15-s − 4.98·16-s + (−2.05 − 3.56i)17-s + (0.594 − 5.06i)18-s + (−0.444 + 0.769i)19-s + ⋯ |
L(s) = 1 | + 1.20·2-s + (−0.747 + 0.664i)3-s + 0.444·4-s + (−0.802 − 1.38i)5-s + (−0.897 + 0.798i)6-s − 0.667·8-s + (0.116 − 0.993i)9-s + (−0.964 − 1.67i)10-s + (0.423 − 0.733i)11-s + (−0.332 + 0.295i)12-s + (0.138 − 0.240i)13-s + (1.52 + 0.505i)15-s − 1.24·16-s + (−0.498 − 0.863i)17-s + (0.140 − 1.19i)18-s + (−0.101 + 0.176i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.719428 - 0.891146i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.719428 - 0.891146i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.29 - 1.15i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 5 | \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.05 + 3.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.444 - 0.769i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.93 + 5.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + (2.38 - 4.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + (-0.0618 - 0.107i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.87T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + (5.32 + 9.21i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + (2.05 + 3.56i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.80 + 8.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.37447965965307122938467019974, −10.06870572235943975539236036878, −8.929943146717076140009410255695, −8.399303079785936027718992507488, −6.68168170424727991975189245820, −5.73951986305941898485664444808, −4.79666396377839468859946073020, −4.34244755339517652840843750407, −3.30205228751117924941101815140, −0.53400593059500691858953867677,
2.27842928313334343828653154111, 3.65017604043491863373875568716, 4.49231979571777572635090336742, 5.79855275105721094286133080386, 6.61805188720198292615774379863, 7.19882610087466827873696826693, 8.341409784588991914384703087442, 9.918935202818988036004435193194, 10.92928096777734709969072281664, 11.64755723541431522188024931047